9.3: Popular Diets
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The concept of functional foods represents initiatives aimed at addressing health problems. Certain diet plans take this concept one step further, by striving to prevent or treat specific conditions. For example, it is widely understood that people with diabetes need to follow a particular diet. Although some of these diet plans may be nutritionally sound, use caution because some diets may be fads or be so extreme that they actually cause health problems. Before experimenting with a diet, discuss your plans with your doctor or a registered dietitian. Throughout this section, we will discuss some of the more popular diets. Some fall under the category of fad diets, while others are backed by scientific evidence. Those that fall into the latter category provide a good foundation to build a solid regimen for optimal health.
The DASH Diet
The Dietary Approaches to Stop Hypertension, or DASH diet, focuses on reducing sodium intake to either 2,300 milligrams per day (as recommended by the Dietary Guidelines for Americans ) or 1,500 milligrams per day. The DASH diet is an evidence-based eating plan that can help reduce high blood pressure. This plan may also decrease the risk of heart attack, stroke, diabetes, osteoporosis, and certain cancers. DASH tips to lower sodium include:
- Using spices instead of salt to add flavor
- Reading sodium content on processed or canned food labels, and choosing low-sodium options
- Removing some sodium from canned foods (such as beans) by rinsing the product before consumption
- Avoiding salt when cooking
DASH dieters eat lots of whole grains and high-fiber fruits and vegetables, and moderate amounts of low-fat dairy products, lean meats, and heart-healthy fish. In addition, DASH limits the use of saturated fats to less than 7 percent of total calories, and restricts the consumption of sweets and alcohol. The DASH diet also calls for consuming less added sugar and drinking fewer sugar-sweetened drinks. It replaces red meat with fish and legumes and calls for increased calcium, magnesium, potassium, and fiber. Also, even though some people on the DASH diet may find it lowers their HDL (good) cholesterol along with their LDL (bad) cholesterol, it still has a positive cumulative effect on heart health.
The Gluten-Free Diet
The gluten-free diet helps people whose bodies cannot tolerate gluten , a protein found in wheat, barley, and rye. One of the most important ways to treat this condition is to avoid the problematic foods, which is not easy. Although following a gluten-free diet is challenging, it is prescribed for patients with gluten intolerance and celiac disease, an autoimmune disorder with a genetic link. People who have celiac disease cannot consume gluten products without damaging their intestinal lining. Eating a gluten-free diet means finding replacements for bread, cereal, pasta, and more. It also means emphasizing fresh fruits, vegetables, and other foods without gluten. However, it is important to note that the gluten-free trend has become something of a fad even for those without a gluten intolerance. Celiac disease is a relatively rare condition found in only 1 percent of the population. Therefore, a gluten-free diet should be followed only with a physician’s recommendation.
Low-Carb Diets
Low-carb diets, which include the Atkins Diet and the South Beach Diet, focus on limiting carbohydrates—such as grains, fruit, and starchy vegetables—to promote weight loss. The theory behind the low-carb diet is that insulin prevents the breakdown of fat by allowing sugar in the form of blood glucose to be used for energy. Proponents of this approach believe that because limiting carbs generally lowers insulin levels, it would then cause the body to burn stored fat instead. They believe this method not only brings about weight loss, but also reduces the risk factors for a number of conditions. However, some studies have shown that people who followed certain low-carb diet plans for two years lost an average of nearly 9 pounds, which is similar to the amount of weight lost on higher carbohydrate diets.
The benefits of this kind of diet include an emphasis on whole, unprocessed foods and a deemphasis of refined carbohydrates, such as white flour, white bread, and white sugar. However, there are a number of downsides. Typically, the first two weeks allow for only 20 grams of carbs per day, which can be dangerously low. In addition, dieters using the low-carb approach tend to consume twice as many saturated fats as people on a diet high in healthy carbohydrates. Low-carb diets are also associated with a higher energy intake, and the notion that “calories don’t count,” which is prevalent in this kind of diet, is not supported by scientific evidence.
The Mediterranean Diet
The traditional Mediterranean diet incorporates many elements of the dietary choices of people living in Greece and southern Italy. The Mediterranean diet focuses on small portions of nutritionally-sound food. This diet features food from plant sources, including vegetables, fruits, whole grains, beans, nuts, seeds, breads and potatoes, and olive oil. It also restricts the consumption of processed foods and recommends eating locally grown foods rich in micronutrients and antioxidants. Other aspects of this eating plan include consuming fish and poultry at least twice per week, eating red meat only a few times per month, having up to seven eggs per week, and drinking red wine in moderation and with meals.  Instead, it incorporates low-fat cheese and dairy products, and it substitutes olive oil, canola oil, and other healthy oils for butter and margarine.
More than fifty years of nutritional and epidemiological research has shown that people who follow the Mediterranean diet have some of the lowest rates of chronic disease and the highest rates of longevity among the populations of the world. Studies have shown that the Mediterranean diet also helps to decrease excess body weight, blood pressure, blood fats, and blood sugar and insulin levels significantly.
The Raw Food Diet
The raw food diet is followed by those who avoid cooking as much as possible in order to take advantage of the full nutrient content of foods. The principle behind raw foodism is that plant foods in their natural state are the most wholesome for the body. The raw food diet is not a weight-loss plan, it is a lifestyle choice. People who practice raw foodism eat only uncooked and non-processed foods, emphasizing whole fruits and vegetables. Staples of the raw food diet include whole grains, beans, dried fruits, seeds and nuts, seaweed, sprouts, and unprocessed produce. As a result, food preparation mostly involves peeling, chopping, blending, straining, and dehydrating fruits and vegetables.
The positive aspects of this eating method include consuming foods that are high in fiber and nutrients, and low in calories and saturated fat. However, the raw food diet offers little in the way of protein, dairy, or fats, which can cause deficiencies of the vitamins A, D, E, and K. In addition, not all foods are healthier uncooked, such as spinach and tomatoes. Also, cooking eliminates potentially harmful microorganisms that can cause foodborne illnesses.
Therefore, people who primarily eat raw foods should thoroughly clean all fruit and vegetables before eating them. Poultry and other meats should always be cooked before eating.
Vegetarian and Vegan Diets
Vegetarian and vegan diets have been followed for thousands of years for different reasons, including as part of a spiritual practice, to show respect for living things, for health reasons, or because of environmental concerns. For many people, being a vegetarian is a logical outgrowth of “thinking green.” When a food system is heavily focused on meat production there are deforestation issues, overgrazing of land and pasturage, and animal abuses. By avoiding animal flesh, vegetarians hope to look after their own health and that of the planet at the same time. Broadly speaking, vegetarians eat beans, grains, and fruits and vegetables, and do not eat red meat, poultry, seafood, or any other animal flesh. Some vegetarians, known as lactovegetarians, will eat dairy products. Others, known as lacto-ovo vegetarians, will eat dairy products and eggs.
Vegetarian diets have a number of benefits. Well-balanced eating plans can lower the risk of a number of chronic conditions, including heart disease, diabetes, and obesity. They also help to promote sustainable agriculture. However, if a vegetarian does not vary his or her food choices, the diet may be insufficient in calcium, iron, omega-3 fatty acids, zinc, and vitamin B 12 . Also, if people who follow these diets do not plan out their meals, they may gravitate toward foods high in fats.
Table \(\PageIndex{1}\) The Pros and Cons of Seven Popular Diets
| Diet | Pros | Cons |
| DASH Diet |
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| Gluten-Free Diet |
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| Low-Carb Diet |
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| Mediterranean Diet |
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| Raw Food Diet |
|
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| Vegetarianism and Veganism |
|
|


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4: Some Basic Applications of Statistical Thermodynamics
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4.1: Interpreting the Partition Function
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Only quantum states whose energy is less than kT can make substantial contributions to the magnitude of a partition function. Very approximately, we can say that the partition function is equal to the number of quantum states for which the energy is less than kT . Each such quantum state will contribute approximately one to the sum that comprises the partition function; the contribution of the corresponding energy level will be approximately equal to its degeneracy.
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4.2: Conditions under which Integrals Approximate Partition Functions
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A common approximation is to substitue integrals for sums. This section looks at the constraints that must be satisfied in order to make the integral a good approximation to the sum.
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4.3: Probability Density Functions from the Energies of Classical-mechanical Models
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We could postulate probability density functions apply to other energies derived from classical-mechanical models for molecular motion. We will see that this can indeed be done. The results correspond to the results that we get from the Boltzmann equation, where we assume for both derivations that many energy levels satisfy ϵ≪kT. The point is that, at a sufficiently high temperature, the behavior predicted by the quantum mechanical model and that predicted from classical mechanics converge.
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4.4: Partition Functions and Average Energies at High Temperatures
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It is important to remember that the use of integrals to approximate Boltzmann-equation sums assumes that there are a large number of energy levels, ϵi , for which ϵi≪kT . If we select a high enough temperature, the energy levels for any motion will always satisfy this condition. The energy levels for translational motion satisfy this condition even at sub-ambient temperatures. This is the reason that Maxwell’s derivation of the probability density function for translational motion is successf
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4.5: Energy Levels for a Three-dimensional Harmonic Oscillator
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One of the earliest applications of quantum mechanics was Einstein’s demonstration that the union of statistical mechanics and quantum mechanics explains the temperature variation of the heat capacities of solid materials. The physical model underlying Einstein’s development is that a monatomic solid consists of atoms vibrating about fixed points in a lattice. The particles of this solid are distinguishable from one another, because the location of each lattice point is uniquely specified.
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4.6: Energy and Heat Capacity of the "Einstein Crystal"
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4.7: Applications of Other Entropy Relationships
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4.8: Problems
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7.3: The Relational Data Model and others
The Relational Data Model
The relational data model was introduced by C. F. Codd in 1970. Currently, it is the most widely used data model.
The relational model has provided the basis for:
- Research on the theory of data/relationship/constraint
- Numerous database design methodologies
- The standard database access language called s tructured query language (SQL)
- Almost all modern commercial database management systems
The relational data model describes the world as “a collection of inter-related relations (or tables).”
Fundamental Concepts in the Relational Data Model
Relation
A relation, also known as a table or file , is a subset of the Cartesian product of a list of domains characterized by a name. And within a table, each row represents a group of related data values. A row , or record, is also known as a tuple . The columns in a table is a field and is also referred to as an attribute. You can also think of it this way: an attribute is used to define the record and a record contains a set of attributes.
Table
A database is composed of multiple tables and each table holds the data. Figure 7.1 shows a database that contains three tables.
Column
A database stores pieces of information or facts in an organized way. Understanding how to use and get the most out of databases requires us to understand that method of organization.
The principal storage units are called columns or fields or attributes . These house the basic components of data into which your content can be broken down. When deciding which fields to create, you need to think generically about your information, for example, drawing out the common components of the information that you will store in the database and avoiding the specifics that distinguish one item from another.
