14.2: Univariate Analysis
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Univariate analysis—or analysis of a single variable—refers to a set of statistical techniques that can describe the general properties of one variable. Univariate statistics include: frequency distribution, central tendency, and dispersion. The frequency distribution of a variable is a summary of the frequency—or percentages—of individual values or ranges of values for that variable. For instance, we can measure how many times a sample of respondents attend religious services—as a gauge of their ‘religiosity’—using a categorical scale: never, once per year, several times per year, about once a month, several times per month, several times per week, and an optional category for ‘did not answer’. If we count the number or percentage of observations within each category—except ‘did not answer’ which is really a missing value rather than a category—and display it in the form of a table, as shown in Figure 14.1, what we have is a frequency distribution. This distribution can also be depicted in the form of a bar chart, as shown on the right panel of Figure 14.1, with the horizontal axis representing each category of that variable and the vertical axis representing the frequency or percentage of observations within each category.
With very large samples, where observations are independent and random, the frequency distribution tends to follow a plot that looks like a bell-shaped curve—a smoothed bar chart of the frequency distribution—similar to that shown in Figure 14.2. Here most observations are clustered toward the centre of the range of values, with fewer and fewer observations clustered toward the extreme ends of the range. Such a curve is called a normal distribution .
Central tendency is an estimate of the centre of a distribution of values. There are three major estimates of central tendency: mean, median, and mode. The arithmetic mean —often simply called the ‘mean’—is the simple average of all values in a given distribution. Consider a set of eight test scores: 15, 22, 21, 18, 36, 15, 25, and 15. The arithmetic mean of these values can be calculated using the sum divided by the number of values . In this example, the mean would be 20.875. Other types of means include geometric mean ( th root of the product of numbers in a distribution) and harmonic mean (the reciprocal of the arithmetic means of the reciprocal of each value in a distribution), but these means are not very popular for statistical analysis of social research data.The second measure of central tendency, the median , is the middle value within a range of values in a distribution. This is computed by sorting all values in a distribution in increasing order and selecting the middle value. In cases where an even number of values in a distribution means there are two middle values, the average of those two values represents the median. In the above example, the sorted values are: 15, 15, 15, 18, 22, 21, 25, and 36. The two middle values are 18 and 22, and hence the median is .
Lastly, the mode is the most frequently occurring value in a distribution of values. In the previous example, the most frequently occurring value is 15, which is the mode of the above set of test scores. Note that any value that is estimated from a sample, such as mean, median, mode, or any of the later estimates are called a statistic .
Dispersion refers to the way values are spread around the central tendency—for example, how tightly or how widely the values are clustered around the mean. Two common measures of dispersion are the range and standard deviation. The range is the difference between the highest and lowest values in a distribution. The range in our previous example is .
The range is particularly sensitive to the presence of outliers. For instance, if the highest value in the above distribution was 85, and the other vales remained the same, the range would be . Standard deviation —the second measure of dispersion—corrects for such outliers by using a formula that takes into account how close or how far each value is from the distribution mean:
where is the standard deviation, is the th observation (or value), is the arithmetic mean, is the total number of observations, and means summation across all observations. The square of the standard deviation is called the variance of a distribution. In a normally distributed frequency distribution, 68% of the observations lie within one standard deviation of the mean , 95% of the observations lie within two standard deviations , and 99.7% of the observations lie within three standard deviations , as shown in Figure 14.2.


5.3: Cyclic structures of monosaccharides
- Draw and inter-convert open-chain and cyclic hemiacetal forms of compounds containing carbonyl and alcohol groups in the same molecule.
- Convert Fisher projections to Howarth projections and to chair conformation in the cases of pyranose forms of monosaccharides.
- Define Howarth projection, anomeric carbon, \(\alpha\) and \(\beta\) configuration of the anomeric carbon, furanose, and pyranose forms, and mutarotation of monosaccharides.
Cyclic hemiacetal
Alcohol (\(\ce{R-OH}\)) and a carbonyl (\(\ce{C=O}\)) groups react with each other and form a hemiacetal group. The hemiacetals are usually unstable and revert to the reactants. However, suppose the alcohol and the carbonyl groups are on the same molecule and can react to form a five- or six-member cyclic hemiacetal. The product is more stable than the reactants, as shown in the following example reaction.
Note that the hemiacetal \(\ce{C}\) is a new chiral center that is produced as a racemic mixture, which is represented by a wavy line in the \(\ce{-OH}\) group. Monosaccharides are polyhydroxy aldehydes or polyhydroxy ketones that often exist as hemiacetal as the dominant form in equilibrium with the open-chain form, as described in the next section.
Fisher projections to Haworth projections
Fisher projections represent the open-chain forms of monosaccharides where the \(\ce{C}\)-chain is a vertical line. Hemiacetals are five or six-member ring structures represented by pentagon or hexagon shapes.
Figure \(\PageIndex{1}\) explains the steps to re-draw Fisher projection in cyclic conformation.
- Rotate the Fisher projection by 90 o clockwise from vertical to horizontal orientation. The groups on the right of Fisher's projection end up above, and those to the left end below in the horizontal orientation.
- Move \(\ce{C's}\)#1, 4, 5, and 6 up by 60 o each to arrive at the cyclic open-chain conformation#1.
- Rotate the \(\ce{C}\)4-\(\ce{C}\)5 bond by 60 o counterclock wise to bring the \(\ce{C}\)#5\(\ce{-OH}\) group in the cycle of \(\ce{C's}\) and \(\ce{C}\)#6 pointing above, i.e., cyclic conformation#1 in Figure \(\PageIndex{1}\).
- \(\ce{C}\)#5\(\ce{-OH}\) and a carbonyl (\(\ce{C=O}\)) of \(\ce{C}\)#1 react to produce the hemiacetal group with the new \(\ce{-OH}\) group on \(\ce{C}\)#1 either pointing away from the hemiacetal \(\ce{O}\), i.e., \(\alpha\)- or point towards the hemiacetal \(\ce{O}\), i.e., \(\beta\)-orientation in the cyclic hemiacetal.
\(\ce{C}\) he \(\ce{C}\) \(\ce{C=O}\) \(\ce{C}\) \(\ce{C}\)#1 is the anomeric \(\ce{C}\)
five or six-membered cyclic hemiacetals of monosaccharides, presented as pentagon or hexagon shape. It is viewed through its edge perpendicular to the plane of the page, such that the anomeric \(\ce{C}\)
Conversion of monosaccharides from open chain to cyclic hemiacetal produced a new chiral center at the anomeric \(\ce{C}\). It is a mixture of two conformations: the \(\ce{-OH}\) group on anomeric \(\ce{C}\) going away from the
is \(\alpha\)- and towards it is \(\beta\)-configuration, as illustrated in the figure on the right with the help of D-glucose example.D-Fructose exists in five-membered and six-membered hemiacetal forms, as shown in Figure \(\PageIndex{2}\).
Furanose is a five-membered, and pyranose is a six-membered cyclic hemiacetal of monosaccharide. These names originate from the names of cyclic ethers, furan, and pyran, shown in the figure on the right. Based on this nomenclature, the class name of \(\alpha\)-D-Glucose is \(\alpha\)-D-glucopyranose and of \(\beta\)-D-glucose is \(\beta\)-D-glucopyranose. D-Fructose exists in pyranose and furanose forms, as shown in Figure \(\PageIndex{2}\).
Chair conformations of pyranose forms
Haworth projections assume a flat ring structure. It is close to the structure of furanose forms (five-membered rings), but pyranose forms (six-membered rings) exist in chair conformations. The figure on the right shows a chair conformation of a six-membered cyclic structure. The points to note are the following.
- Each \(\ce{C}\) has two bonds (shown in black) in the cycle, one axial bond (shown in red) pointing along the axis of the cycle and one equatorial bond (shown in blue) pointing approximately along the equator of the cycle.
- The axial bonds point in opposite directions, up and down, on neighboring \(\ce{C'}\). The same applies to equatorial bonds.
- Bulky group, i.e., any group other than \(\ce{-H}\), is more stable at the equatorial than axial position because of steric strain (push away) from the other two axial positions on the same face of the ring., as illustrated in the figure on the left.
How to draw chair conformations of monosaccharides
Alpha (\(\alpha\)) and axial start with the letter a . Remember
\(\ce{C}\)Since \(\alpha\)\(\ce{-OH}\) is axial, \(\beta\)\(\ce{-OH}\) is equatorial at anomeric \(\ce{C}\). The Howarth projection of \(\beta\)-D-glucose in the figure on the right shows that the bulky groups alternate between up and down positions on \(\ce{C}\)#1 to \(\ce{C}\)#5. It means all of them are either axial or equatorial. Since \(\beta\)\(\ce{-OH}\) is equatorial, all bulky groups in \(\beta\)-D-glucose are equatorial, as shown in Figure \(\PageIndex{3}\).
After learning how to draw \(\beta\)-D-glucopyranose, the other monosaccharides can be drawn easily based on the knowledge of the difference from \(\beta\)-D-glucopyranose. For example, \(\beta\)-D-manopyranose is \(\ce{C}\)#2 epimer, \(\beta\)-D-galactopyranose is \(\ce{C}\)#4 epimer, and their \(\alpha\) forms have an axial \(\ce{-OH}\) at anomeric \(\ce{C}\), as shown in Figure \(\PageIndex{3}\).
Mutarotation
D-Glucose is a chiral compound. A freshly prepared aqueous solution of \(\beta\)-D-glucopyranose has a specific rotation of +112.2. A freshly prepared solution of \(\alpha\)-D-glucopyranose has a specific rotation of +18.7. However, the specific rotation of \(\beta\)-D-glucopyranose and \(\alpha\)-D-glucopyranose gradually changes to +52.5. This is because the open chain and the two cyclic forms establish equilibrium in the solution, as illustrated in Figure \(\PageIndex{4}\).
Open chain and the two cyclic forms of D-glucose exist in equilibrium in solution where \(\beta\)-D-glucose has all bulky group in equatorial position is the most stable, \(\alpha\)-D-glucose with one bulky group in axial position is the second most stable and, and open chain for is the least stable, as reflected by their proportions at equilibrium shown in Figure \(\PageIndex{4}\).
Mutarotation is the change in the optical rotation of carbohydrate solution over time due to the change in the composition of different compound forms to achieve equilibrium composition.


8.5: Classification of Finance Problems
Learning Objectives
In this section, you will learn to:
- Re-examine the types of financial problems and classify them.
- Re-examine the vocabulary words used in describing financial calculations
We'd like to remind the reader that the hardest part of solving a finance problem is determining the category it falls into. So in this section, we will emphasize the classification of problems rather than finding the actual solution. We suggest that the student read each problem carefully and look for the word or words that may give clues to the kind of problem that is presented.
For instance, students often fail to distinguish a lump-sum problem from an annuity. Since the payments are made each period, an annuity problem contains words such as each, every, per etc. One should also be aware that in the case of a lump-sum, only a single deposit is made, while in an annuity numerous deposits are made at equal spaced time intervals. To help interpret the vocabulary used in the problems, we include a glossary at the end of this section.
Students often confuse the present value of an annuity with the future value of an annuity. Don't get too caught up on the words, as either formula can be used to find what should be done now (in the present) and what would be the result in the future.
- The future value formula will be used in the context of investments, where the regular deposits and interest both work together to add to the account balance.
- The present value formula will be used in the context of loans or bank withdraws, where the regular payments and interest work against each other. In a loan, the monthly payments lower the account balance while the interest payments raise the account balance.
CLASSIFICATION OF PROBLEMS AND EQUATIONS
We now list eight problems that form a basis for all finance problems. Read each question and determine which formula would be used to find the missing information.
Example \(\PageIndex{1}\)
Dillon takes out a simple interest loan for $5,000 at 10.5% interest with an agreement to pay it back in 10 months. What is the total amount he will repay at the end of the 10 months?
Classification: Since the loan is a simple interest loan and we are looking for the total amount he will repay, we will use the Accumulated Value of Simple Interest Formula .
Equation: \[\mathrm{A}=5,000(1+0.105(\frac{10}{12}))\]
Example \(\PageIndex{2}\)
If $2,000 is invested at 7% compounded quarterly, what will the final amount be in 5 years?
Classification: Since the $2000 is invested once, this is a lump-sum. The interest is compounded each quarter, so this will use the Accumulated Value of Periodic Compounding Formula .
Equation: \[\mathrm{A}=\$ 2000(1+.07 / 4)^{20}\]
Example \(\PageIndex{3}\)
If $2,000 is invested at 7% compounded continuously, what will the final amount be in 5 years?
Classification: Since the $2000 is invested once, this is a lump-sum. The interest is compounded continuously, so this will use the Accumulated Value of Continuous Compounding Formula .
Equation: \[\mathrm{A}=\$ 2000e^{0.07*5}\]
Example \(\PageIndex{4}\)
How much should be invested at 8% compounded yearly, for the final amount to be $5,000 in five years?
Classification: Since this question is asking for a one-time investment, this is a lump-sum. The interest is compounded each year, so this will use the Accumulated Value of Periodic Compounding Formula . Note that we are given the accumulated value at the end of the 5 years and are looking for the principal.
Equation: \[\$ 5,000=\mathrm{P}(1+.08)^{5}\]
Example \(\PageIndex{5}\)
If $200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?
Classification: Since the money is invested each month, this is an example of an annuity. This is an investment where the monthly payments and the interest both add to the value, so this will use the Future Value of an Annuity Formula.
Equation: \[\mathrm{A}=\frac{\$ 200\left[(1+.085 / 12)^{12*4}-1\right]}{.085 / 12}\]
Example \(\PageIndex{6}\)
How much should be invested each month at 9% for it to accumulate to $8,000 in three years?
Classification: Since the money is invested each month, this is an example of an annuity. This is an investment where the monthly payments and the interest both add to the value, so this will use the Future Value of an Annuity Formula. Note that we are given the accumulated value at the end of the 3 years and are looking for the monthly payment.
Equation: \[\$ 8,000=\frac{m\left[(1+.09 / 12)^{36}-1\right]}{.09 / 12}\]
Example \(\PageIndex{7}\)
Keith wants to buy a new car worth $30,000 and needs to apply for an auto loan. The dealership is offering him financing at 9.25% compounded monthly over the next 5 years. How much will each monthly payment be?
Classification: Since the money will be paid each month, this is an example of an annuity. This is a loan where the monthly payments lower the account balance and the interest raises the account balance, so this will use the Present Value of an Annuity Formula.
Equation: \[\$ 30,000(1+.0925/12)^{12*5}=\frac{m \left[(1+.0925/12)^{12*5}-1\right]}{.0925/12}\]
Example \(\PageIndex{8}\)
Maria is saving money for her retirement. The retirement fund earns 8% interest compounded monthly, and she plans to withdraw $1,500 each month for 25 years after her retirement. How much does she need in this fund before she retires?
Classification: Since the money will be withdrawn each month, this is an example of an annuity. This is a fund where the monthly withdraws lower the account balance and the interest raises the account balance, so this will use the Present Value of an Annuity Formula.
Equation: \[\mathrm{P}(1+.08/12)^{12*25}=\frac{\$1,500\left[(1+.08/12)^{12*25}-1\right]}{.08/12}\]
GLOSSARY: VOCABULARY AND SYMBOLS USED IN FINANCIAL CALCULATIONS
As we’ve seen in these examples, it’s important to read the problems carefully to correctly identify the situation. It is essential to understand to vocabulary for financial problems. Many of the vocabulary words used are listed in the glossary below for easy reference.
|
\(t\) |
Term |
Time period for a loan or investment. In this book \(t\) is represented in years and should be converted into years when it is stated in months or other units. |
|
\(\mathrm{P}\) |
Principal |
Principal is the amount of money borrowed in a loan. If a sum of money is invested for a period of time, the sum invested at the start is the Principal. |
|
\(\mathrm{P}\) |
Present Value |
Value of money at the beginning of the time period. |
|
\(\mathrm{A}\) |
Accumulated Value Future Value |
Value of money at the end of the time period |
|
\(D\) |
Discount |
In loans involving simple interest, a discount occurs if the interest is deducted from the loan amount at the beginning of the loan period, rather than being repaid at the end of the loan period. |
|
\(m\) |
Periodic Payment |
The amount of a constant periodic payment that occurs at regular intervals during the time period under consideration (examples: periodic payments made to repay a loan, regular periodic payments into a bank account as savings, regular periodic payment to a retired person as an annuity,) |
|
\(n\) |
Number of payment periods and compounding periods per year |
In this book, when we consider periodic payments, we will always have the compounding period be the same as the payment period.
|
|
\(nt\) |
Number of periods |
\(nt\) = (number of periods per year)\(\times\)(number of years) \(nt\) gives the total number of payment and compounding periods In some situations we will calculate \(nt\) as the multiplication shown above. In other situations the problem may state \(nt\), such as a problem describing an investment of 18 months duration compounded monthly. In this example: \(nt\) = 18 months and \(n\) = 12; then \(t\) = 1.5 years but \(t\) is not stated explicitly in the problem. The TI-84+ calculators built in TVM solver uses \(N = nt\). |
|
\(r\) |
Annual interest rate Nominal rate |
The stated annual interest rate. This is stated as a percent but converted to decimal form when using financial calculation formulas. If a bank account pays 3% interest compounded quarterly, then 3% is the nominal rate, and it is included in the financial formulas as \(r\) = 0.03 |
|
\(r/n\) |
Interest rate per compounding period |
If a bank account pays 3% interest compounded quarterly, then \(r/n\) = 0.03/4 = 0. 075, corresponding to a rate of 0.75% per quarter. Some Finite Math books use the symbol \(i\) to represent \(r/n\) |
|
\(r_{EFF}\) |
Effective Rate Effective Annual Interest Rate APY Annual Percentage Yield APR Annual Percentage Rate |
The effective rate is the interest rate compounded annually that would give the same interest rate as the compounded rate stated for the investment. The effective rate provides a uniform way for investors or borrowers to compare different interest rates with different compounding periods. |
|
\(I\) |
Interest |
Money paid by a borrower for the use of money borrowed as a loan. Money earned over time when depositing money into a savings account, certificate of deposit, or money market account. When a person deposits money in a bank account, the person depositing the funds is essentially temporarily lending the money to the bank and the bank pays interest to the depositor. |
|
Sinking Fund |
A fund set up by making payments over a period of time into a savings or investment account in order to save to fund a future purchase. Businesses use sinking funds to save for a future purchase of equipment at the end of the savings period by making periodic installment payments into a sinking fund. |
|
|
Annuity |
An annuity is a stream of periodic payments. In this book it refers to a stream of constant periodic payments made at the end of each compounding period for a specific amount of time. In common use the term annuity generally refers to a constant stream of periodic payments received by a person as retirement income, such as from a pension. Annuity payments in general may be made at the end of each payment period (ordinary annuity) or at the start of each period (annuity due). The compounding periods and payment periods do not need to be equal, but in this textbook we only consider situations when these periods are equal. |
|
|
Lump Sum |
A single sum of money paid or deposited at one time, rather than being spread out over time. An example is lottery winnings if the recipient chooses to receive a single “lump sum” one-time payment, instead of periodic payments over a period of time or as. Use of the word lump sum indicates that this is a one time transaction and is not a stream of periodic payments. |
|
|
Loan |
An amount of money that is borrowed with the understanding that the borrower needs to repay the loan to the lender in the future by the end of a period of time that is called the term of the loan. The repayment is most often accomplished through periodic payments until the loan has been completely repaid over the term of the loan. However there are also loans that can be repaid as a single sum at the end of the term of the loan, with interest paid either periodically over the term or in a lump sum at the end of the loan or as a discount at the start of the loan. |


2.2: William Bradford (1590–1657)
William Bradford was born in Austerfield, Yorkshire and reared as a farmer. In 1606, inspired by the preaching of nonconformist minister Richard Clyfton (d. 1616), Bradford joined the Separatist group tied to William Brewster (1568–1644) in Scrooby, Nottinghamshire. As Separatists from the Church of England headed by the English monarch, this group (and similar others) engaged in treason against the English crown. To escape the consequent-enforced secrecy and persecution, the group left England for the Netherlands. In 1609, Bradford joined them there, became a weaver, and started his own business upon inheriting money from his family.
Image \(\PageIndex{1}\): William Bradford
To escape further persecution, the group petitioned for and won a land grant in North America. Bradford was one of the pilgrims who sailed from Southampton, England in 1620 on the Mayflower to settle in the land granted. Their land grant was originally meant to be in Virginia but, due to difficulty navigating in storms, they landed at Plymouth, Massachusetts. William Bradford helped define for themselves and future generations their Puritan settlement and endeavor at Plymouth Plantation. After the death of their elected governor John Carver (1576–1621), Bradford was elected governor. He was re-elected thirty times, serving as governor for almost all but the last five years of his life. He signed the Mayflower Compact that ordered their earthly rule (even as a means to prepare for heavenly rule); held to the Compact’s democratic principles in his governorship; worked to repay the debt to the British investors who funded their project in America; and did much to organize and lead the pilgrims’ lives.