Look at the example of an ID card in Figure 7.2 to see the relationship between fields and their data.
Domain
A domain is the original sets of atomic values used to model data. By atomic value , we mean that each value in the domain is indivisible as far as the relational model is concerned. For example:
- The domain of Marital Status has a set of possibilities: Married, Single, Divorced.
- The domain of Shift has the set of all possible days: {Mon, Tue, Wed…}.
- The domain of Salary is the set of all floating-point numbers greater than 0 and less than 200,000.
- The domain of First Name is the set of character strings that represents names of people.
In summary, a domain is a set of acceptable values that a column is allowed to contain. This is based on various properties and the data type for the column. We will discuss data types in another chapter.
Records
Just as the content of any one document or item needs to be broken down into its constituent bits of data for storage in the fields, the link between them also needs to be available so that they can be reconstituted into their whole form. Records allow us to do this. Records contain fields that are related, such as a customer or an employee. As noted earlier, a tuple is another term used for record.
Records and fields form the basis of all databases. A simple table gives us the clearest picture of how records and fields work together in a database storage project.
The simple table example in Figure 7.3 shows us how fields can hold a range of different sorts of data. This one has:
- A Record ID field: this is an ordinal number; its data type is an integer.
- A PubDate field: this is displayed as day/month/year; its data type is date.
- An Author field: this is displayed as Initial. Surname; its data type is text.
- A Title field text: free text can be entered here.
You can command the database to sift through its data and organize it in a particular way. For example, you can request that a selection of records be limited by date: 1. all before a given date, 2. all after a given date or 3. all between two given dates. Similarly, you can choose to have records sorted by date. Because the field, or record, containing the data is set up as a Date field, the database reads the information in the Date field not just as numbers separated by slashes, but rather, as dates that must be ordered according to a calendar system.
Degree
The degree is the number of attributes in a table. In our example in Figure 7.3, the degree is 4.
Properties of a Table
- A table has a name that is distinct from all other tables in the database.
- There are no duplicate rows; each row is distinct.
- Entries in columns are atomic. The table does not contain repeating groups or multivalued attributes.
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Entries from columns are from the same domain based on their data type
including
:
- number (numeric, integer, float, smallint,…)
- character (string)
- date
- logical (true or false)
- Operations combining different data types are disallowed.
- Each attribute has a distinct name.
- The sequence of columns is insignificant.
- The sequence of rows is insignificant.
Data Warehouse
As organizations have begun to utilize databases as the centerpiece of their operations, the need to fully understand and leverage the data they are collecting has become more and more apparent. However, directly analyzing the data that is needed for day-to-day operations is not a good idea; we do not want to tax the operations of the company more than we need to. Further, organizations also want to analyze data in a historical sense: How does the data we have today compare with the same set of data this time last month, or last year? From these needs arose the concept of the data warehouse.
The concept of the data warehouse is simple: extract data from one or more of the organization’s databases and load it into the data warehouse (which is itself another database) for storage and analysis. However, the execution of this concept is not that simple. A data warehouse should be designed so that it meets the following criteria:
- It uses non-operational data. This means that the data warehouse is using a copy of data from the active databases that the company uses in its day-to-day operations, so the data warehouse must pull data from the existing databases on a regular, scheduled basis.
- The data is time-variant. This means that whenever data is loaded into the data warehouse, it receives a time stamp, which allows for comparisons between different time periods.
- The data is standardized . Because the data in a data warehouse usually comes from several different sources, it is possible that the data does not use the same definitions or units. For example, our Events table in our Student Clubs database lists the event dates using the mm/dd/yyyy format (e.g., 01/10/2013). A table in another database might use the format yy/mm/dd (e.g., 13/01/10) for dates. In order for the data warehouse to match up dates, a standard date format would have to be agreed upon and all data loaded into the data warehouse would have to be converted to use this standard format. This process is called extraction-transformation-load (ETL).
There are two primary schools of thought when designing a data warehouse: bottom-up and top-down. The bottom-up approach starts by creating small data warehouses, called data marts, to solve specific business problems. As these data marts are created, they can be combined into a larger data warehouse. The top-down approach suggests that we should start by creating an enterprise-wide data warehouse and then, as specific business needs are identified, create smaller data marts from the data warehouse.
Key Terms
column: see attribute
degree : number of attributes in a table
domain : the original sets of atomic values used to model data; a set of acceptable values that a column is allowed to contain
field : see attribute
file : see relation
record : contains fields that are related; see tuple
relation : a subset of the Cartesian product of a list of domains characterized by a name; the technical term for table or file
row : see tuple
structured query language (SQL) : the standard database access language
table: see relation
tuple : a technical term for row or record
Terminology Key
Several of the terms used in this chapter are synonymous. In addition to the Key Terms above, please refer to Table 8.1 below. The terms in the Alternative 1 column are most commonly used.


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6.1: Reactions Carbon-Carbon Double Bonds
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Enantioselective reduction of C=C double bonds have important applications in the synthesis of many natural products and pharmaceutically important compounds.
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6.2: Reactions of Ketones
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Enantioselective reduction of C=O double bond in organic synthesis has important application in synthesis of many natural products as well as pharmaceutical products.
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6.3: Reactions of Imines (C=N)
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An important field of investigation for new industrial catalysts is the development of improved catalysts for the reduction of imines to obtain the corresponding chiral amines. These chiral amines are used as key components in many active pharmaceutical intermediates.
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6.4: Problems and Reference
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12.3.1: Rate Law for Dissociative Mechanisms
Derrivation of the Rate Law
A Dissociative (D) mechanism is two steps in which the first step is the dissociation of a ligand. The intermediate, \(\color{blue}{\ce{ML_{n}}}\), has a lower coordination number than that of the reactant, \(\ce{ML_{n}X}\).
\[\begin{array}{rc}
\text{Step 1 (Dissociation):} & \ce{ML_{n}X <=>[k_1][k_{-1}]}\textcolor{blue}{\ce{ML_{n}}}\ce{ + X}\\
\text{Step 2 (Association):} & \textcolor{blue}{\ce{ML_{n}}}\ce{ + Y <=>[k_2][k_{-2}] ML_{n}Y} \\
\hline \text{Overall Reaction:} & \ce{ML_{n}X + Y <=> ML_{n}Y + X}
\end{array} \nonumber \]
If we assume that \(k_{-2}<<k_2\), the rate law for the formation of products is the rate law of the second step.
\[\frac{d\left[\mathrm{ML}_{n}\right]}{d t}=k_{2} \textcolor{blue}{\ce{[ML_{n}]}} \ce{[Y]} \nonumber \]
We can assume that the concentration of the intermediate is small enough that it would be difficult or impossible to measure its concentration. Here, we can evoke the steady-state approximation and assume that the concentration, \(\color{blue}{\ce{[ML_{n}]}}\), is approximately unchanging over the course of reaction. The \(\color{blue}{\ce{[ML_{n}]}}\) depends on \(k_1, k_{-1}\) and \(k_2\). These assumption are expressed mathematically as follows.
\[\frac{d\left[\mathrm{ML}_{n}\right]}{d t}=k_{1}\left[\mathrm{ML}_{n} \mathrm{X}\right]-k_{-1}\left[\mathrm{ML}_{n}\right][\mathrm{X}]-k_{2}\left[\mathrm{ML}_{n}\right][\mathrm{Y}]=0 \nonumber \]
We can then use the equation above to solve for \(\color{blue}{\ce{[ML_{n}]}}\).
\[\textcolor{blue}{\ce{ML_{n}}}=\frac{k_{1}\left[\mathrm{ML}_{n} \mathrm{X}\right]}{k_{-1}[\mathrm{X}]+k_{2}[\mathrm{Y}]} \nonumber \]
Then substitution this into the rate law given above for the second step of the mechanism. This would yield a rate law that depends only on species with measurable concentrations.
\[\frac{d\left[\mathrm{ML}_{n} \mathrm{Y}\right]}{d t}=\frac{k_{2} k_{1}\left[\mathrm{ML}_{n} \mathrm{X}\right][\mathrm{Y}]}{k_{-1}[\mathrm{X}]+k_{2}[\mathrm{Y}]} \nonumber \]
The rate law above suggests that the rate of product formation has an inverse relationship to the concentration of the outgoing ligand, X. This is one piece of evidence that can be used to distinguish a Dissociative mechanism; that is if the rate of reaction decreases as [X] in solution is increased it is evidence of a D mechanism. This rate law also suggests that ther is a complicated dependence of the reaction rate on the concentration of incoming ligand, Y. The importance of [Y] compared to [X] depends mathmatically on the weight of the two values in the demoninator of the rate law. If [X] >> [Y], then a valid approximation is to assume that the \(k_2\ce{[Y]}\) term is relatively small and simplify the rate law by dropping it out completely.
\[\text{If X >> Y, then simplify: } \frac{d\left[\mathrm{ML}_{n} \mathrm{Y}\right]}{d t}=\frac{k_{2} k_{1}\left[\mathrm{ML}_{n} \mathrm{X}\right][\mathrm{Y}]}{k_{-1}[\mathrm{X}]} \nonumber \]
The resulting rate law (above) has a simpler dependence on [Y]; that is when [X] is large, the reaction rate is directly related to [Y].
The opposite extreme is under conditions of high [Y]. This yields a rate law that is first order in its dependence on the reactant metal complex. When [Y] is large, and the \(k_{-1}\ce{[X]}\) term is relatively insignificant, the rate law can be simplified to the pseudo-first order rate law as follows.
\[\text{If Y >> X, then simplify: } \frac{d\left[\mathrm{ML}_{n} \mathrm{Y}\right]}{d t}= k_{1}\ce{[ML_{n}X]} \nonumber \]
The best type of experimental evidence to determine a dissociative mechanism would be a systematic study of the dependence of rate on both [X] and [Y]. However, the dissociative reaction is unique in that it displays this type of saturation kinetics at high concnetration of the incoming ligand, and so the study of how a reaction rate behaves under high [Y] is common.
First-order reactions
Under conditions of high [Y], the a reaction following a D mechanism is said to follow "first order" kinetics. This type of reaction is sometimes called a first order reaction. That means the rate law depends on only one concentration term.
Why does the dissociative mechanism depend on concentrations in this specific way?
The rate depends on one molecule losing a ligand. Once it does so, a second ligand can replace the one that left. However, losing a ligand may be harder to do than gaining a new one. To lose a ligand, a bond must be broken, which costs energy. To gain a new ligand, a bond is made, releasing energy. That first step is harder to do, so it takes longer. It is a bottleneck that slows the reaction down. It is called the rate-determining step.
- The rate-determining step is the slow step of the reaction.
- The rate-determining step controls the rate of the overall reaction; everything else has to wait for that step to happen.