Self-educated particularly in languages—including Hebrew—and an avid reader, Bradford applied his knowledge and skills to recording the history Of Plymouth Plantation. He started this chronicle largely in response to the growth of Non-Separatist settlers in the colony, settlers whom he saw as competing with the Separatists. His history records such important events as the pilgrims’ landing at Plymouth, the Mayflower Compact, the first Thanksgiving, and the Puritan ethic in action as it was put to trial and served as testimony of God’s designs. These designs included the pilgrims’ persecutions, voyage to and landing at Plymouth, suffering starvation and sickness there, as well as experiencing increasing tensions between themselves and the Native Americans. In the Puritan plain style, Bradford offers simple yet monumental truths of their lives.
Image \(\PageIndex{2}\): Landing of the pilgrims at Plymouth 11th Dec. 1620


11.18: Quiz 17-Materials Handling, Storage, Use, Disposal
Indicate whether the statement is true or false.
________ 1. All scrap lumber, waste material, and rubbish shall be removed from the immediate work area as the work progresses if you have employees available and capable lo do the work.
Complete each statement.
2. All materials stored in tiers shall be ________ , racked, ________ , interlocked, or otherwise secured to prevent sliding, falling or collapse.
Standard ________ .
3. ________ and ________ shall be kept clear to provide for the free and safe movement of material handling equipment or employees. Such areas shall be kept in good repair.
Standard ________ .
4. Brick stacks shall not be higher than ________ feet. What has to happen after 4 feet if the stack is loose? ________
Standard ________ .
5. According to Standard 1926.251 (a)(l). How often shall rigging equipment for material handling be inspected? ________
6. According to Standard 1926.252(e). Where shall oily rags be kept until they are removed from the job? ________


10: Right Triangle Trigonometry Angles
Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ´ιγωνo), meaning “triangle”, and metro (µǫτρω´ ), meaning “measure”. Though the ancient Greeks, such as Hipparchus and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. - A.D. 200, its history is much older. For example, the Egyptian scribe Ahmes recorded some rudimentary trigonometric calculations (concerning ratios of sides of pyramids) in the famous Rhind Papyrus sometime around 1650 B.C. Trigonometry is distinguished from elementary geometry in part by its extensive use of certain functions of angles, known as the trigonometric functions. Before discussing those functions, we will review some basic terminology about angles.
-
- 10.1: Angles
- In elementary geometry, angles are always considered to be positive and not larger than \(360^\circ \). You also learned that the sum of the angles in a triangle equals \(180^◦\), and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is \(90^◦\) and the other two angles are acute angles whose sum is \(90^◦\) (i.e. the other two angles are complementary angles).
-
- 10.2: Trigonometric Functions of an Acute Angle
- For a right triangle △ABC, with the right angle at C and with lengths a, b, and c. For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. Recall that the hypotenuse of the triangle is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we can define the six trigonometric functions of A.
-
- 10.3: Applications and Solving Right Triangles
- Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements of angles and small, known distances. Today, trigonometry is widely used in physics, astronomy, engineering, navigation, surveying, and various fields of mathematics and other disciplines. In this section we will see some of the ways in which trigonometry can be applied. Your calculator should be in degree mode for these examples.
-
- 10.4: Trigonometric Functions of Any Angle
- To define the trigonometric functions of any angle - including angles less than 0° or greater than 360° - we need a more general definition of an angle. We say that an angle is formed by rotating a ray OA about the endpoint O (called the vertex), so that the ray is in a new position, denoted by the ray OB. The ray OA is called the initial side of the angle, and OB is the terminal side of the angle.
-
- 10.5: Rotations and Reflections of Angles
- Now that we know how to deal with angles of any measure, we will take a look at how certain geometric operations can help simplify the use of trigonometric functions of any angle, and how some basic relations between those functions can be made. The two operations on which we will concentrate in this section are rotation and reflection.
Thumbnails: Types of angles.
Contributors
-
Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2 .


Page 11.5: Sustainable Agriculture
“ Sustainable agriculture ” was addressed by Congress in the 1990 “Farm Bill”. Under that law, “the term sustainable agriculture means an integrated system of plant and animal production practices having a site-specific application that will, over the long term:
- satisfy human food and fiber needs;
- enhance environmental quality and the natural resource base upon which the agricultural economy depends;
- make the most efficient use of nonrenewable resources and on-farm resources and integrate, where appropriate, natural biological cycles and controls;
- sustain the economic viability of farm operations; and
- enhance the quality of life for farmers and society as a whole.”
Organic Farming is Good for Farmers, Consumers and the Environment
Organic agriculture is an ecological production management system that promotes and enhances biodiversity, biological cycles and soil biological activity. Organic food is produced by farmers who emphasize the use of renewable resources and the conservation of soil and water to enhance environmental quality for future generations. Organic meat, poultry, eggs, and dairy products come from animals that are given no antibiotics or growth hormones. Organic food is produced without using most conventional pesticides, fertilizers made with synthetic ingredients or sewage sludge, or GMOs.
Organic production, with the corresponding practices to maintain soil fertility and soil health is therefore a more benign alternative to conventional, high-value horticulture. The organic food movement has been endorsed by the UN’s Food and Agricultural Organization, which maintains in a 2007 report that organic farming fights hunger, tackles climate change, and is good for farmers, consumers, and the environment. The strongest benefits of organic agriculture are its use of resources that are independent of fossil fuels, are locally available, incur minimal environmental stresses, and are cost effective.
IPM is a Combination of Common-Sense Practices
Integrated Pest Management (IPM) refers to a mix of farmer-driven, ecologically-based pest control practices that seeks to reduce reliance on synthetic chemical pesticides. It involves (a) managing pests (keeping them below economically damaging levels) rather than seeking to eradicate them; (b) relying, to the extent possible, on non-chemical measures to keep pest populations low; and (c) selecting and applying pesticides, when they have to be used, in a way that minimizes adverse effects on beneficial organisms, humans, and the environment. It is commonly understood that applying an IPM approach does not necessarily mean eliminating pesticide use, although this is often the case because pesticides are often over-used for a variety of reasons.
The IPM approach regards pesticides as mainly short-term corrective measures when more ecologically based control measures are not working adequately (sometimes referred to as using pesticides as the “last resort”). In those cases when pesticides are used, they should be selected and applied in such a manner as to minimize the amount of disruption that they cause to the environment, such as using products that are non-persistent and applying them in the most targeted way possible).
Biological Control
Biological control (biocontrol) is the use of one biological species to reduce populations of a different species. There has been a substantial increase in commercialization of biocontrol products, such as beneficial insects, cultivated predators and natural or non-toxic pest control products. Biocontrol is being mainstreamed to major agricultural commodities, such as cotton, corn and most commonly vegetable crops. Biocontrol is also slowly emerging in vector control in public health and in areas that for a long time mainly focused on chemical vector control in mosquito/malaria—and black fly/onchocerciasis—control programs. Successful and commercialized examples of biocontrol include ladybugs to depress aphid populations, parasitic wasps to reduce moth populations, use of the bacterium Bacillus thuringenensis to kill mosquito and moth larvae, and introduction of fungi, such as Trichoderma, to suppress fungal-caused plant diseases, leaf beetle ( Galerucella calmariensis ) to suppress purple loosestrife, a noxious weed (Figure \(\PageIndex{1}\)). In all of these cases, the idea is not to completely destroy the pathogen or pest, but rather to reduce the damage below economically significant values.
Figure \(\PageIndex{1}\). Young larvae of the leaf-beetle feed in and on the developing buds of plants, often destroying them. This may stunt plant growth and delay or prevent flowering. Adults (shown) and older larvae feed on leaves and cause severe defoliation. Leaf-beetles can be used to as a biocontrol for invasive plants such as purple loosestrife.Intercropping Promotes Plant Interactions
Intercropping means growing two or more crops in close proximity to each other during part or all of their life cycles to promote soil improvement, biodiversity, and pest management. Incorporating intercropping principles into an agricultural operation increases diversity and interaction between plants, arthropods, mammals, birds and microorganisms resulting in a more stable crop-ecosystem and a more efficient use of space, water, sunlight, and nutrients (Figure \(\PageIndex{2}\)). This collaborative type of crop management mimics nature and is subject to fewer pest outbreaks, improved nutrient cycling and crop nutrient uptake, and increased water infiltration and moisture retention. Soil quality, water quality and wildlife habitat all benefit.
Figure \(\PageIndex{2}\). Intercropping alyssum (the flowering plants shown center left) with organic romaine lettuce for aphid control.Organic Farming Practices Reduce Unnecessary Input Use
In modern agricultural practices, heavy machinery is used to prepare the seedbed for planting, to control weeds, and to harvest the crop. The use of heavy equipment has many advantages in saving time and labor, but can cause compaction of soil and disruption of the natural soil organisms. The problem with soil compaction is that increased soil density limits root penetration depth and may inhibit proper plant growth.
Alternative practices generally encourage minimal tillage or no tillage methods . With proper planning, this can simultaneously limit compaction, protect soil organisms, reduce costs (if performed correctly), promote water infiltration, and help to prevent topsoil erosion (Figure \(\PageIndex{3}\)).
Figure \(\PageIndex{3}\). Farmers should consider no-till farming as the most important tool to prevent loss of soil moisture.Tillage of fields does help to break up clods that were previously compacted, so best practices may vary at sites with different soil textures and composition. Another aspect of soil tillage is that it may lead to more rapid decomposition of organic matter due to greater soil aeration. Over large areas of farmland, this has the unintended consequence of releasing more carbon and nitrous oxides (greenhouse gases) into the atmosphere, thereby contributing to global warming effects. In no-till farming, carbon can actually become sequestered into the soil. Thus, no-till farming may be advantageous to sustainability issues on the local scale and the global scale. No-till systems of conservation farming have proved a major success in Latin America and are being used in South Asia and Africa.
Crop Rotation
Crop rotations are planned sequences of crops over time on the same field. Rotating crops provides productivity benefits by improving soil nutrient levels and breaking crop pest cycles. Farmers may also choose to rotate crops in order to reduce their production risk through diversification or to manage scarce resources, such as labor, during planting and harvesting timing. This strategy reduces the pesticide costs by naturally breaking the cycle of weeds, insects and diseases. Also, grass and legumes in a rotation protect water quality by preventing excess nutrients or chemicals from entering water supplies.
AN ALTERNATIVE TO SPRAYING: BOLLWORM CONTROL IN SHANDONG
Farmers in Shandong (China) have been using innovative methods to control bollworm infestation in cotton when this insect became resistant to most pesticides. Among the control measures implemented were:
- The use of pest resistant cultivars and interplanting of cotton with wheat or maize.
- Use of lamps and poplar twigs to trap and kill adults to lessen the number of adults.
- If pesticides were used, they were applied on parts of cotton plant’s stem rather than by spraying the whole field (to protect natural enemies of the bollworm).
These and some additional biological control tools have proved to be effective in controlling insect populations and insect resistance, protecting surroundings and lowering costs.