- Once the rate-determining step has occurred, everything else follows very quickly.
No collision is necessary for the metal complex to lose a ligand. Instead, a bond in the metal complex has to break. That just takes time and energy. As a result, concentration of the incoming ligand matters very little.
We should think a little more about energy requirements, available energy and reaction rate. It takes a certain amount of energy to break a bond. Over any given period of time, a specific amount of energy is available in the surroundings to use. That energy is not available uniformly. Some molecules will get more energy from their surroundings and others will get less. There will be a statistical distribution, like a bell curve, of energy available in different molecules. That means bond-breaking events are governed by statistics.
Figure \(\PageIndex{2}\): The relationship between temperature, energy available, and energy barrier. The black line represents energy needed to start the reaction, also called the energy barrier or the activation barrier. The blue curve is the distribution of available energy in a group of molecules at a cooler temperature. The yellow curve is for a group of molecules that is a little warmer, and the red curve even warmer.
In figure \(\PageIndex{2}\), most of the molecules at the low temperature (blue) do not have enough energy to begin the reaction. A small portion do, and so the reaction will proceed, but very slowly. In the yellow curve, there is more energy available, and so a large fraction of molecules have the energy necessary to begin the reaction. In the red curve, the vast majority have sufficient energy to react. Thus, one of the factors governing how quickly a reaction will happen is the energy needed, or activation barrier. A second factor is the energy available, as indicated by the temperature.
Of course, even if there is enough energy for the reaction, the reaction might not occur yet. Energy is necessary but not sufficient to start a reaction. There are also statistical factors in terms of whether a molecule has its energy allotted into the right places, or in some cases, whether two molecules that need to react together are oriented properly.
Suppose at a given temperature it takes a specific amount of time for half the molecules to gain enough energy so that they can undergo the reaction. That amount of time is called the half life of the reaction. After one half life, half the molecules have reacted and half remain. After a second half life, half the remaining molecules (another quarter, for three quarters of the original material in all) have also reacted, and a quarter still remain. After a third half life, half the remaining ones (another eighth, making it seven eighths reacted in total) will have reacted, leaving an eighth of the original material behind.
- Exponential decay is based on a statistical distribution of energy availability.
- The concept of half life is related to exponential decay.
- It takes a fixed period of time for a half of the metal complex obtain enough energy to dissociate.
Thus, the time it takes for the reaction to happen does not really depend on the concentration of anything.
However, the change in concentration over time -- the quantity that we can usually measure most easily -- depends on the original concentration, and for that reason the concentration of the metal complex appears in the rate law.
Figure \(\PageIndex{3}\): The reactions in the top row and bottom row are proceeding with the same half-life as we move from left to right. However, the top row starts out more concentrated than the bottom row. As a result, the concentrations in the top row are changing more quickly than in the bottom row.
Suppose the half life for a particular case of ligand substitution is one second. After a half life, a 1 M solution becomes 0.5 M, so the rate of change in concentration per time is 0.5M/s. But after the same half life, a 0.5 M solution becomes 0.25 M, so the change in concentration is 0.25 M/s.
If a first order reaction has a half-life of 120 seconds, how much of the original material is left after
a) four minutes? b) six minutes? c) eight minutes? d) ten minutes?
- Answer a
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4 minutes = 240 seconds = 2 x 120 second = 2 half lives.
Material left = 50% x 50% = 0.5 x 0.5 = 0.25 = 25% left
- Answer b
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6 minutes = 360 seconds = 3 x 120 second = 3 half lives.
Material left = 0.5 x 0.5 x 0.5 = 0.125 = 12.5% left
- Answer c
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8 minutes = 480 seconds = 4 x 120 second = 4 half lives.
Material left = 0.5 x 0.5 x 0.5 x 0.5 = 0.0625 = 6.25% left
- Answer d
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10 minutes = 600 seconds = 5 x 120 second = 5 half lives.
Material left = 0.5 x 0.5 x 0.5 x 0.5 x 0.5 = 0.03125 = 3.125% left
Given the first-order dissociative rate law above, what would happen to the reaction rate for substitution in each of the following cases?
- the concentration of ligand is doubled, and the concentration of metal complex is doubled
- the concentration of ligand is tripled, and the concentration of metal complex is halved
- the concentration of ligand is doubled, and the concentration of metal complex is tripled
- the concentration of ligand is halved, and the concentration of metal complex is halved
- Answer a
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Dissociative Rate Law: Rate = [ML n ], if ML n is the complex. There is no dependence on [X], if X is the new ligand.
Rate will double: Rate = 2 x [ML n ] 0 , if [ML n ] 0 is the original concentration.
- Answer b
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Rate will be halved: Rate = 0.5 x [ML n ] 0 .
- Answer c
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Rate will triple: Rate = 3 x [ML n ] 0 .
- Answer d
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Rate will be halved: Rate = 0.5 x [ML n ] 0 .
Plot graphs of initial rate vs concentration to show what you would see in dissociative substitution.
a) The concentration of metal complex, [ML n ], is held constant at 0.1 mol/L and the concentration of ligand is changed from 0.5 mol/L to 1 mol/L and then to 1.5 mol/L.
b) The concentration of new ligand, [X], is held constant at 0.1 mol/L and the concentration of metal complex is changed from 0.5 mol/L to 1 mol/L and then to 1.5 mol/L
Given the following sets of initial rate data, determine whether each case represents a dissociative substitution. [MLn] = concentration of the coordination complex; [X] = concentration of incoming ligand.
a)
b)
In the following data, the concentration of the metal complex and the incoming ligand were held constant, but more of the departing ligand was added to the solution.
a) Explain what the data says about rate dependence on this concentration.
b) Explain this rate dependence in terms of the reaction.
- Answer a
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The rate of the reaction is depressed when the concentration of the departing ligand is increased.
- Answer b
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This dependence could indicate an equilibrium in the dissociative step. The more departing ligand is added, the more the equilibrium is pushed back towards the original metal complex. With less dissociated metal complex around, the entering ligand cannot form the new complex as quickly.
In certain solvents, such as THF, acetonitrile and pyridine, the rate law for substitution often appears to be Rate = k 1 [ML n ] + k 2 [ML n ][X], in which X is the incoming ligand and ML n is the metal complex.
a) What do these solvents have in common?
b) What is a possible explanation for this rate law?
c) This rate law has been shown to be consistent with an entirely associative mechanism. How is that possible?
- Answer a
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These solvents all have lone pairs. They could be Lewis bases or nucleophiles.
- Answer b
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It looks like two competing mechanisms. On term suggests a dissociative mechanism, whereas the other term suggests a dissociative mechanism. They could be happening in competition with each other.
- Answer c
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On the other hand, it could be that there is one mechanism with two different nucleophiles. If the incoming ligand is the nucleophile, the term on the right shows up in the rate law. If the solvent is the nucleophile, forming a third complex, the term on the left shows up in the rate law. That's because we would typically change the amount of metal complex and the amount of ligand that we add to the solution in order to determine the rate law, but we wouldn't normally be able to change the concentration of the solvent, so it would be a constant. (How could you confirm this explanation in an experiment?)


2.6: Powers with Arbitrary Real Exponents. Irrationals
In complete fields, one can define \(a^{r}\) for any \(a>0\) and \(r \in E^{1}\) (for \(r \in N,\) see §§5-6, Example \((\mathrm{f}) ) .\) First of all, we have the following theorem.
Given \(a \geq 0\) in a complete field \(F,\) and a natural number \(n \in E^{1}\), there always is a unique element \(p \in F, p \geq 0,\) such that
\[p^{n}=a.\]
It is called the \(n\) th root o f \(a,\) denoted
\[\sqrt[n]{a} \text{ or } a^{1 / n}.\]
( Note that \(\sqrt[n]{a} \geq 0,\) by definition .)
- Proof
-
A direct proof, from the completeness axiom, is sketched in Problems 1 and 2 below. We shall give a simpler proof in Chapter 4,§9, Example (a). At present, we omit it and temporarily take Theorem 1 for granted. Hence we obtain the following result.
Every complete field \(F\) ( such as \(E^{1}\)) has irrational elements, i.e., elements that are not rational .
In particular , \(\sqrt{2}\) is irrational .
- Proof
-
By Theorem 1, \(F\) has the element
\[p=\sqrt{2} \text{ with } p^{2}=2\]
Seeking a contradiction, suppose \(\sqrt{2}\) is rational, i.e.,
\[\sqrt{2}=\frac{m}{n}\]
for some \(m, n \in N\) in lowest terms (see §7, final note).
Then \(m\) and \(n\) are not both even (otherwise, reduction by 2 would yield a smaller \(n ) .\) From \(m / n=\sqrt{2},\) we obtain
\[m^{2}=2 n^{2};\]
so \(m^{2}\) is even .
Only even elements have even squares, however. Thus \(m\) itself must be even; i.e., \(m=2 r\) for some \(r \in N .\) It follows that
\[4 r^{2}=m^{2}=2 n^{2}, \text{ i.e., } 2 r^{2}=n^{2}\]
and, by the same argument, \(n\) must be even .
This contradicts the fact that \(m\) and \(n\) are not both even, and this contradiction shows that \(\sqrt{2}\) must be irrational. \(\square\)
Note 1. Similarly, one can prove the irrationality of \(\sqrt{a}\) where \(a \in N\) and \(a\) is not the square of a natural. See Problem 3 below for a hint.
Note 2. Theorem 2 shows that the field \(R\) of all rationals is not complete (for it contains no irrationals), even though it is Archimedean (see Problem 6 ). Thus the Archimedean property does not impleteness (but see Theorem 1 of §10).
Next, we define \(a^{r}\) for any rational number \(r>0\).
Given \(a \geq 0\) in a complete field \(F,\) and a rational number
\[r=\frac{m}{n} \quad\left(m, n \in N \subseteq E^{1}\right)\]
we define
\[a^{r}=\sqrt[n]{a^{m}}.\]
Here we must clarify two facts.
(1) If \(n=1,\) we have
\[a^{r}=a^{m / 1}=\sqrt[1]{a^{m}}=a^{m}.\]
If \(m=1,\) we get
\[a^{r}=a^{1 / n}=\sqrt[n]{a}.\]
Thus Definition 1 agrees with our previous definitions of \(a^{m}\) and \(\sqrt[n]{a}\) \((m, n \in N) .\)
(2) If \(r\) is written as a fraction in two different ways ,
\[r=\frac{m}{n}=\frac{p}{q},\]
then, as is easily seen,
\[\sqrt[n]{a^{m}}=\sqrt[q]{a^{p}}=a^{r},\]
and so our definition is unambiguous (independent of the particular representation of \(r ) .\)
Indeed,
\[\frac{m}{n}=\frac{p}{q} \text{ implies } m q=n p,\]
whence
\[a^{m q}=a^{p n},\]
i.e.,
\[\left(a^{m}\right)^{q}=\left(a^{p}\right)^{n};\]
cf. §§5-6, Problem 6.
By definition, however,
\[\left(\sqrt[n]{a^{m}}\right)^{n}=a^{m} \text{ and } \left(\sqrt[q]{a^{p}}\right)^{q}=a^{p}.\]
Substituting this in \(\left(a^{m}\right)^{q}=\left(a^{p}\right)^{n},\) we get
\[\left(\sqrt[n]{a^{m}}\right)^{n q}=\left(\sqrt[q]{a^{p}}\right)^{n q},\]
whence
\[\sqrt[n]{a^{m}}=\sqrt[q]{a^{p}}.\]
Thus Definition 1 is valid, indeed.
By using the results of Problems 4 and 6 of §§5-6, the reader will easily obtain analogous formulas for powers with positive rational exponents, namely,
\[\begin{aligned} a^{r} a^{s} &=a^{r+s} ;\left(a^{r}\right)^{s}=a^{r s} ;(a b)^{r}=a^{r} b^{r} ; a^{r}<a^{s} \text { if } 0<a<1 \text { and } r>s \\ a &<b \text { iff } a^{r}<b^{r}(a, b, r>0) ; a^{r}>a^{s} \text { if } a>1 \text { and } r>s ; 1^{r}=1 \end{aligned}\]
Henceforth we assume these formulas known, for rational \(r, s>0\).
Next, we define \(a^{r}\) for any real \(r>0\) and any element \(a>1\) in a complete field \(F .\)
Let \(A_{a r}\) denote the set of all members of \(F\) of the form \(a^{x},\) with \(x \in R\) and \(0<x \leq r ;\) i.e.,
\[A_{ar}=\{a^{x} | 0<x \leq r, x \text{ rational}\}.\]
By the density of rationals in \(E^{1}\) (Theorem 3 of §10), such rationals \(x\) do exist; thus \(A_{\text {ar}} \neq \emptyset\).
Moreover, \(A_{a r}\) is right bounded in \(F .\) Indeed, fix any rational number \(y>r\). By the formulas in \((1),\) we have, for any positive rational \(x \leq r\),
\[a^{y}=a^{x+(y-x)}=a^{x} a^{y-x}>a^{x}\]
since \(a>1\) and \(y-x>0\) implies
\[a^{y-x}>1.\]
Thus \(a^{y}\) is an upper bound of all \(a^{x}\) in \(A_{a r}\).
Hence, by the assumed completeness of \(F,\) sup \(A_{ar}\) exists . So we may define
\[a^{r}=\sup A_{a r}.\]
We also put
\[a^{-r}=\frac{1}{a^{r}}.\]
If \(0<a<1\) (so that \(\frac{1}{a}>1 ),\) we put
\[a^{r}=\left(\frac{1}{a}\right)^{-r} \text{ and } a^{-r}=\frac{1}{a^{r}},\]
where
\[\left(\frac{1}{a}\right)^{r}=\sup A_{1 / a, r},\]
as above.
Summing up, we have the following definitions.
Given \(a>0\) in a complete field \(F\), and \(r \in E^{1}\), we define the following.(i) If \(r>0\) and \(a>1\), then
\[a^{r}=\sup A_{a r}=\sup \left\{a^{x} | 0<x \leq r, x \text { rational }\right\}\]
(ii) If \(r>0\) and \(0<a<1\), then \(a^{r}=\frac{1}{(1 / a)^{r}}\), also written \((1 / a)^{-r}.\)
(iii) \(a^{-r}=1 / a^{r}\). (This defines powers with negative exponents as well.)
We also define \(0^{r}=0\) for any real \(r>0\), and \(a^{0}=1\) for any \(a \in F\), \(a \neq 0\); \(0^{0}\) remains undefined.
The power \(a^{r}\) is also defined if \(a<0\) and \(r\) is a rational \(\frac{m}{n}\) with \(n\) because \(a^{r}=\sqrt[n]{a^{m}}\) has sense in this case. (Why?) This does not work for other values of \(r\). Therefore, in general, we assume \(a>0\).
Again, it is easy to show that the formulas in (1) remain also valid for powers with real exponents (see Problems } 8-13 below), provided \(F\) is complete.


1.6: Chapter 1 References
References
- (ASCE 2016) American Society of Civil Engineers, 2013 Report Card on America’s Infrastructure, http://www.infrastructurereportcard.org/ (accessed 1/5/2016).
- (DHS 2016) Department of Homeland Security, ‘National Infrastructure Protection Plan,’ http://www.dhs.gov/national-infrastr...rotection-plan (accessed 1/5/2016).
- (Hendrickson 2016) Hendrickson, Chris, ‘Project Management for Construction,’ http://pmbook.ce.cmu.edu/ (accessed 1/6/2016).
- (Hendrickson 2016b) Hendrickson, Chris and H. Scott Matthews,’ Civil Infrastructure Planning, Investment and Pricing’, http://faculty.ce.cmu.edu/textbooks/cspbook/ (accessed 1/6/2016).
- (Matthews 2016) Matthews, H. Scott, Chris Hendrickson and Deanna Matthews, ‘Lifecycle Assessment: Quantitative Approaches to Decisions that Matter,’ http://www.lcatextbook.com/ (accessed 1/6/2016).
- (NRC 2009) National Resource Council, ‘Sustainable Critical Infrastructure Systems: A Framework for Meeting 21st Century Imperatives,’ National Academies Press, 2009, DOI: 10.17226/12638).


16.6: Blood Typing
By the end of this section, you will be able to:
- Describe the two basic physiological consequences of transfusion of incompatible blood
- Compare and contrast ABO and Rh blood groups
- Identify which blood groups may be safely transfused into patients with different ABO types
Blood transfusions in humans were risky procedures until the discovery of the major human blood groups by Karl Landsteiner, an Austrian biologist and physician, in 1900. Until that point, physicians did not understand that death sometimes followed blood transfusions, when the type of donor blood infused into the patient was incompatible with the patient’s own blood. Blood groups are determined by the presence or absence of specific marker molecules on the plasma membranes of erythrocytes. With their discovery, it became possible for the first time to match patient-donor blood types and prevent transfusion reactions and deaths.
Antigens, Antibodies, and Transfusion Reactions
Antigens are substances that the body does not recognize as belonging to the “self” and that therefore trigger a defensive response from the leukocytes of the immune system (covered more thoroughly in another chapter). Here, we will focus on the role of immunity in blood transfusion reactions.
Recall that all cells have molecules embedded in their plasma membranes that help with cell recognition and identification. Red blood cells have several surface proteins involved in this function; the combination of those present (or absent) determines one's blood type. Because incompatible blood types will trigger a defensive response from the leukocytes, the blood-type related surface proteins are referred to as antigens when discussing blood types.
Following the infusion of incompatible blood, erythrocytes with antigens not present on the person's own erythrocytes appear in the bloodstream and trigger an immune response. Proteins called antibodies (immunoglobulins), which are produced by certain B lymphocytes called plasma cells, attach to the antigens on the plasma membranes of the infused erythrocytes and cause them to adhere to one another. Because the arms of the Y-shaped antibodies attach randomly to more than one non-self erythrocyte surface, they form clumps of erythrocytes. This process is called agglutination .
- The clumps of erythrocytes may block small blood vessels throughout the body, depriving tissues of oxygen and nutrients.
- As the erythrocyte clumps are degraded, in a process called hemolysis , their hemoglobin is released into the bloodstream. This hemoglobin travels to the kidneys, which are responsible for filtration of the blood. However, the load of hemoglobin released can easily overwhelm the kidney’s capacity to clear it, and the patient can quickly develop kidney failure.
More than 50 antigens have been identified on erythrocyte membranes, but the most significant in terms of their potential harm to patients are classified in two groups: the ABO blood group and the Rh blood group.
The ABO Blood Group
Although the ABO blood group name consists of three letters, ABO blood typing designates the presence or absence of just two antigens, designated as A and B; both are glycoproteins.  People can also have both A and B antigens on their erythrocytes, in which case they are blood type AB. People with neither A nor B antigens are designated blood type O. ABO blood types are genetically determined.
Normally the body must be exposed to a foreign antigen before an antibody can be produced. This is not the case for the ABO blood group. Individuals with type A blood—without any prior exposure to incompatible blood—have preformed antibodies to the B antigen circulating in their blood plasma. These antibodies, referred to as anti-B antibodies, will cause agglutination and hemolysis if they ever encounter erythrocytes with B antigens. Similarly, an individual with type B blood has pre-formed anti-A antibodies. Individuals with type AB blood, which has both antigens, do not have preformed antibodies to either of these. People with type O blood lack antigens A and B on their erythrocytes, but both anti-A and anti-B antibodies circulate in their blood plasma.
Rh Blood Groups
The Rh blood group is classified according to the presence or absence of a second erythrocyte antigen identified as Rh. (It was first discovered in a type of primate known as a rhesus macaque, often used in research because its blood is similar to that of humans.) Although dozens of Rh antigens have been identified, only one, designated D, is clinically important. Those who have the Rh D antigen present on their erythrocytes—about 85 percent of Americans—are described as Rh positive (Rh + ) and those who lack it are Rh negative (Rh − ). Note that the Rh group is distinct from the ABO group, so any individual, no matter their ABO blood type, may have or lack this Rh antigen. When identifying a patient’s blood type, the Rh group is designated by adding the word positive or negative to the ABO type. For example, A positive (A + ) means ABO group A blood with the Rh antigen present, and AB negative (AB − ) means ABO group AB blood without the Rh antigen.
In contrast to the ABO group antibodies, which are preformed, antibodies to the Rh antigen are produced only in Rh − individuals after exposure to the antigen. This process, called sensitization, occurs following a transfusion with Rh-incompatible blood or, more commonly, with the birth of an Rh + baby to an Rh − mother. Problems are rare in a first pregnancy, since the baby’s Rh + cells rarely cross the placenta (the organ of gas and nutrient exchange between the baby and the mother). However, during or immediately after birth, the Rh − mother can be exposed to the baby’s Rh + red blood cells (Figure \(\PageIndex{1}\)). Research has shown that this occurs in about 13−14 percent of such pregnancies. After exposure, the mother’s immune system begins to generate anti-Rh antibodies. If the mother should then conceive another Rh + baby, the Rh antibodies she has produced can cross the placenta into the fetal bloodstream and destroy the fetal RBCs. This condition, known as hemolytic disease of the newborn (HDN) or erythroblastosis fetalis, may cause anemia in mild cases, but the agglutination and hemolysis can be so severe that without treatment the fetus may die in the womb or shortly after birth.