The Future of the Sustainable Agriculture Concept
Many in the agricultural community have adopted the sense of urgency and direction pointed to by the sustainable agriculture concept. Sustainability has become an integral component of many government, commercial, and non-profit agriculture research efforts, and it is beginning to be woven into agricultural policy. Increasing numbers of farmers and ranchers have embarked on their own paths to sustainability, incorporating integrated and innovative approaches into their own enterprises.
Contributors and Attributions
- Essentials of Environmental Science by Kamala Doršner is licensed under CC BY 4.0. Modified from the original by Matthew R. Fisher.


10.4: Human Resource Management and Compliance
- How does the human resources compliance role of HR add value to an organization?
Human resources compliance is an area that traces back to the very origin of the human resources function—to administrative and regulatory functions. Compliance continues to be a very important area that HR manages, and there are numerous regulations and laws that govern the employment relationship. HR professionals must be able to understand and navigate these laws to help their organizations remain compliant and avoid having to pay fines or penalties. The additional threat of reputational harm to the organization is another reason that HR needs to be aware and alert to any potential gaps in compliance.
Some of the most common examples of laws and regulations that govern the employer-employee relationship include the following (SHRM.org):
- Age Discrimination in Employment Act (ADEA)
- Americans with Disabilities Act (ADA)
- Fair Labor Standards Act (FLSA)
- Family and Medical Leave Act (FMLA)
- National Labor Relations Act (NLRA)
- Worker Adjustment and Retraining Notification Act (WARN)
The Age Discrimination in Employment Act (ADEA) of 1967 protects individuals who are 40 years of age or older from employment discrimination based on age. These protections apply to both employees and job applicants. It also makes it unlawful to discriminate based on age with respect to any terms of employment, such as hiring, firing, promotion, layoff, compensation, benefits, job assignments, and training.
The Americans with Disabilities Act (ADA) of 1990 prohibits private employers, state and local governments, employment agencies, and labor unions from discriminating against qualified individuals with disabilities. The ADA defines an individual with a disability as a person who:
- has a mental or physical impairment that substantially limits one or more major life activities
- has a record of such impairment or
- is regarded as having such impairment. An employer is required to make a reasonable accommodation to the known disability of a qualified applicant or employee if it would not impose an “undue hardship” on the operation of the employer’s business
The Fair Labor Standards Act (FLSA) of 1938 establishes the minimum wage, overtime pay, recordkeeping, and youth employment standards affecting full-time and part-time workers in the private sector and in federal, state, and local governments. Special rules apply to state and local government employment involving fire protection and law enforcement activities, volunteer services, and compensatory time off instead of cash overtime pay.
The Family and Medical Leave Act (FMLA) of 1993 entitles eligible employees to take up to 12 weeks of unpaid, job-protected leave in a 12-month period for specified family and medical reasons. FMLA applies to all public agencies, including state, local, and federal employers, local education agencies (schools), and private-sector employers who employed 50 or more employees in 20 or more workweeks in the current or preceding calendar year, including joint employers and successors of covered employers.
The National Labor Relations Act (NLRA) of 1947 extends rights to many private-sector employees, including the right to organize and bargain with their employer collectively. Employees covered by the act are protected from certain types of employers and union misconduct and have the right to attempt to form a union where none exists.
The Worker Adjustment and Retraining Notification Act (WARN) of 1988 generally covers employers with 100 or more employees, not counting those who have worked less than six months in the last 12 months and those who work an average of less than 20 hours a week. Regular federal, state, and local government entities that provide public services are not covered. WARN protects workers, their families, and communities by requiring employers to provide notification 60 calendar days in advance of plant closings and mass layoffs.
These are just a few of the key regulatory federal statutes, regulations, and guidance that human resources professionals need to understand to confirm organizational compliance. For additional information on HR compliance resources, the Society of Human Resource Management (SHRM) at SHRM.org maintains a plethora of resources for the HR professional and the businesses that they support.
To ensure the successful management and oversight of the many compliance rules and regulations, the human resources team must utilize best practices to inform and hold employees accountable to HR compliance practices. Some of these best practices include education and training, documentation, and audit. Each of these is described in greater detail and will help HR achieve its important goal of maintaining HR compliance for the organization.
Education and training in the areas of compliance and labor law are critical to ensure that all applicable laws and regulations are being followed. These laws can change from year to year, so the HR professionals in the organization need to ensure that they are engaged in ongoing education and training. It is not just imperative for the HR professional to receive training. In many organizations, managers receive training on key rules and regulations (such as FMLA or ADA, to name a few) so that they have a foundation of knowledge when dealing with employee situations and potential risk areas. Human resources and management need to partner to ensure alignment on compliance issues—especially when there is a risk that an employee situation treads into compliance regulation territory. See Table 10.1 for a partial list of federal labor laws by number of employees, as displayed on the Society for Human Resource Management website. For a full list, go to this website .
Refer to Table 10.1: Federal Labor Laws by Number of Employees.
| American Taxpayer Relief Act of 2012 |
| Consumer Credit Protection Act of 1968 |
| Employee Polygraph Protection Act of 1988 |
| Employee Retirement Income Security Act of 1974 (ERISA) |
| Equal Pay Act of 1963 |
| Fair and Accurate Credit Transaction Act of 2003 (FACT) |
| Fair Credit Reporting Act of 1969 |
| Fair Labor Standards Act of 1938 |
| Federal Insurance Contributions Act of 1935 (Social Security) (FICA) |
| Health Insurance Portability and Accountability Act of 1996 (if a company offers benefits) (HIPPA) |
| Immigration Reform and Control Act of 1986 |
|
These federal laws cover all employees of all organizations. Several other factors may apply in determining employer coverage, such as whether the employer is public or private, whether the employer offers health insurance, and whether the employer uses a third party to conduct background checks. Source: SHRM, https://www.shrm.org/ , accessed 4 April 2024. |
Table 10.1 (Attribution: Copyright Rice University, OpenStax, under CC-BY 4.0 license)
Documentation of the rules and regulations—in the form of an employee handbook—can be one of the most important resources that HR can provide to the organization to mitigate compliance risk. The handbook should be updated regularly and should detail the organization’s policies and procedures and how business is to be conducted. Legal counsel should review any such documentation before it is distributed to ensure that it is up-to-date and appropriate for the audience.
Scheduling HR compliance audits should be part of the company’s overall strategy to avoid legal risk. Noncompliance can cause enormous financial and reputational risk to a company, so it is important to have audits that test the organization’s controls and preparedness. When the HR function takes the lead in implementing audits and other best practices, they create real value for the organization.
- What are some of the key regulations that guide the compliance work of human resource management?
- What does an employee handbook provide to an organization?


11.7: Assessing Motor Strength
A brief musculoskeletal assessment is performed as part of the neurological assessment to determine the neurological stimulation of bilateral strength. Read more details about muscles and musculoskeletal assessment in the “Musculoskeletal Assessment” chapter. Unequal extremity motor strength can indicate underlying neurological disease or injury. Assessing motor strength includes comparing bilateral hand grasps, upper extremity strength, and lower extremity strength. Keep in mind that extremities on the dominant side are usually slightly stronger than the nondominant side.
Hand Grasps
To perform a hand grasp test, extend two fingers on both hands toward the patient. Ask the patient to squeeze both of your hands and compare for similar bilateral strength. See Figure \(\PageIndex{1}\) [1] for an image of assessing hand grasp strength.
Upper Extremity Strength
To test upper extremity strength, ask the patient to extend their forearms with palms facing upwards. Place your hands on their inner forearms and ask them to pull their arms toward them while you provide resistance. An expected finding is the patient strongly bilaterally pulls against resistance with both arms.
An alternative test is to ask the patient to put their hands in the air with their palms facing you. Place your palms against theirs and ask them to push while you provide resistance. See Figure \(\PageIndex{2}\) [2] for an image of assessing upper body strength.
Lower Body Strength
To assess lower body strength while the patient is in a seated position, place your hands behind their calves. Ask them to pull backwards with their lower legs while you provide resistance in the opposite direction.
Alternative tests are to place your hands on the patient’s lower thighs and ask them to lift their legs upwards while you provide downward resistance, or place your hands on the top of their feet and ask them to pull their toes upwards while you provide resistance. In a similar manner, you can also place your hands on the dorsal part of their feet and ask them to press downwards “like pressing the gas pedal of a car,” while providing resistance. Compare lower extremity strength on both sides. See Figure \(\PageIndex{3}\) [3] for images of assessing lower extremity strength.
- "Neuro Exam Image 38.png" by Meredith Pomietlo for Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- "Neuro Exam Image 41.png" and "Neuro Exam Image 39.jpg" by Meredith Pomietlo for Chippewa Valley Technical College are licensed under CC BY 4.0 ↵
- "Musculoskeletal Exam Image 2.png," "Neuro Exam image 6.png," and "Musculoskeletal Exam Image 7.png" by Meredith Pomietlo for Chippewa Valley Technical College are licensed under CC BY 4.0 ↵


15.11: Violence and Abuse
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Violence and abuse are among the most disconcerting of the challenges that today’s families face. Abuse can occur between spouses, between parent and child, as well as between other family members. The frequency of violence among families is difficult to determine because many cases of spousal abuse and child abuse go unreported. In any case, studies have shown that abuse (reported or not) has a major impact on families and society as a whole. 72
Adolescent Child Abuse
Children and teens are among the most helpless victims of abuse. In 2010, there were more than 3.3 million reports of child abuse involving an estimated 5.9 million children (Child Help 2011). Three-fifths of child abuse reports are made by professionals, including teachers, law enforcement personnel, and social services staff. The rest are made by anonymous sources, other relatives, parents, friends, and neighbors.  Some children suffer from a combination of these forms of abuse. The majority (81.2 %) of perpetrators are parents; 6.2 percent are other relatives. 73
Does Corporal Punishment Constitute Child Abuse?
Physical abuse in children may come in the form of beating, kicking, throwing, choking, hitting with objects, burning, or other methods. Injury inflicted by such behavior is considered abuse even if the parent or caregiver did not intend to harm the child. Other types of physical contact that are characterized as discipline (spanking, for example) are not considered abuse as long as no injury results (Child Welfare Information Gateway 2008).
This issue is rather controversial among modern-day people in the United States. While some parents feel that physical discipline, or corporal punishment, is an effective way to respond to bad behavior, others feel that it is a form of abuse. According to a poll conducted by ABC News, 65 percent of respondents approve of spanking and 50 percent said that they sometimes spank their child. But in the U.S., the majority of mental health professionals, as well as other professionals such as physicians and child welfare personnel, do not support the use of physical punishment.
Tendency toward physical punishment may be affected by culture and education. Those who live in the South are more likely than those who live in other regions to spank their child. Those who do not have a college education are also more likely to spank their child (Crandall 2011). Currently, 23 states officially allow spanking in the school system; however, many parents may object and school officials must follow a set of clear guidelines when administering this type of punishment (Crandall 2011).