A drug known as RhoGAM, short for Rh immune globulin, can temporarily prevent the development of Rh antibodies in the Rh − mother, thereby averting this potentially serious disease for the fetus. RhoGAM antibodies destroy any fetal Rh + erythrocytes that may cross the placental barrier. RhoGAM is normally administered to Rh − mothers during weeks 26−28 of pregnancy and within 72 hours following birth. It has proven remarkably effective in decreasing the incidence of HDN. Earlier we noted that the incidence of HDN in an Rh + subsequent pregnancy to an Rh − mother is about 13–14 percent without preventive treatment. Since the introduction of RhoGAM in 1968, the incidence has dropped to about 0.1 percent in the United States.
Determining ABO Blood Types
Clinicians are able to determine a patient’s blood type quickly and easily using commercially prepared antibodies. An unknown blood sample is allocated into separate wells. Into one well a small amount of anti-A antibody is added, and to another a small amount of anti-B antibody. If the antigen is present, the antibodies will cause visible agglutination of the cells (Figure \(\PageIndex{2}\)). The blood should also be tested for Rh antibodies.
ABO Transfusion Protocols
To avoid transfusion reactions, it is best to transfuse only matching blood types; that is, a type B + recipient should ideally receive blood only from a type B + donor and so on. That said, in emergency situations, when acute hemorrhage threatens the patient’s life, there may not be time for cross matching to identify blood type. In these cases, blood from a universal donor —an individual with type O − blood—may be transfused. Recall that type O erythrocytes do not display A or B antigens. Thus, anti-A or anti-B antibodies that might be circulating in the patient’s blood plasma will not encounter any erythrocyte surface antigens on the donated blood and therefore will not be provoked into a response. One problem with this designation of universal donor is if the O − individual had prior exposure to Rh antigen, Rh antibodies may be present in the donated blood. Also, introducing type O blood into an individual with type A, B, or AB blood will nevertheless introduce antibodies against both A and B antigens, as these are always circulating in the type O blood plasma. This may cause problems for the recipient, but because the volume of blood transfused is much lower than the volume of the patient’s own blood, the adverse effects of the relatively few infused plasma antibodies are typically limited. Rh factor also plays a role. If Rh − individuals receiving blood have had prior exposure to Rh antigen, antibodies for this antigen may be present in the blood and trigger agglutination to some degree. Although it is always preferable to cross match a patient’s blood before transfusing, in a true life-threatening emergency situation, this is not always possible, and these procedures may be implemented.
 Donated blood is also screened for common and serious blood-bourne illnesses like HIV and hepatitis (among others) to prevent the transmission of disease when a patient receives a blood transfusion from an anonymous donor.
A patient with blood type AB + is known as the universal recipient . This patient can theoretically receive any type of blood, because the patient’s own blood—having both A and B antigens on the erythrocyte surface—does not produce anti-A or anti-B antibodies. In addition, an Rh + patient can receive both Rh + and Rh − blood. However, keep in mind that the donor’s blood will contain circulating antibodies, again with possible negative implications. Figure \(\PageIndex{3}\) summarizes the characteristics related to each ABO blood type including compatibilities.
At the scene of multiple-vehicle accidents, military engagements, and natural or human-caused disasters, many victims may suffer simultaneously from acute hemorrhage, yet type O blood may not be immediately available. In these circumstances, medics may at least try to replace some of the volume of blood that has been lost. This is done by intravenous administration of a saline solution that provides fluids and electrolytes in proportions equivalent to those of normal blood plasma. Research is ongoing to develop a safe and effective artificial blood that would carry out the oxygen-carrying function of blood without the RBCs, enabling transfusions in the field without concern for incompatibility. These blood substitutes normally contain hemoglobin as well as other types of oxygen carriers.
Concept Review
Antigens are non-self molecules, usually large proteins, which provoke an immune response. In transfusion reactions, antibodies attach to antigens on the surfaces of erythrocytes and cause agglutination and hemolysis. ABO blood group antigens are designated A and B. People with type A blood have A antigens on their erythrocytes, whereas those with type B blood have B antigens. Those with AB blood have both A and B antigens, and those with type O blood have neither A nor B antigens. The blood plasma contains preformed antibodies against the antigens not present on a person’s erythrocytes.
A second group of blood antigens is the Rh group, the most important of which is Rh D. People with Rh − blood do not have this antigen on their erythrocytes, whereas those who are Rh + do. About 85 percent of Americans are Rh + . When a woman who is Rh − becomes pregnant with an Rh + fetus, her body may begin to produce anti-Rh antibodies. If she subsequently becomes pregnant with a second Rh + fetus and is not treated preventively with RhoGAM, the fetus will be at risk for an antigen-antibody reaction, including agglutination and hemolysis. This is known as hemolytic disease of the newborn.
Cross matching to determine blood type is necessary before transfusing blood, unless the patient is experiencing hemorrhage that is an immediate threat to life, in which case type O − blood may be transfused.
Review Questions
Q. The process in which antibodies attach to antigens, causing the formation of masses of linked cells, is called ________.
A. sensitization
B. coagulation
C. agglutination
D. hemolysis
- Answer
-
Answer: C
Q. People with ABO blood type O ________.
A. have both antigens A and B on their erythrocytes
B. lack both antigens A and B on their erythrocytes
C. have neither anti-A nor anti-B antibodies circulating in their blood plasma
D. are considered universal recipients
- Answer
-
Answer: B
Q. Hemolytic disease of the newborn is a risk during a subsequent pregnancy in which ________.
 Following a motor vehicle accident, a patient is rushed to the emergency department with multiple traumatic injuries, causing severe bleeding. The patient’s condition is critical, and there is no time for determining his blood type. What type of blood is transfused, and why?
- Answer
-
A. In emergency situations, blood type O − will be infused until cross matching can be done. Blood type O − is called the universal donor blood because the erythrocytes have neither A nor B antigens on their surface, and the Rh factor is negative.
Q. In preparation for a scheduled surgery, a patient visits the hospital lab for a blood draw. The technician collects a blood sample and performs a test to determine its type. She places a sample of the patient’s blood in two wells. To the first well she adds anti-A antibody. To the second she adds anti-B antibody. Both samples visibly agglutinate. Has the technician made an error, or is this a normal response? If normal, what blood type does this indicate?
- Answer
-
A. The lab technician has not made an error. Blood type AB has both A and B surface antigens, and neither anti-A nor anti-B antibodies circulating in the plasma. When anti-A antibodies (added to the first well) contact A antigens on AB erythrocytes, they will cause agglutination. Similarly, when anti-B antibodies contact B antigens on AB erythrocytes, they will cause agglutination.
Glossary
- ABO blood group
- blood-type classification based on the presence or absence of A and B glycoproteins on the erythrocyte membrane surface
- agglutination
- clustering of cells into masses linked by antibodies
- cross matching
- blood test for identification of blood type using antibodies and small samples of blood
- hemolysis
- destruction (lysis) of erythrocytes and the release of their hemoglobin into circulation
- hemolytic disease of the newborn (HDN)
- (also, erythroblastosis fetalis) disorder causing agglutination and hemolysis in an Rh + fetus or newborn of an Rh − mother
- Rh blood group
- blood-type classification based on the presence or absence of the antigen Rh on the erythrocyte membrane surface
- universal donor
- individual with type O − blood
- universal recipient
- individual with type AB + blood
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


12: Mendel's Experiments and Heredity
Gregor Johann Mendel was a German-speaking Moravian scientist and Augustinian friar who gained posthumous fame as the founder of the modern science of genetics. Though farmers had known for centuries that crossbreeding of animals and plants could favor certain desirable traits, Mendel's experiments established many of the rules of heredity, now referred to as the laws of Mendelian inheritance.
-
- 12.0: Prelude to Mendel's Experiments and Heredity
- Genetics is the study of heredity. Johann Gregor Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. Mendel selected a simple biological system and conducted methodical, quantitative analyses using large sample sizes. Because of Mendel’s work, the fundamental principles of heredity were revealed.
-
- 12.1: Mendel’s Experiments and the Laws of Probability
- In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted faithfully from parents to offspring independently of other traits and in dominant and recessive patterns.
-
- 12.2: Characteristics and Traits
- The genetic makeup of peas consists of two similar or homologous copies of each chromosome, one from each parent. Each pair of homologous chromosomes has the same linear order of genes; hence peas are diploid organisms. The same is true for many other plants and for virtually all animals. Diploid organisms utilize meiosis to produce haploid gametes, which contain one copy of each homologous chromosome that unite at fertilization to create a diploid zygote.
-
- 12.3: Laws of Inheritance
- Mendel generalized the results of his pea-plant experiments into four postulates, some of which are sometimes called “laws,” that describe the basis of dominant and recessive inheritance in diploid organisms. As you have learned, more complex extensions of Mendelism exist that do not exhibit the same F2 phenotypic ratios (3:1). Nevertheless, these laws summarize the basics of classical genetics.
Thumbnail: Pea plants were used by Gregor Mendel to discover some fundamental laws of genetics. (Flicker-Christian Guthier-CC:A).


30.13: Transport of Water and Solutes in Plants - Water and Solute Potential
- Describe the water and solute potential in plants
Water Potential
Plants are phenomenal hydraulic engineers. Using only the basic laws of physics and the simple manipulation of potential energy, plants can move water to the top of a 116-meter-tall tree. Plants can also use hydraulics to generate enough force to split rocks and buckle sidewalks. Water potential is critical for moving water to leaves so that photosynthesis can take place.
Water potential is a measure of the potential energy in water, or the difference in potential energy between a given water sample and pure water (at atmospheric pressure and ambient temperature). Water potential is denoted by the Greek letter ψ (psi) and is expressed in units of pressure (pressure is a form of energy) called megapascals (MPa). The potential of pure water (Ψ w pure H2O ) is designated a value of zero (even though pure water contains plenty of potential energy, that energy is ignored). Water potential values for the water in a plant root, stem, or leaf are, therefore, expressed in relation to Ψ w pure H2O .
The water potential in plant solutions is influenced by solute concentration, pressure, gravity, and factors called matrix effects. Water potential can be broken down into its individual components using the following equation:
Ψ system = Ψ total = Ψ s + Ψ p + Ψ g + Ψ m
where
- Ψ s = solute potential
- Ψ p , = pressure potential
- Ψ g , = gravity potential
- Ψ m = matric potential
“System” can refer to the water potential of the soil water (Ψ soil ), root water (Ψ root ), stem water (Ψ stem ), leaf water (Ψ leaf ), or the water in the atmosphere (Ψ atmosphere ), whichever aqueous system is under consideration. As the individual components change, they raise or lower the total water potential of a system. When this happens, water moves to equilibrate, moving from the system or compartment with a higher water potential to the system or compartment with a lower water potential. This brings the difference in water potential between the two systems (Δ) back to zero (Δ = 0). Therefore, for water to move through the plant from the soil to the air (a process called transpiration), the conditions must exist as such:
Ψ soil > Ψ root > Ψ stem > Ψ leaf > Ψ atmosphere .