Decades of research have yielded more than 500 studies examining the impact of physical punishment on children (Gershoff & Grogan-Kaylor, 2016). Within the past 15 years, several meta-analyses have attempted to synthesize this body of research. In a highly publicized meta-analysis, Gershoff (2002) concluded that physical punishment is not only ineffective, but also harmful. It may lead to aggression by the victim, particularly in those who are spanked at a young age (Berlin 2009).
Debates about parental use of physical punishment have been ongoing in the USA for decades. Calls to “move beyond” the research, or to “end the debate,” have become commonplace (e.g., Durrant & Ensom, 2017; MacMillan & Mikton, 2017). Three questions, it seems, sit at the center of these debates. Is physical punishment of children a Human Rights Issue? Is physical punishment effective? Is physical punishment harmful? 75
Contributors and Attributions
72. Lifespan Development – Module 7: Adolescence by Lumen Learning is licensed under CC BY 4.0
73. Introduction to Sociology 2e by OpenStax is licensed under CC BY 4.0
75. Challenges Families Face by OpenStax is licensed under CC BY 4.0
Physical punishment of children by US parents: moving beyond debate to promote children’s health and well-being by Cindy Miller Perrin and Robin Perrin is licensed under CC BY 4.0


 Why not?
Prepare a table identifying several energy transitions that take place during the typical operation of an automobile.
Explain the difference between heat capacity and specific heat of a substance.
Calculate the heat capacity, in joules and in calories per degree, of the following:
- 28.4 g of water
- 1.00 oz of lead
Calculate the heat capacity, in joules and in calories per degree, of the following:
- 45.8 g of nitrogen gas
- 1.00 pound of aluminum metal
How much heat, in joules and in calories, must be added to a 75.0–g iron block with a specific heat of 0.449 J/g °C to increase its temperature from 25 °C to its melting temperature of 1535 °C?
How much heat, in joules and in calories, is required to heat a 28.4-g (1-oz) ice cube from −23.0 °C to −1.0 °C?
How much would the temperature of 275 g of water increase if 36.5 kJ of heat were added?
If 14.5 kJ of heat were added to 485 g of liquid water, how much would its temperature increase?
A piece of unknown substance weighs 44.7 g and requires 2110 J to increase its temperature from 23.2 °C to 89.6 °C.
- What is the specific heat of the substance?
- If it is one of the substances found in Table 5.1, what is its likely identity?
A piece of unknown solid substance weighs 437.2 g, and requires 8460 J to increase its temperature from 19.3 °C to 68.9 °C.
- What is the specific heat of the substance?
- If it is one of the substances found in Table 5.1, what is its likely identity?
An aluminum kettle weighs 1.05 kg.
- What is the heat capacity of the kettle?
- How much heat is required to increase the temperature of this kettle from 23.0 °C to 99.0 °C?
- How much heat is required to heat this kettle from 23.0 °C to 99.0 °C if it contains 1.25 L of water (density of 0.997 g/mL and a specific heat of 4.184 J/g °C)?
Most people find waterbeds uncomfortable unless the water temperature is maintained at about 85 °F. Unless it is heated, a waterbed that contains 892 L of water cools from 85 °F to 72 °F in 24 hours. Estimate the amount of electrical energy required over 24 hours, in kWh, to keep the bed from cooling. Note that 1 kilowatt-hour (kWh) = 3.6 10 6 J, and assume that the density of water is 1.0 g/mL (independent of temperature). What other assumptions did you make? How did they affect your calculated result (i.e., were they likely to yield “positive” or “negative” errors)?
A 500-mL bottle of water at room temperature and a 2-L bottle of water at the same temperature were placed in a refrigerator. After 30 minutes, the 500-mL bottle of water had cooled to the temperature of the refrigerator. An hour later, the 2-L of water had cooled to the same temperature. When asked which sample of water lost the most heat, one student replied that both bottles lost the same amount of heat because they started at the same temperature and finished at the same temperature. A second student thought that the 2-L bottle of water lost more heat because there was more water. A third student believed that the 500-mL bottle of water lost more heat because it cooled more quickly. A fourth student thought that it was not possible to tell because we do not know the initial temperature and the final temperature of the water. Indicate which of these answers is correct and describe the error in each of the other answers.
Would the amount of heat measured for the reaction in Example 5.5 be greater, lesser, or remain the same if we used a calorimeter that was a poorer insulator than a coffee cup calorimeter? Explain your answer.
Would the amount of heat absorbed by the dissolution in Example 5.6 appear greater, lesser, or remain the same if the experimenter used a calorimeter that was a poorer insulator than a coffee cup calorimeter? Explain your answer.
Would the amount of heat absorbed by the dissolution in Example 5.6 appear greater, lesser, or remain the same if the heat capacity of the calorimeter were taken into account? Explain your answer.
How many milliliters of water at 23 °C with a density of 1.00 g/mL must be mixed with 180 mL (about 6 oz) of coffee at 95 °C so that the resulting combination will have a temperature of 60 °C? Assume that coffee and water have the same density and the same specific heat.
How much will the temperature of a cup (180 g) of coffee at 95 °C be reduced when a 45 g silver spoon (specific heat 0.24 J/g °C) at 25 °C is placed in the coffee and the two are allowed to reach the same temperature? Assume that the coffee has the same density and specific heat as water.
A 45-g aluminum spoon (specific heat 0.88 J/g °C) at 24 °C is placed in 180 mL (180 g) of coffee at 85 °C and the temperature of the two become equal.
- What is the final temperature when the two become equal? Assume that coffee has the same specific heat as water.
- The first time a student solved this problem she got an answer of 88 °C. Explain why this is clearly an incorrect answer.
The temperature of the cooling water as it leaves the hot engine of an automobile is 240 °F. After it passes through the radiator it has a temperature of 175 °F. Calculate the amount of heat transferred from the engine to the surroundings by one gallon of water with a specific heat of 4.184 J/g °C.
A 70.0-g piece of metal at 80.0 °C is placed in 100 g of water at 22.0 °C contained in a calorimeter like that shown in Figure 5.12. The metal and water come to the same temperature at 24.6 °C. How much heat did the metal give up to the water? What is the specific heat of the metal?
If a reaction produces 1.506 kJ of heat, which is trapped in 30.0 g of water initially at 26.5 °C in a calorimeter like that in Figure 5.12, what is the resulting temperature of the water?
A 0.500-g sample of KCl is added to 50.0 g of water in a calorimeter (Figure 5.12). If the temperature decreases by 1.05 °C, what is the approximate amount of heat involved in the dissolution of the KCl, assuming the specific heat of the resulting solution is 4.18 J/g °C? Is the reaction exothermic or endothermic?
Dissolving 3.0 g of CaCl 2 ( s ) in 150.0 g of water in a calorimeter (Figure 5.12) at 22.4 °C causes the temperature to rise to 25.8 °C. What is the approximate amount of heat involved in the dissolution, assuming the specific heat of the resulting solution is 4.18 J/g °C? Is the reaction exothermic or endothermic?
When 50.0 g of 0.200 M NaCl( aq ) at 24.1 °C is added to 100.0 g of 0.100 M AgNO 3 ( aq ) at 24.1 °C in a calorimeter, the temperature increases to 25.2 °C as AgCl( s ) forms. Assuming the specific heat of the solution and products is 4.20 J/g °C, calculate the approximate amount of heat in joules produced.
The addition of 3.15 g of Ba(OH) 2 ·8H 2 O to a solution of 1.52 g of NH 4 SCN in 100 g of water in a calorimeter caused the temperature to fall by 3.1 °C. Assuming the specific heat of the solution and products is 4.20 J/g °C, calculate the approximate amount of heat absorbed by the reaction, which can be represented by the following equation:
Ba(OH) 2 ·8H 2 O( s ) + 2NH 4 SCN( aq ) ⟶ Ba(SCN) 2 ( aq ) + 2NH 3 ( aq ) + 10H 2 O( l )
The reaction of 50 mL of acid and 50 mL of base described in Example 5.5 increased the temperature of the solution by 6.9 ºC. How much would the temperature have increased if 100 mL of acid and 100 mL of base had been used in the same calorimeter starting at the same temperature of 22.0 ºC? Explain your answer.
If the 3.21 g of NH 4 NO 3 in Example 5.6 were dissolved in 100.0 g of water under the same conditions, how much would the temperature change? Explain your answer.
When 1.0 g of fructose, C 6 H 12 O 6 ( s ), a sugar commonly found in fruits, is burned in oxygen in a bomb calorimeter, the temperature of the calorimeter increases by 1.58 °C. If the heat capacity of the calorimeter and its contents is 9.90 kJ/°C, what is q for this combustion?
When a 0.740-g sample of trinitrotoluene (TNT), C 7 H 5 N 2 O 6 , is burned in a bomb calorimeter, the temperature increases from 23.4 °C to 26.9 °C. The heat capacity of the calorimeter is 534 J/°C, and it contains 675 mL of water. How much heat was produced by the combustion of the TNT sample?
One method of generating electricity is by burning coal to heat water, which produces steam that drives an electric generator. To determine the rate at which coal is to be fed into the burner in this type of plant, the heat of combustion per ton of coal must be determined using a bomb calorimeter. When 1.00 g of coal is burned in a bomb calorimeter (Figure 5.17), the temperature increases by 1.48 °C. If the heat capacity of the calorimeter is 21.6 kJ/°C, determine the heat produced by combustion of a ton of coal (2.000 10 3 pounds).
The amount of fat recommended for someone with a daily diet of 2000 Calories is 65 g. What percent of the calories in this diet would be supplied by this amount of fat if the average number of Calories for fat is
9.1 Calories/g?
A teaspoon of the carbohydrate sucrose (common sugar) contains 16 Calories (16 kcal). What is the mass of one teaspoon of sucrose if the average number of Calories for carbohydrates is 4.1 Calories/g?
What is the maximum mass of carbohydrate in a 6-oz serving of diet soda that contains less than 1 Calorie per can if the average number of Calories for carbohydrates is 4.1 Calories/g?
A pint of premium ice cream can contain 1100 Calories. What mass of fat, in grams and pounds, must be produced in the body to store an extra 1.1 10
3
Calories if the average number of Calories for fat is
9.1 Calories/g?
A serving of a breakfast cereal contains 3 g of protein, 18 g of carbohydrates, and 6 g of fat. What is the Calorie content of a serving of this cereal if the average number of Calories for fat is 9.1 Calories/g, for carbohydrates is 4.1 Calories/g, and for protein is 4.1 Calories/g?
Which is the least expensive source of energy in kilojoules per dollar: a box of breakfast cereal that weighs 32 ounces and costs $4.23, or a liter of isooctane (density, 0.6919 g/mL) that costs $0.45? Compare the nutritional value of the cereal with the heat produced by combustion of the isooctane under standard conditions. A 1.0-ounce serving of the cereal provides 130 Calories.
Explain how the heat measured in Example 5.5 differs from the enthalpy change for the exothermic reaction described by the following equation:
Using the data in the check your learning section of Example 5.5, calculate Δ H in kJ/mol of AgNO 3 ( aq ) for the reaction:
Calculate the enthalpy of solution (Δ H for the dissolution) per mole of NH 4 NO 3 under the conditions described in Example 5.6.
Calculate Δ
H
for the reaction described by the equation. (
Hint
: Use the value for the approximate amount of heat absorbed by the reaction that you calculated in a previous exercise.)
Calculate the enthalpy of solution (Δ H for the dissolution) per mole of CaCl 2 (refer to Exercise 5.25).
Although the gas used in an oxyacetylene torch (Figure 5.7) is essentially pure acetylene, the heat produced by combustion of one mole of acetylene in such a torch is likely not equal to the enthalpy of combustion of acetylene listed in Table 5.2. Considering the conditions for which the tabulated data are reported, suggest an explanation.
How much heat is produced by burning 4.00 moles of acetylene under standard state conditions?
How much heat is produced by combustion of 125 g of methanol under standard state conditions?
How many moles of isooctane must be burned to produce 100 kJ of heat under standard state conditions?
What mass of carbon monoxide must be burned to produce 175 kJ of heat under standard state conditions?
When 2.50 g of methane burns in oxygen, 125 kJ of heat is produced. What is the enthalpy of combustion per mole of methane under these conditions?
How much heat is produced when 100 mL of 0.250 M HCl (density, 1.00 g/mL) and 200 mL of 0.150 M NaOH (density, 1.00 g/mL) are mixed?
If both solutions are at the same temperature and the specific heat of the products is 4.19 J/g °C, how much will the temperature increase? What assumption did you make in your calculation?
A sample of 0.562 g of carbon is burned in oxygen in a bomb calorimeter, producing carbon dioxide. Assume both the reactants and products are under standard state conditions, and that the heat released is directly proportional to the enthalpy of combustion of graphite. The temperature of the calorimeter increases from 26.74 °C to 27.93 °C. What is the heat capacity of the calorimeter and its contents?
Before the introduction of chlorofluorocarbons, sulfur dioxide (enthalpy of vaporization, 6.00 kcal/mol) was used in household refrigerators. What mass of SO 2 must be evaporated to remove as much heat as evaporation of 1.00 kg of CCl 2 F 2 (enthalpy of vaporization is 17.4 kJ/mol)?
The vaporization reactions for SO 2 and CCl 2 F 2 are and respectively.
Homes may be heated by pumping hot water through radiators. What mass of water will provide the same amount of heat when cooled from 95.0 to 35.0 °C, as the heat provided when 100 g of steam is cooled from 110 °C to 100 °C.
Which of the enthalpies of combustion in Table 5.2 the table are also standard enthalpies of formation?
Does the standard enthalpy of formation of H 2 O( g ) differ from Δ H ° for the reaction
Joseph Priestly prepared oxygen in 1774 by heating red mercury(II) oxide with sunlight focused through a lens. How much heat is required to decompose exactly 1 mole of red HgO( s ) to Hg( l ) and O 2 ( g ) under standard conditions?
How many kilojoules of heat will be released when exactly 1 mole of manganese, Mn, is burned to form Mn 3 O 4 ( s ) at standard state conditions?
How many kilojoules of heat will be released when exactly 1 mole of iron, Fe, is burned to form Fe 2 O 3 ( s ) at standard state conditions?
The following sequence of reactions occurs in the commercial production of aqueous nitric acid:
Determine the total energy change for the production of one mole of aqueous nitric acid by this process.
Both graphite and diamond burn.
For the conversion of graphite to diamond:
Which produces more heat, the combustion of graphite or the combustion of diamond?
From the molar heats of formation in Appendix G, determine how much heat is required to evaporate one mole of water:
Which produces more heat?
or
for the phase change
Calculate for the process
from the following information:
Calculate for the process
from the following information:
Calculate Δ H for the process
from the following information:
Calculate for the process
from the following information:
Calculate the standard molar enthalpy of formation of NO(
g
) from the following data:
Using the data in Appendix G, calculate the standard enthalpy change for each of the following reactions:
Using the data in Appendix G, calculate the standard enthalpy change for each of the following reactions:
The following reactions can be used to prepare samples of metals. Determine the enthalpy change under standard state conditions for each.
The decomposition of hydrogen peroxide, H
2
O
2
, has been used to provide thrust in the control jets of various space vehicles. Using the data in Appendix G, determine how much heat is produced by the decomposition of exactly 1 mole of H
2
O
2
under standard conditions.
Calculate the enthalpy of combustion of propane, C 3 H 8 ( g ), for the formation of H 2 O( g ) and CO 2 ( g ). The enthalpy of formation of propane is −104 kJ/mol.
Calculate the enthalpy of combustion of butane, C 4 H 10 ( g ) for the formation of H 2 O( g ) and CO 2 ( g ). The enthalpy of formation of butane is −126 kJ/mol.
Both propane and butane are used as gaseous fuels. Which compound produces more heat per gram when burned?
The white pigment TiO 2 is prepared by the reaction of titanium tetrachloride, TiCl 4 , with water vapor in the gas phase:
How much heat is evolved in the production of exactly 1 mole of TiO 2 ( s ) under standard state conditions?
Water gas, a mixture of H 2 and CO, is an important industrial fuel produced by the reaction of steam with red hot coke, essentially pure carbon:
-
Under the conditions of the reaction, methanol forms as a gas. Calculate for this reaction and for the condensation of gaseous methanol to liquid methanol.
(c) Calculate the heat of combustion of 1 mole of liquid methanol to H 2 O( g ) and CO 2 ( g ).
In the early days of automobiles, illumination at night was provided by burning acetylene, C
2
H
2
. Though no longer used as auto headlamps, acetylene is still used as a source of light by some cave explorers. The acetylene is (was) prepared in the lamp by the reaction of water with calcium carbide, CaC
2
:
Calculate the standard enthalpy of the reaction. The of CaC
2
is −15.14 kcal/mol.
From the data in Table 5.2, determine which of the following fuels produces the greatest amount of heat per gram when burned under standard conditions: CO( g ), CH 4 ( g ), or C 2 H 2 ( g ).
The enthalpy of combustion of hard coal averages −35 kJ/g, that of gasoline, 1.28 10 5 kJ/gal. How many kilograms of hard coal provide the same amount of heat as is available from 1.0 gallon of gasoline? Assume that the density of gasoline is 0.692 g/mL (the same as the density of isooctane).
Ethanol, C 2 H 5 OH, is used as a fuel for motor vehicles, particularly in Brazil.
- Write the balanced equation for the combustion of ethanol to CO 2 ( g ) and H 2 O( g ), and, using the data in Appendix G, calculate the enthalpy of combustion of 1 mole of ethanol.
- The density of ethanol is 0.7893 g/mL. Calculate the enthalpy of combustion of exactly 1 L of ethanol.
- Assuming that an automobile’s mileage is directly proportional to the heat of combustion of the fuel, calculate how much farther an automobile could be expected to travel on 1 L of gasoline than on 1 L of ethanol. Assume that gasoline has the heat of combustion and the density of n–octane, C 8 H 18 density = 0.7025 g/mL).
Among the substances that react with oxygen and that have been considered as potential rocket fuels are diborane [B 2 H 6 , produces B 2 O 3 ( s ) and H 2 O( g )], methane [CH 4 , produces CO 2 ( g ) and H 2 O( g )], and hydrazine [N 2 H 4 , produces N 2 ( g ) and H 2 O( g )]. On the basis of the heat released by 1.00 g of each substance in its reaction with oxygen, which of these compounds offers the best possibility as a rocket fuel? The
How much heat is produced when 1.25 g of chromium metal reacts with oxygen gas under standard conditions?
Ethylene, C 2 H 4 , a byproduct from the fractional distillation of petroleum, is fourth among the 50 chemical compounds produced commercially in the largest quantities. About 80% of synthetic ethanol is manufactured from ethylene by its reaction with water in the presence of a suitable catalyst.
Using the data in the table in Appendix G, calculate Δ H ° for the reaction.
The oxidation of the sugar glucose, C
6
H
12
O
6
, is described by the following equation:
The metabolism of glucose gives the same products, although the glucose reacts with oxygen in a series of steps in the body.
- How much heat in kilojoules can be produced by the metabolism of 1.0 g of glucose?
- How many Calories can be produced by the metabolism of 1.0 g of glucose?
Propane, C 3 H 8 , is a hydrocarbon that is commonly used as a fuel.
- Write a balanced equation for the complete combustion of propane gas.
- Calculate the volume of air at 25 °C and 1.00 atmosphere that is needed to completely combust 25.0 grams of propane. Assume that air is 21.0 percent O 2 by volume. (Hint: We will see how to do this calculation in a later chapter on gases—for now use the information that 1.00 L of air at 25 °C and 1.00 atm contains 0.275 g of O 2 .)
- The heat of combustion of propane is −2,219.2 kJ/mol. Calculate the heat of formation, of propane given that of H 2 O( l ) = −285.8 kJ/mol and of CO 2 ( g ) = −393.5 kJ/mol.
- Assuming that all of the heat released in burning 25.0 grams of propane is transferred to 4.00 kilograms of water, calculate the increase in temperature of the water.
During a recent winter month in Sheboygan, Wisconsin, it was necessary to obtain 3500 kWh of heat provided by a natural gas furnace with 89% efficiency to keep a small house warm (the efficiency of a gas furnace is the percent of the heat produced by combustion that is transferred into the house).
-
(f) How many kilowatt–hours (1 kWh = 3.6 10
6
J) of electricity would be required to provide the heat necessary to heat the house? Note electricity is 100% efficient in producing heat inside a house.
(g) Although electricity is 100% efficient in producing heat inside a house, production and distribution of electricity is not 100% efficient. The efficiency of production and distribution of electricity produced in a coal-fired power plant is about 40%. A certain type of coal provides 2.26 kWh per pound upon combustion. What mass of this coal in kilograms will be required to produce the electrical energy necessary to heat the house if the efficiency of generation and distribution is 40%?


2.4: Examples
Consider an extra-heavy hockey puck, of mass 5.0 kg, traveling between points A and B in a straight line at a speed of 10.0 m/s. At point B, it gets hit by a hockey stick which delivers an impulse of \(\Delta\vec{p}\) to the puck. Using meter sticks and protractors, make scale drawings of this process on the whiteboards, and answer the following two questions:
-
In figure 1, the amount of impulse delivered is \(\Delta p\)=30 kg m/s at an angle of 30\(\circ\), as shown.