Water only moves in response to Δ, not in response to the individual components. However, because the individual components influence the total Ψ system , a plant can control water movement by manipulating the individual components (especially Ψ s ).
Solute Potential
Solute potential (Ψ s ), also called osmotic potential, is negative in a plant cell and zero in distilled water. Typical values for cell cytoplasm are –0.5 to –1.0 MPa. Solutes reduce water potential (resulting in a negative Ψ w ) by consuming some of the potential energy available in the water. Solute molecules can dissolve in water because water molecules can bind to them via hydrogen bonds; a hydrophobic molecule like oil, which cannot bind to water, cannot go into solution. The energy in the hydrogen bonds between solute molecules and water is no longer available to do work in the system because it is tied up in the bond. In other words, the amount of available potential energy is reduced when solutes are added to an aqueous system. Thus, Ψ s decreases with increasing solute concentration. Because Ψ s is one of the four components of Ψ system or Ψ total , a decrease in Ψ s will cause a decrease in Ψ total . The internal water potential of a plant cell is more negative than pure water because of the cytoplasm’s high solute content. Because of this difference in water potential, water will move from the soil into a plant’s root cells via the process of osmosis. This is why solute potential is sometimes called osmotic potential.
Plant cells can metabolically manipulate Ψ s (and by extension, Ψ total ) by adding or removing solute molecules. Therefore, plants have control over Ψ total via their ability to exert metabolic control over Ψ s .
Key Points
- Plants use water potential to transport water to the leaves so that photosynthesis can take place.
- Water potential is a measure of the potential energy in water as well as the difference between the potential in a given water sample and pure water.
- Water potential is represented by the equation Ψ system = Ψ total = Ψ s + Ψ p + Ψ g + Ψ m .
- Water always moves from the system with a higher water potential to the system with a lower water potential.
- Solute potential (Ψ s ) decreases with increasing solute concentration; a decrease in Ψs causes a decrease in the total water potential.
- The internal water potential of a plant cell is more negative than pure water; this causes water to move from the soil into plant roots via osmosis..
Key Terms
- solute potential : (osmotic potential) pressure which needs to be applied to a solution to prevent the inward flow of water across a semipermeable membrane
- transpiration : the loss of water by evaporation in terrestrial plants, especially through the stomata; accompanied by a corresponding uptake from the roots
- water potential : the potential energy of water per unit volume; designated by ψ


25.7: Extending Debye-Hückel Theory to Higher Concentrations
The equation for \(\log \gamma _{\pm }\) predicted from Debye–Hückel limiting law is:
\[\log _{10}\gamma _{\pm }=-Az_{j}^{2}{\frac {\sqrt {I}}{1+Ba_{0}{\sqrt {I}}}} \label{DH} \]
It gives satisfactory agreement with experimental measurements for low electrolyte concentrations, typically less than \(10^{−3} mol/L\). Deviations from the theory occur at higher concentrations and with electrolytes that produce ions of higher charges, particularly asymmetrical electrolytes. These deviations occur because the model is oversimplified, so there is little to be gained by making small adjustments to the model. Instead, we must challenge the individual assumptions of the model:
- Ions do not interact with each other . Ion association may take place, particularly with ions of higher charge. This was followed up in detail by Niels Bjerrum. The Bjerrum length is the separation at which the electrostatic interaction between two ions is comparable in magnitude to \(kT\).
- Complete dissociation . A weak electrolyte is one that is not fully dissociated. As such it has a dissociation constant. The dissociation constant can be used to calculate the extent of dissociation and hence, make the necessary correction needed to calculate activity coefficients.
- Ions are spherical point charges that cannot be polarizable . Ions, as all other atoms and molecules, have a finite size. Many ions, such as the nitrate ion \(\ce{NO3^{−}}\), are not spherical. Polyatomic ions are polarizable.
- The solvent composition does not matter . The solvent is not a structureless medium but is made up of molecules. The water molecules in aqueous solution are both dipolar and polarizable. Both cations and anions have a strong primary solvation shell and a weaker secondary solvation shell. Ion–solvent interactions are ignored in Debye–Hückel theory.
- Ionic radius is negligible . At higher concentrations, the ionic radius becomes comparable to the radius of the ionic atmosphere.
Most extensions to the Debye–Hückel theory are empirical in nature. They usually allow the Debye–Hückel equation to be followed at low concentration and add further terms in some power of the ionic strength to fit experimental observations. Several approaches have been proposed to extend the validity of the Debye–Hückel theory.
Extended Debye-Hückel Equation
One such approach is the Extended Debye-Hückel Equation:
\[- \log(\gamma) = \dfrac{A|z_+z_-|\sqrt{I}}{1 + Ba\sqrt{I}} \nonumber \]
where \(\gamma\) is the activity coefficient, \(z\) is the integer charge of the ion \(\mu\) is the ionic strength of the aqueous solution, and \(a\), is the size or effective diameter of the ion in angstrom. The effective hydrated radius of the ion, \(a\) is the radius of the ion and its closely bound water molecules. Large ions and less highly charged ions bind water less tightly and have smaller hydrated radii than smaller, more highly charged ions. Typical values are 3 Å for ions such as H+,Cl-,CN-, and HCOO-. The effective diameter for the hydronium ion is 9 Å. \ (A\) and \(B\) are constants with values of respectively 0.5085 and 0.3281 at 25°C in water.
Other approaches include the Davies equation, Pitzer equations and specific ion interaction theory.
Davis Equation
The Davies equation is an empirical extension of Debye–Hückel theory which can be used to calculate activity coefficients of electrolyte solutions at relatively high concentrations at 25 °C. The equation, originally published in 1938, was refined by fitting to experimental data. The final form of the equation gives the mean molal activity coefficient f± of an electrolyte that dissociates into ions having charges z1 and z2 as a function of ionic strength I:
\[ -\log f_{\pm }=0.5z_{1}z_{2}\left({\frac {\sqrt {I}}{1+{\sqrt {I}}}}-0.30I\right). \nonumber \]
The second term, 0.30 \(I\), goes to zero as the ionic strength goes to zero, so the equation reduces to the Debye–Hückel equation at low concentration. However, as concentration increases, the second term becomes increasingly important, so the Davies equation can be used for solutions too concentrated to allow the use of the Debye–Hückel equation. For 1:1 electrolytes the difference between measured values and those calculated with this equation is about 2% of the value for 0.1 M solutions. The calculations become less precise for electrolytes that dissociate into ions with higher charges. Further discrepancies will arise if there is association between the ions, with the formation of ion pairs, such as \(\ce{Mg^{2+}SO^{2-}4}\).
Plot of activity coefficients calculated using the Davies equation.
Pitzer Equations
Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterize interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate.
An expression is obtained for the mean activity coefficient.
\[\ln \gamma _{\pm }= \dfrac {p\ln \gamma _{M}+q\ln \gamma _{X}}{p+q} \nonumber \]
\[\ln \gamma _{\pm }=|z^{+}z^{-}|f^{\gamma }+m\left({\frac {2pq}{p+q}}\right)B_{MX}^{\gamma }+m^{2}\left[2{\frac {(pq)^{3/2}}{p+q}}\right]C_{MX}^{\gamma } \nonumber \]
These equations were applied to an extensive range of experimental data at 25 °C with excellent agreement to about 6 mol kg−1 for various types of electrolyte. The treatment can be extended to mixed electrolytes and to include association equilibria. Values for the parameters β(0), β(1) and C for inorganic and organic acids, bases and salts have been tabulated. Temperature and pressure variation is also discussed.
Specific ion interaction theory
Specific ion Interaction Theory (SIT theory) is a theory used to estimate single-ion activity coefficients in electrolyte solutions at relatively high concentrations. It does so by taking into consideration interaction coefficients between the various ions present in solution. Interaction coefficients are determined from equilibrium constant values obtained with solutions at various ionic strengths. The determination of SIT interaction coefficients also yields the value of the equilibrium constant at infinite dilution.
The activity coefficient of the jth ion in solution is written as \(γ_j\) when concentrations are on the molal concentration scale and as yj when concentrations are on the molar concentration scale. (The molality scale is preferred in thermodynamics because molal concentrations are independent of temperature). The basic idea of SIT theory is that the activity coefficient can be expressed as
\[\log \gamma _{j}=-z_{j}^{2}{\frac {0.51{\sqrt {I}}}{1+1.5{\sqrt {I}}}}+\sum _{k}\epsilon _{jk}m_{k} \nonumber \]
where z is the electrical charge on the ion, \(I\) is the ionic strength, \(ε\) and \(b\) are interaction coefficients and \(m\) are concentrations. The summation extends over the other ions present in solution, which includes the ions produced by the background electrolyte. The first term in these expressions comes from Debye-Hückel theory. The second term shows how the contributions from "interaction" are dependent on concentration. Thus, the interaction coefficients are used as corrections to Debye-Hückel theory when concentrations are higher than the region of validity of that theory.


15.4: Coupled Equilibria
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Learning Objectives
By the end of this section, you will be able to:
- Describe examples of systems involving two (or more) coupled chemical equilibria
- Calculate reactant and product concentrations for coupled equilibrium systems
As discussed in preceding chapters on equilibrium, coupled equilibria involve two or more separate chemical reactions that share one or more reactants or products. This section of this chapter will address solubility equilibria coupled with acid-base and complex-formation reactions.
An environmentally relevant example illustrating the coupling of solubility and acid-base equilibria is the impact of ocean acidification on the health of the ocean’s coral reefs. These reefs are built upon skeletons of sparingly soluble calcium carbonate excreted by colonies of corals (small marine invertebrates). The relevant dissolution equilibrium is
\[\ce{CaCO_3(s) <=> Ca^{2+}(aq) + CO_3^{-2}(aq)} \quad K_{\text {sp }}=8.7 \times 10^{-9} \nonumber \]
Rising concentrations of atmospheric carbon dioxide contribute to an increased acidity of ocean waters due to the dissolution, hydrolysis, and acid ionization of carbon dioxide:
\[\begin{align*}
\ce{CO_2(g) &<=> CO_2(aq)} \\[4pt]
\ce{CO_2(aq) + H2O(l) &<=> H_2 CO_3(aq)} \\[4pt]
\ce{H_2 CO_3(aq) + H2O(l) &<=> HCO_3^{-}(aq) + H3O^{+}(aq)} \quad K_{ a 1}=4.3 \times 10^{-7} \\[4pt]
\ce{HCO_3^{-}(aq) + H2O(l) &<=> CO_3^{2-}(aq) + H3O^{+}(aq)} \quad K_{ a 2}=4.7 \times 10^{-11}
\end{align*} \]
Inspection of these equilibria shows the carbonate ion is involved in the calcium carbonate dissolution and the acid hydrolysis of bicarbonate ion. Combining the dissolution equation with the reverse of the acid hydrolysis equation yields
\[\ce{CaCO_3(s) + H3O^{+}(aq) <=> Ca^{2+}(aq) + HCO_3^{-}(aq) + H2O(l)} \quad \quad K=K_{\text {sp }} / K_{ a 2}=180 \nonumber \]
The equilibrium constant for this net reaction is much greater than the K sp for calcium carbonate, indicating its solubility is markedly increased in acidic solutions. As rising carbon dioxide levels in the atmosphere increase the acidity of ocean waters, the calcium carbonate skeletons of coral reefs become more prone to dissolution and subsequently less healthy (Figure \(\PageIndex{1}\)).