- What is the final momentum of the hockey puck, \(p_f\)?
-
In figure 2, we measure the final momentum \(p_f\) to be 100 kg m/s, moving at an angle of 40\(^\circ\), as shown.
- What was the impulse transferred to the hockey puck, \(\Delta p\)?
- At what angle did the impulse act on the puck?
Suppose a cart with mass \(m\) is moving to the right at a speed of \(v_0\). It collides with a cart of mass \(2m\), which is initially at rest. After the collision, the first cart rebounds to the left with a speed of \(\frac{1}{3}v_0\).
- How much impulse did the first cart deliver to the second? Use \(v_f\) as the final speed of the second cart.
- How much impulse did the second cart deliver to the first?
- Assuming that these two impulse transfers are the same, determine the final speed \(v_f\) in terms of the initial speed \(v_0\).
I drop a golf ball of mass \(m = 46\) g onto the floor, and right before it hits it has a velocity of \(v_{yi} = -2.2\) m/s.
- What is the momentum of the ball right before it hits the ground?
- If the speed of the ball after it bounces, \(v_{yf}\) , is 75\% of the speed right before it hits, what was the impulse the floor delivered to the ball?
- If the ball interacts with the floor for a time period \(\Delta t\), find an expression for the ratio between the average force the floor acted on the ball with to the weight of the ball \(mg\), in terms of \(v_{yi}\) , \(v_{yf}\) , \(g\), and \(\Delta t\).
- Calculate this ratio using \(\Delta t\) = 150 ms and the other parameters of the problem.
The graph shows a collision between two carts (possibly equipped with magnets so that they repel each other before they actually touch) on an air track. The inertia (mass) of cart 1 is 1 kg. Note: this is a position vs. time graph!
- What are the initial velocities of the carts?
- What are the final velocities of the carts?
- What is the mass of the second cart?
- Does the air track appear to be level? Why? (Hint: does the graph show any evidence of acceleration, for either cart, outside of the collision region?)
- At the collision time, is the change in velocity (\(\Delta v\) )of the first cart positive or negative? How about the second cart? (Justify your answers.)
- For the system consisting of the two carts, what is its initial (total) momentum? What is its final momentum?
- Imagine now that one of the magnets is reversed, so when the carts collide they stick to each other. What would then be the final momentum of the system? What would be its final velocity?
Figure \(\PageIndex{1}\): A collision between two carts
Solution
(a) All the velocities are to be calculated by picking an easy straight part of each curve and calculating
\[ v=\frac{\Delta x}{\Delta t} \nonumber \]
for suitable intervals. In this way one gets
\begin{aligned}
&v_{1 i}=-1 \: \frac{\mathrm{m}}{\mathrm{s}}\\
&v_{2 i}=0.5 \: \frac{\mathrm{m}}{\mathrm{s}}
\end{aligned}
(b) Similarly, one gets
\begin{aligned}
&v_{1 f}=1 \: \frac{\mathrm{m}}{\mathrm{s}}\\
&v_{2 f}=-0.5 \: \frac{\mathrm{m}}{\mathrm{s}}
\end{aligned}
(c) Use this equation, or equivalent (conservation of momentum is OK)
\begin{aligned}
&\frac{m_{2}}{m_{1}}=-\frac{\Delta v_{1}}{\Delta v_{2}}\\
&\frac{m_{2}}{m_{1}}=-\frac{1-(-1)}{-0.5-0.5}=2
\end{aligned}
so the mass of the second cart is 2 kg.
(d) Yes, the track appears to be level because the carts do not show any evidence of acceleration outside of the collision region (the position vs. time curves are straight lines outside of the region approximately given by 4.5 s < \(t\) < 5.5 s).
(e) The change in velocity of the first cart is positive. You can see this either graphically (the curve is like a parabola that opens upwards, i.e., concave), or algebraically (the cart’s velocity increases, going from −1 m/s to 1 m/s)
Similarly, the change in velocity of the second cart is negative. The curve is like a parabola that opens downwards, i.e., convex; or, algebraically, the cart’s velocity decreases, going from 0.5 m/s to −0.5 m/s.
(f) The initial momentum of the system is
\[ p_{i}=m_{1} v_{1 i}+m_{2} v_{2 i}=(1 \: \mathrm{kg}) \times\left(-1 \: \frac{\mathrm{m}}{\mathrm{s}}\right)+(2 \: \mathrm{kg}) \times\left(0.5 \: \frac{\mathrm{m}}{\mathrm{s}}\right)=0 \nonumber \]
The final momentum is
\[ p_{f}=m_{1} v_{1 f}+m_{2} v_{2 f}=(1 \: \mathrm{kg}) \times\left(1 \: \frac{\mathrm{m}}{\mathrm{s}}\right)+(2 \: \mathrm{kg}) \times\left(-0.5 \: \frac{\mathrm{m}}{\mathrm{s}}\right)=0 \nonumber \]
You could also just say that the final momentum should be the same as the initial momentum, since the system appears to be isolated.
(g) The momentum should be conserved in this case as well, so \(p_f\) = 0. The velocity would be
\[ v_{f}=\frac{p_{f}}{m_{1}+m_{2}}=0 \nonumber .\]


26.4: Electrolyte Balance
By the end of this section, you will be able to:
- List the role of the six most important electrolytes in the body
- Name the disorders associated with abnormally high and low levels of the six electrolytes
- Identify the predominant extracellular anion
- Describe the role of aldosterone on the level of water in the body
The body contains a large variety of ions, or electrolytes, which perform a variety of functions. Some ions assist in the transmission of electrical impulses along cell membranes in neurons and muscles. Other ions help to stabilize protein structures in enzymes. Still others aid in releasing hormones from endocrine glands. All of the ions in plasma contribute to the osmotic balance that controls the movement of water between cells and their environment.
Electrolytes in living systems include sodium, potassium, chloride, bicarbonate, calcium, phosphate, magnesium, copper, zinc, iron, manganese, molybdenum, copper, and chromium. In terms of body functioning, six electrolytes are most important: sodium, potassium, chloride, bicarbonate, calcium, and phosphate.
Roles of Electrolytes
These six ions aid in nerve excitability, endocrine secretion, membrane permeability, buffering body fluids, and controlling the movement of fluids between compartments. These ions enter the body through the digestive tract. More than 90 percent of the calcium and phosphate that enters the body is incorporated into bones and teeth, with bone serving as a mineral reserve for these ions. In the event that calcium and phosphate are needed for other functions, bone tissue can be broken down to supply the blood and other tissues with these minerals. Phosphate is a normal constituent of nucleic acids; hence, blood levels of phosphate will increase whenever nucleic acids are broken down.
Excretion of ions occurs mainly through the kidneys, with lesser amounts lost in sweat and in feces. Excessive sweating may cause a significant loss, especially of sodium and chloride. Severe vomiting or diarrhea will cause a loss of chloride and bicarbonate ions. Adjustments in respiratory and renal functions allow the body to regulate the levels of these ions in the ECF.
Table \(\PageIndex{1}\) lists the reference values for blood plasma, cerebrospinal fluid (CSF), and urine for the six ions addressed in this section. In a clinical setting, sodium, potassium, and chloride are typically analyzed in a routine urine sample. In contrast, calcium and phosphate analysis requires a collection of urine across a 24-hour period, because the output of these ions can vary considerably over the course of a day. Urine values reflect the rates of excretion of these ions. Bicarbonate is the one ion that is not normally excreted in urine; instead, it is conserved by the kidneys for use in the body’s buffering systems.
| Name | Chemical symbol | Plasma | CSF | Urine |
|---|---|---|---|---|
| Sodium | Na + | 136.00–146.00 (mM) | 138.00–150.00 (mM) | 40.00–220.00 (mM) |
| Potassium | K + | 3.50–5.00 (mM) | 0.35–3.5 (mM) | 25.00–125.00 (mM) |
| Chloride | Cl - | 98.00–107.00 (mM) | 118.00–132.00 (mM) | 110.00–250.00 (mM) |
| Bicarbonate | HCO 3 - | 22.00–29.00 (mM) | ------ | ------ |
| Calcium | Ca ++ | 2.15–2.55 (mmol/day) | ------ | Up to 7.49 (mmol/day) |
| Phosphate | 0.81–1.45 (mmol/day) | ------ | 12.90–42.00 (mmol/day) |
Sodium
Sodium is the major cation of the extracellular fluid. It is responsible for one-half of the osmotic pressure gradient that exists between the interior of cells and their surrounding environment. People eating a typical Western diet, which is very high in NaCl, routinely take in 130 to 160 mmol/day of sodium, but humans require only 1 to 2 mmol/day. This excess sodium appears to be a major factor in hypertension (high blood pressure) in some people. Excretion of sodium is accomplished primarily by the kidneys. Sodium is freely filtered through the glomerular capillaries of the kidneys, and although much of the filtered sodium is reabsorbed in the proximal convoluted tubule, some remains in the filtrate and urine, and is normally excreted.
Hyponatremia is a lower-than-normal concentration of sodium, usually associated with excess water accumulation in the body, which dilutes the sodium. An absolute loss of sodium may be due to a decreased intake of the ion coupled with its continual excretion in the urine. An abnormal loss of sodium from the body can result from several conditions, including excessive sweating, vomiting, or diarrhea; the use of diuretics; excessive production of urine, which can occur in diabetes; and acidosis, either metabolic acidosis or diabetic ketoacidosis.
A relative decrease in blood sodium can occur because of an imbalance of sodium in one of the body’s other fluid compartments, like IF, or from a dilution of sodium due to water retention related to edema or congestive heart failure. At the cellular level, hyponatremia results in increased entry of water into cells by osmosis, because the concentration of solutes within the cell exceeds the concentration of solutes in the now-diluted ECF. The excess water causes swelling of the cells; the swelling of red blood cells—decreasing their oxygen-carrying efficiency and making them potentially too large to fit through capillaries—along with the swelling of neurons in the brain can result in brain damage or even death.
Hypernatremia is an abnormal increase of blood sodium. It can result from water loss from the blood, resulting in the hemoconcentration of all blood constituents. Hormonal imbalances involving ADH and aldosterone may also result in higher-than-normal sodium values.
Potassium
Potassium is the major intracellular cation. It helps establish the resting membrane potential in neurons and muscle fibers after membrane depolarization and action potentials. In contrast to sodium, potassium has very little effect on osmotic pressure. The low levels of potassium in blood and CSF are due to the sodium-potassium pumps in cell membranes, which maintain the normal potassium concentration gradients between the ICF and ECF. The recommendation for daily intake/consumption of potassium is 4700 mg. Potassium is excreted, both actively and passively, through the renal tubules, especially the distal convoluted tubule and collecting ducts. Potassium participates in the exchange with sodium in the renal tubules under the influence of aldosterone, which also relies on basolateral sodium-potassium pumps.
Hypokalemia is an abnormally low potassium blood level. Similar to the situation with hyponatremia, hypokalemia can occur because of either an absolute reduction of potassium in the body or a relative reduction of potassium in the blood due to the redistribution of potassium. An absolute loss of potassium can arise from decreased intake, frequently related to starvation. It can also come about from vomiting, diarrhea, or alkalosis.
Some insulin-dependent diabetic patients experience a relative reduction of potassium in the blood from the redistribution of potassium. When insulin is administered and glucose is taken up by cells, potassium passes through the cell membrane along with glucose, decreasing the amount of potassium in the blood and IF, which can cause hyperpolarization of the cell membranes of neurons, reducing their responses to stimuli.
Hyperkalemia , an elevated potassium blood level, also can impair the function of skeletal muscles, the nervous system, and the heart. Hyperkalemia can result from increased dietary intake of potassium. In such a situation, potassium from the blood ends up in the ECF in abnormally high concentrations. This can result in a partial depolarization (excitation) of the plasma membrane of skeletal muscle fibers, neurons, and cardiac cells of the heart, and can also lead to an inability of cells to repolarize. For the heart, this means that it won’t relax after a contraction, and will effectively “seize” and stop pumping blood, which is fatal within minutes. Because of such effects on the nervous system, a person with hyperkalemia may also exhibit mental confusion, numbness, and weakened respiratory muscles.
Chloride
Chloride is the predominant extracellular anion. Chloride is a major contributor to the osmotic pressure gradient between the ICF and ECF, and plays an important role in maintaining proper hydration. Chloride functions to balance cations in the ECF, maintaining the electrical neutrality of this fluid. The paths of secretion and reabsorption of chloride ions in the renal system follow the paths of sodium ions.
Hypochloremia , or lower-than-normal blood chloride levels, can occur because of defective renal tubular absorption. Vomiting, diarrhea, and metabolic acidosis can also lead to hypochloremia. Hyperchloremia , or higher-than-normal blood chloride levels, can occur due to dehydration, excessive intake of dietary salt (NaCl) or swallowing of sea water, aspirin intoxication, congestive heart failure, and the hereditary, chronic lung disease, cystic fibrosis. In people who have cystic fibrosis, chloride levels in sweat are two to five times those of normal levels, and analysis of sweat is often used in the diagnosis of the disease.
Interactive Link
Read this article for an explanation of the effect of seawater on humans. What effect does drinking seawater have on the body?
Bicarbonate
Bicarbonate is the second most abundant anion in the blood. Its principal function is to maintain your body’s acid-base balance by being part of buffer systems. This role will be discussed in a different section.
Bicarbonate ions result from a chemical reaction that starts with carbon dioxide (CO 2 ) and water, two molecules that are produced at the end of aerobic metabolism. Only a small amount of CO 2 can be dissolved in body fluids. Thus, over 90 percent of the CO 2 is converted into bicarbonate ions, HCO 3 – , through the following reactions:
The bidirectional arrows indicate that the reactions can go in either direction, depending on the concentrations of the reactants and products. Carbon dioxide is produced in large amounts in tissues that have a high metabolic rate. Carbon dioxide is converted into bicarbonate in the cytoplasm of red blood cells through the action of an enzyme called carbonic anhydrase. Bicarbonate is transported in the blood. Once in the lungs, the reactions reverse direction, and CO 2 is regenerated from bicarbonate to be exhaled as metabolic waste.
Calcium
About two pounds of calcium in your body are bound up in bone, which provides hardness to the bone and serves as a mineral reserve for calcium and its salts for the rest of the tissues. Teeth also have a high concentration of calcium within them. A little more than one-half of blood calcium is bound to proteins, leaving the rest in its ionized form. Calcium ions, Ca 2 + , are necessary for muscle contraction, enzyme activity, and blood coagulation. In addition, calcium helps to stabilize cell membranes and is essential for the release of neurotransmitters from neurons and of hormones from endocrine glands.
Calcium is absorbed through the intestines under the influence of activated vitamin D. A deficiency of vitamin D leads to a decrease in absorbed calcium and, eventually, a depletion of calcium stores from the skeletal system, potentially leading to rickets in children and osteomalacia in adults, contributing to osteoporosis.
Hypocalcemia , or abnormally low calcium blood levels, is seen in hypoparathyroidism, which may follow the removal of the thyroid gland, because the four nodules of the parathyroid gland are embedded in it. Hypercalcemia , or abnormally high calcium blood levels, is seen in primary hyperparathyroidism. Some malignancies may also result in hypercalcemia.
Phosphate
Phosphate is present in the body in three ionic forms: , , and . The most common form is . Bone and teeth bind up 85 percent of the body’s phosphate as part of calcium-phosphate salts. Phosphate is found in phospholipids, such as those that make up the cell membrane, and in ATP, nucleotides, and buffers.
Hypophosphatemia , or abnormally low phosphate blood levels, occurs with heavy use of antacids, during alcohol withdrawal, and during malnourishment. In the face of phosphate depletion, the kidneys usually conserve phosphate, but during starvation, this conservation is impaired greatly. Hyperphosphatemia , or abnormally increased levels of phosphates in the blood, occurs if there is decreased renal function or in cases of acute lymphocytic leukemia. Additionally, because phosphate is a major constituent of the ICF, any significant destruction of cells can result in dumping of phosphate into the ECF.
Regulation of Sodium and Potassium
Sodium is reabsorbed from the renal filtrate, and potassium is excreted into the filtrate in the renal collecting tubule. The control of this exchange is governed principally by two hormones—aldosterone and angiotensin II.
Aldosterone
Recall that aldosterone increases the excretion of potassium and the reabsorption of sodium in the distal tubule. Aldosterone is released if blood levels of potassium increase, if blood levels of sodium severely decrease, or if blood pressure decreases. Its net effect is to conserve and increase water levels in the plasma by reducing the excretion of sodium, and thus water, from the kidneys. In a negative feedback loop, increased osmolality of the ECF (which follows aldosterone-stimulated sodium absorption) inhibits the release of the hormone (Figure \(\PageIndex{2}\) ).
Angiotensin II
Angiotensin II causes vasoconstriction and an increase in systemic blood pressure. This action increases the glomerular filtration rate, resulting in more material filtered out of the glomerular capillaries and into Bowman’s capsule. Angiotensin II also signals an increase in the release of aldosterone from the adrenal cortex.
In the distal convoluted tubules and collecting ducts of the kidneys, aldosterone stimulates the synthesis and activation of the sodium-potassium pump (Figure \(\PageIndex{3}\) ). Sodium passes from the filtrate, into and through the cells of the tubules and ducts, into the ECF and then into capillaries. Water follows the sodium due to osmosis. Thus, aldosterone causes an increase in blood sodium levels and blood volume. Aldosterone’s effect on potassium is the reverse of that of sodium; under its influence, excess potassium is pumped into the renal filtrate for excretion from the body.
Regulation of Calcium and Phosphate
Calcium and phosphate are both regulated through the actions of three hormones: parathyroid hormone (PTH), dihydroxyvitamin D (calcitriol), and calcitonin. All three are released or synthesized in response to the blood levels of calcium.
PTH is released from the parathyroid gland in response to a decrease in the concentration of blood calcium. The hormone activates osteoclasts to break down bone matrix and release inorganic calcium-phosphate salts. PTH also increases the gastrointestinal absorption of dietary calcium by converting vitamin D into dihydroxyvitamin D (calcitriol), an active form of vitamin D that intestinal epithelial cells require to absorb calcium.
PTH raises blood calcium levels by inhibiting the loss of calcium through the kidneys. PTH also increases the loss of phosphate through the kidneys.
Calcitonin is released from the thyroid gland in response to elevated blood levels of calcium. The hormone increases the activity of osteoblasts, which remove calcium from the blood and incorporate calcium into the bony matrix.


6.9: Pressure Variations for Macroscopic Samples
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At \(300\) K, the standard deviation of \(N_2\) speeds is about 40% of the average speed. Clearly the relative variation among molecular speeds in a sample of ordinary gas is very large. Why do we not observe macroscopic effects from this variation? In particular, if we measure the pressure at a small area of the container wall, why do we not observe pressure variations that reflect the wide variety of speeds with which molecules strike the wall?
Qualitatively, the answer is obvious. A single molecule whose scalar velocity is \(v\) contributes \(P_1\left(v\right)={mv^2}/{3V}\) to the pressure on the walls of its container. (See problem 20.) When we measure pressure, we measure an average squared velocity. Even if we measure the pressure over a very small area and a very short time, the
number of molecules striking the wall during the time of the measurement is very large. Consequently, the average speed of the molecules hitting the wall during any one such measurement is very close to the average speed in any other such measurement.
We are now able to treat this question quantitatively. For \(N_2\) gas at \(300\) K and \(1\) bar, roughly \(\mathrm{3\times }{\mathrm{10}}^{\mathrm{15}}\) molecules collide with a square millimeter of wall every microsecond. (See problem 12.) The standard deviation of the velocity of an \(N_2\) molecule is \(201\ \mathrm{m\ }{\mathrm{s}}^{\mathrm{-1}}\). Using the central limit theorem, the standard deviation of the average of \(3\times {10}^{15}\) molecular speeds is
\[\frac{201\, m\,s^{-1}}{\sqrt{3 \times 10^{15}}} \approx 4 \times 10^{-6} \mathrm{ms}^{-1} \nonumber \]
The distribution of the average of \(3\times {10}^{15}\) molecular speeds is very narrow indeed.
Similarly, when molecular velocities follow the Maxwell-Boltzmann distribution function, we can show that the expected value of the pressure for a single-molecule collision is \(\left\langle P_1\left(v\right)\right\rangle =kT/V\). (See problem 21.) The variance of the distribution of these individual pressure measurements is \(\sigma^2_{P_1\left(v\right)}={2k^2T^2}/{3V^2}\), so that the magnitude of the standard deviation is comparable to that of the average:
\[\sigma_{P_{1}(v)} /\left\langle P_{1}(v)\right\rangle=\sqrt{2 / 3} \nonumber \]
For the distribution of averages of \(3\times {10}^{15}\) pressure contributions, we find
\[\begin{aligned} P_{a v g} &=\left\langle P_{1}(v)\right\rangle \\ &=\sqrt{3 / 2} \sigma_{P_{1}(v)} \end{aligned} \nonumber \]
\[\sigma_{avg}=\dfrac{\sigma_{P_1\left(v\right)}}{\sqrt{3\times 10^{15}}} \nonumber \]
and
\[\frac{\sigma_{avg}}{P_{avg}}\approx 1.5\times {10}^{-8} \nonumber \]