Link to Learning
Learn more about ocean acidification and how it affects other marine creatures.
This site has detailed information about how ocean acidification specifically affects coral reefs.
The dramatic increase in solubility with increasing acidity described above for calcium carbonate is typical of salts containing basic anions (e.g., carbonate, fluoride, hydroxide, sulfide). Another familiar example is the formation of dental cavities in tooth enamel. The major mineral component of enamel is calcium hydroxyapatite (Figure \(\PageIndex{2}\)), a sparingly soluble ionic compound whose dissolution equilibrium is
\[\ce{Ca5(PO4)3OH(s) <=> 5 Ca^{2+}(aq) +3 PO4^{3-}(aq) + OH^{-}(aq)} \nonumber \]
This compound dissolved to yield two different basic ions: triprotic phosphate ions
\[\begin{align*}
\ce{PO_4^{3-}(aq) + H3O^{+}(aq) &-> H_2 PO_4^{2-}(aq) + H2O(l)} \\[4pt]
\ce{H_2 PO_4^{2-}(aq) + H3O^{+}(aq) &-> H_2 PO_4^{-}(aq) + H2O(l)} \\[4pt]
\ce{H_2 PO_4^{-}(aq) + H3O^{+}(aq) &-> H_3 PO_4(aq) + H2O(l)}
\end{align*} \]
and monoprotic hydroxide ions:
\[\ce{OH^{-}(aq) + H3O^{+} -> 2 H2O} \nonumber \]
Of the two basic productions, the hydroxide is, of course, by far the stronger base (it’s the strongest base that can exist in aqueous solution), and so it is the dominant factor providing the compound an acid-dependent solubility. Dental cavities form when the acid waste of bacteria growing on the surface of teeth hastens the dissolution of tooth enamel by reacting completely with the strong base hydroxide, shifting the hydroxyapatite solubility equilibrium to the right. Some toothpastes and mouth rinses contain added NaF or SnF 2 that make enamel more acid resistant by replacing the strong base hydroxide with the weak base fluoride:
\[\ce{NaF + Ca5(PO4)3OH <=> Ca5(PO4)3F + Na^{+} + OH^{-}} \nonumber \]
The weak base fluoride ion reacts only partially with the bacterial acid waste, resulting in a less extensive shift in the solubility equilibrium and an increased resistance to acid dissolution. See the Chemistry in Everyday Life feature on the role of fluoride in preventing tooth decay for more information.
Chemistry in Everyday Life: Role of Fluoride in Preventing Tooth Decay
As we saw previously, fluoride ions help protect our teeth by reacting with hydroxylapatite to form fluorapatite, Ca 5 (PO 4 ) 3 F. Since it lacks a hydroxide ion, fluorapatite is more resistant to attacks by acids in our mouths and is thus less soluble, protecting our teeth. Scientists discovered that naturally fluorinated water could be beneficial to your teeth, and so it became common practice to add fluoride to drinking water. Toothpastes and mouthwashes also contain amounts of fluoride (Figure \(\PageIndex{3}\)).
Unfortunately, excess fluoride can negate its advantages. Natural sources of drinking water in various parts of the world have varying concentrations of fluoride, and places where that concentration is high are prone to certain health risks when there is no other source of drinking water. The most serious side effect of excess fluoride is the bone disease, skeletal fluorosis. When excess fluoride is in the body, it can cause the joints to stiffen and the bones to thicken. It can severely impact mobility and can negatively affect the thyroid gland. Skeletal fluorosis is a condition that over 2.7 million people suffer from across the world. So while fluoride can protect our teeth from decay, the US Environmental Protection Agency sets a maximum level of 4 ppm (4 mg/L) of fluoride in drinking water in the US. Fluoride levels in water are not regulated in all countries, so fluorosis is a problem in areas with high levels of fluoride in the groundwater.
The solubility of ionic compounds may also be increased when dissolution is coupled to the formation of a complex ion. For example, aluminum hydroxide dissolves in a solution of sodium hydroxide or another strong base because of the formation of the complex ion
The equations for the dissolution of aluminum hydroxide, the formation of the complex ion, and the combined (net) equation are shown below. As indicated by the relatively large value of K for the net reaction, coupling complex formation with dissolution drastically increases the solubility of Al(OH) 3 .
\[\begin{align*}
\ce{Al ( OH )_3(s) &<=> Al^{3+}(aq) +3 OH^{-}(aq)} & K_{ sp }=2 \times 10^{-32} \\[4pt]
\ce{Al^{3+}(aq) +4 OH^{-}(aq) &<=> Al ( OH )_4^{-}(aq) }& K_{ f }=1.1 \times 10^{33} \\[4pt]
\text { Net: } \ce{Al ( OH )_3(s) + OH^{-}(aq) &<=> Al ( OH )_4^{-}(aq)} & K=K_{ sp } K_{ f }=22
\end{align*} \nonumber \]
Example \(\PageIndex{1}\): Increased Solubility in Acidic Solutions
Compute and compare the molar solubilities for aluminum hydroxide, Al(OH) 3 , dissolved in (a) pure water and (b) a buffer containing 0.100 M acetic acid and 0.100 M sodium acetate.
Solution
(a) The molar solubility of aluminum hydroxide in water is computed considering the dissolution equilibrium only as demonstrated in several previous examples:
\[\ce{Al ( OH )_3(s) <=> Al^{3+}(aq) +3 OH^{-}(aq)} \quad \quad K_{ sp }=2 \times 10^{-32} \nonumber]
\[\text{molar solubility in water} =\left[ \ce{Al^{3+}} \right]= \left(\dfrac{2 \times 10^{-32}}{27}\right)^{1/4} =5 \times 10^{-9} ~\text{M} \nonumber \]
(b) The concentration of hydroxide ion of the buffered solution is conveniently calculated by the Henderson-Hasselbalch equation:
\[\begin{align*}
\text{pH} &= pK_{ a }+\log \dfrac{\left[ \ce{CH3COO^{-}} \right] }{ \left[\ce{CH3COOH} \right]} \\[4pt]
&=4.74+\log \left(\dfrac{0.100}{0.100}\right)=4.74
\end{align*} \nonumber \]
At this pH, the concentration of hydroxide ion is
\[\begin{align*}
& \text{pOH} =14.00-4.74=9.26 \\[4pt]
& {\left[ \ce{OH^{-}} \right]=10^{-9.26}=5.5 \times 10^{-10}}
\end{align*} \nonumber \]
The solubility of \(\ce{Al(OH)3}\) in this buffer is then calculated from its solubility product expression:
\[K_{ sp }=\left[ \ce{Al^{3+}}\right]\left[ \ce{OH^{-}}\right]^3 \nonumber \]
Via
\[\begin{align*} \text { molar solubility in buffer }&=\left[ \ce{Al^{3+}} \right] \\[4pt] &= \dfrac{K_{ sp } }{\left[ \ce{OH^{-}} \right]^3} \\[4pt] &=\dfrac{2 \times 10^{-32}}{(5.5 \times 10^{-10})^3} \\[4pt] &= 1.2 \times 10^{-4}~\text{M}
\end{align*} \]
Compared to pure water, the solubility of aluminum hydroxide in this mildly acidic buffer is approximately ten million times greater (though still relatively low).
Exercise \(\PageIndex{1}\)
What is the solubility of aluminum hydroxide in a buffer comprised of 0.100 M formic acid and 0.100 M sodium formate?
- Answer
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0.1 M
Example \(\PageIndex{2}\): Multiple Equilibria
Unexposed silver halides are removed from photographic film when they react with sodium thiosulfate (Na 2 S 2 O 3 , called hypo) to form the complex ion \(\ce{Ag(S2O3)2^{3-}}\) with \(K_f = 4.7 \times 10^{13}\).
What mass of Na 2 S 2 O 3 is required to prepare 1.00 L of a solution that will dissolve 1.00 g of AgBr by the formation of \(\ce{Ag(S2O3)2^{3-}}\)
Solution
Two equilibria are involved when silver bromide dissolves in an aqueous thiosulfate solution containing the ion:
- dissolution : \[\ce{AgBr(s) <=> Ag^{+}(aq) + Br^{-}(aq)} \quad \quad K_{ sp }=5.0 \times 10^{-13} \nonumber \]
- complexation : \[ \nonumber \ce{Ag^{+}(aq) +2 S2O3^{2-}(aq) <=> Ag ( S2O3)_2^{3-}(aq)}\quad \quad K_{ f }=4.7 \times 10^{13} \nonumber \]
Combining these two equilibrium equations yields
\[\ce{AgBr(s) +2S2O3^{2-}(aq) <=> Ag(S2O3)2^{3-}(aq) + Br^{-}(aq)} \nonumber \]
with
\[K=\frac{\left[ \ce{Ag(S2O3)2^3} \right][ \ce{Br^{-}} ]}{\left[ \ce{S2O3^{-2}} \right]^2}=K_{ sp } K_{ f }=24 \nonumber \]
The concentration of bromide resulting from dissolution of 1.00 g of AgBr in 1.00 L of solution is
\[\left[ Br^{-}\right]=\frac{1.00 ~\text{g} ~\ce{AgBr} \times \dfrac{1 ~ \text{mol} ~\ce{AgBr} }{187.77\, \text{g / mol} } \times \dfrac{1 ~ \text{mol} ~\ce{Br^{-}}}{1~\text{mol} ~\ce{AgBr^{-}}}}{1.00~\text{L} }=0.00532~\text{M} \nonumber \]
The stoichiometry of the dissolution equilibrium indicates the same concentration of aqueous silver ion will result, 0.00532 M, and the very large value of ensures that essentially all the dissolved silver ion will be complexed by thiosulfate ion:
\[\left[ \ce{Ag(S2O3)2^{3-}}\right]=0.00532 ~\text{M} \nonumber \]
Rearranging the K expression for the combined equilibrium equations and solving for the concentration of thiosulfate ion yields
\[\left[ \ce{S2O3^{2-}} \right]=\frac{\left[ \ce{Ag(S2O3)2^{3-}} \right]\left[ \ce{Br^{-}} \right]}{K}=\frac{(0.00532 M)(0.00532~\text{M})}{24}=0.0011 ~ \text{M} \nonumber \]
Finally, the total mass of \(\ce{Na2S2O3}\) required to provide enough thiosulfate to yield the concentrations cited above can be calculated.
Mass of \(\ce{Na2S2O3}\) required to yield 0.00532 M \(\ce{Ag(2S2O3)2^{3-}}\)
\[0.00532 \dfrac{ \text{mol} ~\ce{Ag(S2O3)2^{3-}}}{1.00~\text{L}} \times \dfrac{2~\text{mol}~\ce{S2O3^{2-}}}{1~\text{mol} ~\ce{Ag(S2O3)2^{3-}}} \times \dfrac{1~\text{mol}~\ce{Na2S2O3}}{1 ~\text{mol}~\ce{S2O3^{2-}}} \times \dfrac{158.1 ~ \text{g}~\ce{Na2S2O3}}{1 ~\text{mol}~\ce{Na2S2O3}}=1.68~\text{g} \nonumber \]
Mass of \(\ce{Na2S2O3}\) required to yield 0.00110 M \(\ce{S2O3^{2-}}\)
\[0.0011 \frac{ \text{mol} ~\ce{S2O3^{2-}}}{1.00~\text{L} } \times \frac{1~\text{mol} ~ \ce{S2O3^{2-}}}{1 ~ \text{mol} ~\ce{Na2S2O3}} \times \frac{158.1 ~\text{g}~\ce{Na2S2O3}}{1~\text{mol} ~\ce{Na2S2O3}}=0.17~\text{g} \nonumber \]
The mass of required to dissolve 1.00 g of AgBr in 1.00 L of water is thus 1.68 g + 0.17 g = 1.85 g
Exercise \(\PageIndex{2}\)
AgCl( s ), silver chloride, has a very low solubility:
\[\ce{AgCl (s) <=> Ag^{+}(aq) + Cl^{-}(aq)} \quad\quad K_{ so }=1.6 \times 10^{-10} \nonumber \]
Adding ammonia significantly increases the solubility of AgCl because a complex ion is formed:
\[\ce{Ag^{+}(aq) +2 NH_3(aq) <=> Ag(NH3)2^{+}(aq)} \quad\quad K_{ f }=1.7 \times 10^7 \nonumber \]
What mass of \(\ce{NH3}\) is required to prepare 1.00 L of solution that will dissolve 2.00 g of AgCl by formation of \(\ce{Ag(NH3)^{2+}}\)?
- Answer
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1.00 L of a solution prepared with 4.81 g NH 3 dissolves 2.0 g of \(\ce{AgCl}\).


12.29: Conclusion
Over the eleventh and twelfth centuries, a rise in agricultural production led to an increase in Western Europe’s wealth and population. The chaos of the tenth and eleventh centuries had brought about the origin of a feudal system dominated by knights. Feudal Europe was thus able to respond to the Byzantine Empire’s requests for help when its field army was annihilated by Saljuq Turks, resulting in the First Crusade and establishment of a set of Crusader States in the Eastern Mediterranean, but these Crusader states were gradually conquered by Muslim powers over the next two centuries. An army of Crusaders would eventually defeat and break up the Byzantine Empire, and although that empire would be re-established, it was never strong enough to resist the pressure of the Ottoman Turks, who finally conquered it in 1453.
As Western Europe grew in population and urbanized, the urban cathedral schools became the center of an increase in intellectual activity over the twelfth century known as the Twelfth-Century Renaissance. Thinkers of the Twelfth-Century Renaissance used the philosophy of the Ancient Greeks and Arabs to understand the world, and in the thirteenth century the intellectual movement known as scholasticism would seek to reconcile Christianity with Arabic and Ancient Greek Philosophy. By the end of the Middle Ages, Europe’s intellectuals would seek to study the writings of the Ancient Greeks and Romans, and not the commentators of the previous thousand years. This movement was known as Renaissance humanism.
Europe’s states—with the notable exception of the Holy Roman Empire—gradually consolidated, but the fourteenth and fifteenth centuries saw increasing interstate warfare. The states that emerged out of this endemic war, however, were more militarily powerful and more centralized. At the same time, sailors in the service of Spain and Portugal were exploring the Atlantic and West Africa until the close of the fifteenth century, when Western Europeans discovered the existence of the continents of the Western Hemisphere.


1.8: Modes of Reasoning- Dialectical versus Analytic
What is Dialectical Reasoning ?
- The process of arriving at the truth by stating a thesis, developing a contradictory antithesis based on concrete possibilities, and combining them into a coherent synthesis, often after numerous variations and iterations.
- A method of “argument” or exposition that systematically weighs an idea with a view to the resolution of its real or apparent contradictions.
How to Engage in Abstract or Analytic Reasoning
Analytic argumentation differs markedly from Dialectical. The following pertains to the Analytic method only. Keep it in mind.
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Analytic reasoning commences with a first principle or assumption. On this foundation, an “argument” is built.
- Assumptions must be reasonable and generally true.
- An argument is not a debate. Do not start with an oppositional counter-statement . It is not about “winning.”
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Do not spar with the argument. First try to understand it.
- Taking a contrary position will not assist you in understanding the proposition or argument better.
- Arguments are often nuanced, not black and white.
- An argument is not an opinion. The latter is subjective.
- In college (and later in life), an opinion is not necessarily a “right” to which a student is “entitled.” Sound reasoning, i.e., a good argument, trumps opinion.
- Any position you take must be based on sound argument.
The Talmudic method invariably prefers to pose questions in a concrete rather than an abstract form.
– Rabbi Adin Even-Israel Steinsaltz
Reference Guide to the Talmud , p. 131 (2014)


9.4.5: Tropical Desert Climate
The tropical desert is an environment of extremes: it is the driest and hottest place on earth. Rainfall is sporadic and in some years no measurable precipitation falls at all. The terribly dry conditions of the deserts is due to the year-round influence of subtropical high pressure and continentality.
Geographical Distribution
Deserts are typically found in continental interiors of the subtropics and on the lee side of mountains in the midlatitudes. Cool coastal deserts are found where cold water upwells along a coast, stabilizing the air and preventing moisture formation like that near coastal Chile. Vast deserts cover much of north Africa (The Sahara), Saudi Arabia to Iran, Pakistan and western India. Tropical deserts are found in Baja California and interior Mexico in North America.
Controlling Factors
The most important controlling factor of the climate characteristics of the tropical desert climate is the year-round presence of subtropical high pressure. The descending air of the subtropical high adiabatically warms causing the air to dry out and inhibit condensation. Aridity also arises as distance from moisture sources increases. Leeward situations places one in the rain shadow which also promotes dry conditions. Cool coastal deserts are found along coasts where cold water is upwelling.
Distinguishing Characteristics
Temperature
The tropical desert has the highest mean annual temperature of any climate on Earth. The high temperatures are a result of the high sun angles throughout the year and having the highest percentage of sunshine of any climate. No month has an average temperature below 18 o C (64.4 o F) and many places have consecutive average monthly temperatures in the mid 30 o s Celsius (90 o F). Daytime temperatures can reach 50 o C (120 o F) at low elevation inland deserts.
The sky in the tropical desert remains cloud-free due to the subsiding air of dominant high pressure resulting in large amounts of insolation. The cloudless skies during the day lets insolation in, but also lets much heat out at night. Without the absorptive blanket of clouds, longwave radiation emitted from the Earth readily escapes to space, chilling the nighttime desert air. The high energy input during the day and large loss at night results in an extremely large daily temperature range.
Precipitation
Precipitation in the tropical desert is very irregular and unreliable. Low latitude deserts average less than 25 cm (10 in) in a year. An entire year's worth of rain may fall in one downpour. The continental location of many tropical deserts places them far from a source of moisture, the ocean. Combine continentality with the strong subsidence of the subtropical high and you have one of the driest places on earth. Air subsiding from the subtropical high is adiabatically warmed which reduces the relative humidity of the air. Relative humidity can drop to 10% or less. The extremely low relative humidity causes evaporation of what little surface water there is. The subsiding air also promotes atmospheric stability, further inhibiting precipitation.
Climatologists describe the desert as an "arid" climate. An arid climate, as defined on the basis of the soil moisture balance, is one in which the annual precipitation is less than half of the annual potential evapotranspiration. In the tropical desert the only substantial source of surface water other than exotic streams is an oasis , where the groundwater table is near the surface.


7.19: Assignment- Analyze Yearly Trends
In this assignment, we will pull together everything you have learned about Excel so far to analyze trends in sales data over several years.
To complete this assignment, you will need the assignment you completed last module that you saved as BA132_LastName_SalesData.xlsx (or you can download the original assignment here ) and a new workbook you can download here . Follow the directions, then submit your assignment. If you get stuck on a step, review this module and ask your classmates for help in the discussion forum.
- Open both workbooks. Save the new Module 7 file to the Rowan folder on your desktop as BA132_LastName_YearlyTrends.xlsx , replacing “LastName” with your own last name. (Example: BA132_Hywater_Memo) It is a good idea to save your work periodically.
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First, we need to combine the information into a single workbook. Add a new worksheet to the Module 7 file and copy the table from the Module 6 assignment to that new tab. Name the new tab “2014.” Be sure to set the number type to currency.
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Add another new worksheet. Now we want to set up the spacing for the table we are about to make. Merge the 1 and 2 cells of column A, B, C, D, and E individually (so A1 and A2 merge, B1 and B2 merge, etc). Then name the cells as shown in the list below and center the text in those cells.
A = Month
B = 2012 Sales
C = 2013 Sales
D = 2014 Sales
E = Total by Month
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Copy the following columns of data over in order: “2012 Sales Total,” “2013 Sales Total,” and “2014 Sales Total.” Remember, you can just copy and paste each column in order, using the columns you have already labeled. When you AutoFit the column to the data width, your header cells will change.
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Now apply Autosum to the sales data for both the total by month and by year. When you have the totals, apply bold to them to make them stand out more and type “Totals” in the cell under December.
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Next to the ‘Totals by Month’ column, type ‘Above $50k’ in the column title. In the first cell in the new column use an IF function to identify True or False if the totals by month are greater than $50,000 or not. Remember, after you create the first IF functions in January, you can Flash fill the other cells by dragging the bottom right of the highlighted cell down the other cells.
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Return to the individual tabs for each year. For each year of data, sort the sales total “Z to A” so that the month with the highest sales is at the top. When you chose the “Sort” option, be sure to “Expand the Selection.” That will make sure that the months change as well. The screenshot below shows you only what 2012 should look like, but when you finish you should see October as the highest in 2012, August for 2013, and May for 2014.
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Finally, go back to Sheet4 and create a clustered column chart from the three year columns of your data.
- Save your work and submit it in your course online.
Contributors and Attributions
- Assignment: Analyzing Yearly Trends. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution