6.6 Complex research designs
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In Chapter 5 and in this chapter so far, we have restricted our discussion to the simplest possible research designs – cases where we are dealing with two variables with two values each. To conclude our discussion of statistical hypothesis testing, we will look at two cases of more complex designs – one with two variables that each have more than two values, and one with more than two variables.
6.6.1 Variables with more than two values
n our case studies involving the English possessive constructions, the dependent variable (TYPE OF) POSSESSIVE was treated as binary – we assumed that it had two values, S -POSSESSIVE and OF -POSSESSIVE. The dependent variables were more complex: the cardinal variable LENGTH obviously has a potentially infinite number of values and the ordinal variable ANIMACY was treated as having ten values in our annotation scheme. The nominal value DISCOURSE STATUS, was treated like a binary variable (although potentially it has an infinite number of values, too).
Frequently, perhaps even typically, corpus linguistic research questions will be more complex, and we will be confronted with designs where both the dependent and the independent variable will have (or be treated as having) more than two values. Since we are most likely to deal with nominal variables in corpus linguistics, we will discuss in detail an example where both variables are nominal.
In the preceding chapters we treated as S -POSSESSIVE constructions where the modifier is a possessive pronoun as well as constructions where the modifier is a proper name or a noun with a possessive clitic. Given that the proportion of pronouns and nouns in general varies across language varieties (Biber et al. 1999), we might be interested to see whether the same is true for these three variants of the s -possessive. Our dependent variable MODIFIER OF s -POSSESSIVE would then have three values. The independent variable VARIETY, being heavily dependent on the model of language varieties we adopt – or rather, on the nature of the text categories included in our corpus – has an indefinite number of values. To keep things simple, let us distinguish just four broad text categories recognized in the British National Corpus (and many other corpora): SPOKEN, FICTION, NEWSPAPER, and ACADEMIC. This gives us a four-by-three design.
Searching the BNC Baby for words tagged as possessive pronouns and for words tagged unambiguously as proper names or common nouns yields the observed frequencies shown in the first line of each row in Table 6.14.
Table 6.14: Types of modifiers in the s-possessive in different text categories
The expected frequencies and the χ 2 components are arrived at in the same way as for the two-by-two tables in the preceding chapter. First, for each cell, the sum of the column in which the cell is located is multiplied by the sum of the row in which it is located, and the result is divided by the table sum. For example, for the top left cell, we get the expected frequency
the expected frequencies are shown in the second line of each cell. Next, for each cell, we calculate the χ 2 component. For example, for the top left cell, we get
the corresponding values are shown in the third line of each cell. Adding up the individual χ 2 components gives us a χ 2 value of 3950.89.
Using the formula given in (5) above, Table 6.14 has (4−1) × (3−1) = 6 degrees of freedom. As the χ 2 table in Section 14.1, the required value for a significance level of 0.001 at 6 degrees of freedom is 22.46; the χ 2 value for Table 6.14 is much higher than this, thus, our results are highly significant. We could summarize our findings as follows: “The frequency of pronouns, proper names and nouns as modifiers of the s -possessive differs highly significantly across text categories (χ 2 = 473.73, df = 12, p < 0.001)”.
Recall that the mere fact of a significant association does not tell us anything about the strength of that association – we need a measure of effect size. In the preceding chapter, ϕ was introduced as an effect size for two-by-two tables (see 7). For larger tables, there is a generalized version of ϕ , referred to as Cramer’s V (or, occasionally, as Cramer’s ϕ or ϕ ′ ), which is calculated as follows (N is the table sum, k is the number of rows or columns, whichever is smaller):
(20)
For our table, this gives us:
Recall that the square of a correlation coefficient tells us the proportion of the variance captured by our design, which, in this case, is 0.0275. In other words, VARIETY explains less than three percent of the distribution of s -possessor modifier types across language varieties; or “This study has shown a very weak but highly significant influence of language variety on the realization of s -possessor modifiers as pronouns, proper names or common nouns (χ 2 = 473.73, df = 12, p < 0.001, r = 0.0275).”
Despite the weakness of the effect, this result confirms our expectation that general preferences for pronominal vs. nominal reference across language varieties is also reflected in preferences for types of modifiers in the s -possessive. However, with the increased size of the contingency table, it becomes more difficult to determine exactly where the effect is coming from. More precisely, it is no longer obvious at a glance which of the intersections of our two variables contribute to the overall significance of the result in what way and to what extent.
To determine in what way a particular intersection contributes to the overall result, we need to compare the observed and expected frequencies in each cell. For example, there are 9593 cases of s -possessives with pronominal modifiers in spoken language, where 8378.38 are expected, showing that pronominal modifiers are more frequent in spoken language than expected by chance. In contrast, there are 8533 such modifiers in academic language, where 8959.86 are expected, showing that they are less frequent in academic language than expected by chance. This comparison is no different from that which we make for two-by-two tables, but with increasing degrees of freedom, the pattern becomes less predictable. It would be useful to visualize the relation between observed and expected frequencies for the entire table in a way that would allow us to take them in at a glance
To determine to what extent a particular intersection contributes to the overall result, we need to look at the size of the χ2 components – the larger the component, the greater its contribution to the overall χ2 value. In fact, we can do more than simply compare the χ2 components to each other – we can determine for each component, whether it, in itself, is statistically significant. In order to do so, we first imagine that the large contingency table (in our case, the 4-by-3 table) consists of a series of tables with a single cell each, each containing the result for a single intersection of our variables.
We now treat the χ 2 component as a χ 2 value in its own right, checking it for statistical significance in the same way as the overall χ 2 value. In order to do so, we first need to determine the degrees of freedom for our one-cell tables – obviously, this can only be 1. Checking the table of critical χ 2 values in Section 14.1, we find, for example, that the χ 2 component for the intersection PRONOUN ∩ SPOKEN, which is 176.08, is higher than the critical value 10.83, suggesting that this intersection’s contribution is significant at p < 0.001.
However, matters are slightly more complex: by looking at each intersection separately, we are essentially treating each cell as an independent result – in our case, it is as if we had performed twelve tests instead of just one. Now, recall that levels of significance are based on probabilities of error – for example, p = 0.05 means, roughly, that there is a five percent likelihood that a result is due to chance. Obviously, the more tests we perform, the more likely it becomes that one of the results will, indeed, be due to chance – for example, if we had performed twenty tests, we would expect one of them to yield a significant result at the 5-percent level, because 20 × 0.05 = 1.00.
To avoid this situation, we have to correct the levels of significance when performing multiple tests on the same set of data. The simplest way of doing so is the so-called Bonferroni correction, which consists in dividing the conventionally agreed-upon significance levels by the number of tests we are performing. In the case of Table 6.14, this means dividing them by twelve, giving us significance levels of 0.05 / 12 = 0.004167 (significant), 0.01 / 12 = 0.000833 (very significant), and 0.001 / 12 = 0.000083 (highly significant). 6 Our table does not give the critical χ 2 - values for these levels, but the value for the the intersection PRONOUN ∩ SPOKEN, 176.08, is larger than the value required for the next smaller level (0.00001, with a critical value of 24.28), so we can be certain that the contribution of this intersection is, indeed, highly significant. Again, it would be useful to summarize the degrees of significance in such a way that they can be assessed at a glance.
There is no standard way of representing the way in, and degree to, which each cell of a complex table contributes to the overall result, but the representation in Table 6.15 seems reasonable: in each cell, the first line contains either a plus (for “more frequent than expected”) or a minus (for “less frequent than expected”); the second line contains the χ 2 -component, and the third line contains the (corrected) level of significance (using the standard convention of representing them by asterisks – one for each level of significance).
This table presents the complex results at a single glance; they can now be interpreted. Some patterns now become obvious: For example, spoken language and fiction are most similar to each other – they both favor pronominal modifiers, while proper names and common nouns are disfavored, and the χ 2 -components for these preferences are very similar. Also, if we posit a kind of gradient of referent familiarity from pronouns over proper names to nouns, we can place spoken language and fiction at one end, academic language at the other, and newspaper language somewhere in the middle.
6.6.2 Designs with more than two variables
Note that from a certain perspective the design in Table 6.15 is flawed: the variable VARIETY actually conflates at least two variables that are theoretically independent: MEDIUM (with the variables SPOKEN and WRITTEN, and DISCOURSE DOMAIN (in our design with the variables NEWS (RECOUNTING OF ACTUAL EVENTS), FICTION (RECOUNTING OF IMAGINARY EVENTS) and ACADEMIC (RECOUNTING OF SCIENTIFIC IDEAS, PROCEDURES, AND RESULTS). These two variables are independent in that there is both written and spoken language to be found in each of these discourse domains. They are conflated in our variable VARIETY in that one of the four values is spoken and the other three are written language, and in that the text category SPOKEN is not differentiated by topic. There may be reasons to ignore this conflation a priori , as we have done – for example, our model may explicitly assume that differentiation by topic happens only in the written domain. But even then, it would be useful to treat MEDIUM and DISCOURSE DOMAIN as independent variables, just in case our model is wrong in assuming this.
Table 6.15: Types of modifiers in the s -possessive in different text categories: Contributions of the individual intersections
In contrast to all examples of research designs we have discussed so far, which involved just two variables and were thus bivariate , this design would be multivariate : there is more than one independent variable whose influence on the dependent variable we wish to assess. Such multivariate research designs are often useful (or even necessary) even in cases where the variables in our design are not conflations of more basic variables.
In the study of language use, we will often – perhaps even typically – be confronted with a fragment of reality that is too complex to model in terms of just two variables.
In some cases, this may be obvious from the outset: we may suspect from previous research that a particular linguistic phenomenon depends on a range of factors, as in the case of the choice between the s - and the of -possessive, which we saw in the preceding chapters had long been hypothesized to be influenced by the animacy, the length and/or the givenness of the modifier.
In other cases, the multivariate nature of the phenomenon under investigation may emerge in the course of pursuing an initially bivariate design. For example, we may find that the independent variable under investigation has a statistically significant influence on our dependent variable, but that the effect size is very small, suggesting that the distribution of the phenomenon in our sample is conditioned by more than one influencing factor.
Even if we are pursuing a well-motivated bivariate research design and find a significant influence with a strong effect size, it may be useful to take additional potential influencing factors into account: since corpus data are typically unbalanced, there may be hidden correlations between the variable under investigation and other variables, that distort the distribution of the phenomenon in a way that suggests a significant influence where no such influence actually exists.
The next subsection will use the latter case to demonstrate the potential shortcomings of bivariate designs and the subsection following it will present a solution. Note that this solution is considerably more complex than the statistical procedures we have looked at so far and while it will be presented in sufficient detail to enable the reader in principle to apply it themselves, some additional reading will be highly advisable.
<IP_ADDRESS> A danger of bivariate designs
In recent years, attention has turned to sociolinguistic factors potentially influencing the choice between the s -possessive and the of -construction. It has long been known that the level of formality has an influence (Jucker 1993, also Grafmiller 2014), but recently, more traditional sociolinguistic variables like Sex and Age have been investigated. The results suggest that the latter has an influence, while the former does not – for example, Jankowski & Tagliamonte (2014) find no influence of sex, but find that age has an influence under some conditions, with young speakers using the s-genitive more frequently than old speakers for organizations and places; Shih et al. (2015) find a similar, more general influence of age.
Let us take a look at the influence of SEX and AGE on the choice between the two possessives in the spoken part of the BNC. Since it is known that women tend to use pronouns more than men do (see Case Study <IP_ADDRESS> in Chapter 10), let us exclude possessive pronouns and operationalize the S -POSSESSIVE as “all tokens tagged POS in the BNC”, which will capture the possessive clitic ’s and zero possessives (on common nouns ending in alveolar fricatives). Since the spoken part of the BNC is too large to identify of -possessives manually, let us operationalize them somewhat crudely as “all uses of the preposition of ”; this encompasses not just of -possessives, but also the quantifying and partitive of -constructions that we manually excluded in the preceding chapters, the complementation of adjectives like aware and afraid , verbs like consist and dispose , etc. On the one hand, this makes our case study less precise, on the other hand, any preference for of - constructions may just be a reflex of a general preference for the preposition of , in which case we would be excluding relevant data by focusing on of -constructions. Anyway, our main point will be one concerning statistical methodology, so it does not matter too much either way.
So, let us query all tokens tagged as possessives (POS) or the preposition of (PRF) in the spoken part of the BNC, discarding all hits for which the information about speaker sex or speaker age is missing. Let us further exclude the age range 0–14, as it may include children who have not fully acquired the grammar of the language, and the age range 60+ as too unspecific. To keep the design simple, let us recode all age classes between 15 and 44 years of age as YOUNG and the age range 45–59 als OLD (I fall into the latter, just in case someone thinks this category label discriminates people in their prime). Let us further accept the categorization of speakers into male and female that the makers of the BNC provide.
Table 6.16 shows the intersections of CONSTRUCTION and SEX in the results of this query.
Unlike the studies mentioned above, we find a clear influence of SEX ON CONSTRUCTION , with female speakers preferring the s -possessive and male speakers preferring the of -construction(s). The difference is highly significant, although the effect size is rather weak (χ 2 = 773.55, df = 1, p < 0.001, ϕ = 0.1061).
Next, let us look at the intersections of CONSTRUCTION and SEX in the results of our query, which are shown in Table 6.17.
Like previous studies, we find a significant effect of age, with younger speakers preferring the s -possessive and older speakers preferring the of -construction(s). Again, the difference is highly significant, but the effect is extremely weak (χ 2 = 58.73, df = 1, p < 0.001, ϕ = 0.02922).
We might now be satisfied that both speaker age and speaker sex have an influence on the choice between the two constructions. However, there is a potential problem that we need to take into account: the values of the variables SEX and AGE and their intersections are not necessarily distributed evenly in the subpart of the BNC used here; although the makers of the corpus were careful to include a broad range of speakers of all ages, sexes (and class memberships, ignored in our study), they did not attempt to balance all these demographic variables, let alone their intersections. So let us look at the intersection of SEX and AGE in the results of our query. These are shown in Table 6.18.
Table 6.16: The influence of SEX on the choice between the two possessives
Table 6.17: The influence of AGE on the choice between the two possessives
There are significantly fewer hits produced by old women and significantly more produced by young women in our sample, and, conversely, significantly fewer hits produced by young men and significantly more produced by old men. This overrepresentation of young women and old men is not limited to our sample, but characterizes the spoken part of the BNC in general, which should intrigue feminists and psychoanalysts; for us, it suffices to know that the asymmetries in our sample are highly significant, with an effect size larger than that of that in the preceding two tables (χ 2 = 2142.72, df = 1, p < 0.001, ϕ = 0.1765).
Table 6.18: SEX and AGE in the BNC
This correlation in the corpus of OLD and MALE on the one hand and YOUNG and FEMALE on the other may well be enough to distort the results such that a linguistic behavior typical for female speakers may be wrongly attributed to young speakers (or vice versa), and, correspondingly, a linguistic behavior typical for male speakers may be wrongly interpreted to old speakers (or vice versa). More generally, the danger of bivariate designs is that a variable we have chosen for investigation is correlated with one or more variables ignored in our research design, whose influence thus remains hidden. A very general precaution against this possibility is to make sure that the corpus (or our sample) is balanced with respect to all potentially confounding variables. In reality, this is difficult to achieve and may in fact be undesirable, since we might, for example, want our corpus (or sample) to reflect the real-world correlation of speaker variables).
Therefore, we need a way of including multiple independent variables in our research designs even if we are just interested in a single independent variable, but all the more so if we are interested in the influence of several independent variables. It may be the case, for example, that both SEX and AGE influence the choice between ’ s and of , either in that the two effects add up, or in that they interact in more complex ways.
<IP_ADDRESS> Configural frequency analysis
There is a range of multivariate statistical methods that are routinely used in corpus linguistics, such as the ANOVA mentioned at the end of the previous chapter for situations where the dependent variable is measured in terms of cardinal numbers, and various versions of logistic regression for situations where the dependent variable is ordinal or nominal.
In this book, I will introduce multivariate designs using Configural Frequency Analysis (CFA), a straightforward extension of the χ2 test to designs with more an two nominal variables. This method has been used in psychology and psychiatry since the 1970s, and while it has never become very wide-spread, it has, in my opinion, a number of didactic advantages over other methods, when it comes to understanding multivariate research designs. Most importantly, it is conceptually very simple (if you understand the χ2 test, you should be able to understand CFA), and the results are very transparent (they are presented as observed and expected frequencies of intersections of variables.
This does not mean that CFA is useful only as a didactic tool – it has been applied fruitfully to linguistic research issues, for example, in the study of language disorders (Lautsch et al. 1988), educational linguistics (Fujioka & Kennedy 1997), psycholinguistics (Hsu et al. 2000) and social psychology (Christmann et al. 2000). An early suggestion to apply it to corpus data is found in Schmilz (1983), but the first actual such applications that I am aware of are Gries (2002; 2004). Since Gries introduced the method to corpus linguistics, it has become a minor but nevertheless well-established corpus-linguistic research tool for a range of linguistic phenomena (see, e.g., Stefanowitsch & Gries 2005; 2008; Liu 2010; Goschler et al. 2013; Hoffmann 2014; Hilpert 2015; and others).
As hinted at above, in its simplest variant, configural frequency analysis is simply a χ2 test on a contingency table with more than two dimensions. There is no logical limit to the number of dimensions, but if we insist on calculating this statistic manually (rather than, more realistically, letting a specialized software package do it for us), then a three-dimensional table is already quite complex to deal with. Thus, we will not go beyond three dimensions here or in the case studies in the second part of this book.
A three-dimensional contingency table would have the form of a cube, as shown in Figure 6.2. The smaller cube represents the cells on the far side of the big cube seen from the same perspective and the smallest cube represents the cell in the middle of the whole cube). As before, cells are labeled by subscripts: the first subscript stands for the values and totals of the dependent variable, the second for those of the first independent variable, and the third for those of the second independent variable.
While this kind of visualization is quite useful in grasping the notion of a three-dimensional contingency table, it would be awkward to use it as a basis for recording observed frequencies or calculating the expected frequencies. Thus, a possible two-dimensional representation is shown in Table 6.19. In this table, the first independent variable is shown in the rows, and the second independent variable is shown in the three blocks of three columns (these may be thought of as three “slices” of the cube in Figure 6.2), and the dependent variable is shown in the columns themselves.
Figure 6.2: A three-dimensional contingency table
Table 6.19: A two-dimensional representation of a three-dimensional contingency table
Given this representation, the expected frequencies for each intersection of the three variables can now be calculated in a way that is similar (but not identical) to that used for two-dimensional tables. Table 6.20 shows the formulas for each cell as well as those marginal sums needed for the calculation.
Table 6.20: Calculating expected frequencies in a three-dimensional contingency table
Once we have calculated the expected frequencies, we proceed exactly as before: we derive each cell’s χ2 component by the standard formula ( O − E ) 2 / E and then add up these components to give us an overall χ2 value for the table, which can then be checked for significance. The degrees of freedom of a three-dimensional contingency table are calculated by the following formula (where k is the number of values of each variable and the subscripts refer to the variables themselves):
(21) df = (k 1 × k 2 × k 3 ) − (k 1 + k 2 + k 3 ) + 2
In our case, each variable has two values, thus we get(2×2×2)−(2+2+2)+2 = 4. More interestingly, we can also look at the individual cells to determine whether their contribution to the overall value is significant). In this case, as before, each cell has one degree of freedom and the significance levels have to be adjusted for multiple testing. In CFA, an intersection of variables whose observed frequency is significantly higher than expected is referred to as a type and one whose observed frequency is significantly lower is referred to as an antitype (but if we do not like this terminology, we do not have to use it and can keep talking about “more or less frequent than expected”, as we do with bivariate χ2 tests)
Let us apply this method to the question described in the previous sub-section. Table 6.21 shows the observed and expected frequencies of the two possessive constructions by SPEAKER AGE and SPEAKER SEX, as well as the corresponding χ 2 components. 7 For expository purposes, the table also shows for each cell the significance level of these components (which is “highly significant” for almost all of them), and the direction of deviation from the expected frequencies (i.e., whether the intersection is a “type”, marked by a plus sign, or an “antitype”, marked by a minus sign).
Table 6.21: Sex, Age and Possessives Multivariate (BNC Spoken)
Adding up the χ 2 components yields an overall χ 2 value of 2980.10, which, at four degrees of freedom, is highly significant. This tells us something we already expected from the individual pairwise comparisons of the three variables in the preceding section: there is a significant relationship among them. Of course, what we are interested in is what this relationship is and to answer this question, we need to look at the contributions to χ 2 .
The result is very interesting. A careful inspection of the individual cells shows that age does not , in fact, have a significant influence. Young women use the s -possessive more frequently than expected and old women use it less (in the latter case, non-significantly), but young women also use the of -constructions significantly more frequently than expected and old women use it less. Crucially, young men use the s -possessive less frequently than expected, and old men use it more, but young men also use the of -construction less frequently than expected and old men use it more.
In other words, young women and old men use more of both constructions than young men and old women. A closer look at the contributions to χ 2 tells us that SEX, however, does still have an influence on the choice between the two constructions even when AGE is taken into account: for young women, the overuse of the s -possessive is more pronounced than that of the of -construction, while for old women, underuse of the s -possessive is less pronounced than underuse of the of -construction. In other words, taking into account that old women are underrepresented in the corpus compared to young women, there is a clear preference of all women for the s -possessive. Conversely, young men’s underuse of the s -possessive is less pronounced than that of the of -construction, while old men’s overuse of the s -possessive is less pronounced than their overuse of the of -construction. In other words, taking into account than young men are underrepresented in the corpus compared to old men, there is a clear preference of all men for the of -construction.
Armed with this new insight from our multivariate analysis, let us return to bivariate analysis, looking at each of the two variables while keeping the other constant. Table 6.22 shows the results of the four bivariate analysis this yields. Since we are performing four tests on the same set of data within the same research question, we have to correct for multiple testing by dividing the usual critical p -values by four, giving us p < 0.0125 for “significant”, p < 0.0025 for “very significant” and p < 0.00025 for “highly significant”. The exact p -values for each table are shown below, as is the effect size.
Table 6.22: Possessives, age and sex (BNC Spoken)
Tables 6.22a and 6.22b show that the effect of SEX on the choice between the two constructions is highly significant both in the group of OLD speakers and in the group of YOUNG speakers, with effect sizes similar to those we found for the bivariate analysis for SPEAKER SEX in Table 6.16 in the preceding section. This effect seems to be genuine, or at least, it is not influenced by the hidden variable AGE (it may be influenced by CLASS or some other variable we have not included in our design).
In contrast, Tables 6.22a and 6.22b show that the effect of AGE that we saw in Table 6.17 in the preceding section disappears completely for women, with a p -value not even significant at uncorrected levels of significance. For men, it is still discernible, but only barely, with a p -value that indicates a very significant relationship at corrected levels of significance, but with an effect size that is close to zero. In other words, it may not be a genuine effect at all, but simply a consequence of the fact that the intersections of AGE and SEX are unevenly distributed in the corpus.
This section is intended to impress on the reader one thing: that looking at one potential variable influencing some phenomenon that we are interested in may not be enough. Multivariate research designs are becoming the norm rather than the exception, and rightly so. Excluding the danger of hidden variables is just one advantage of such designs – in many cases, it is sensible to include several independent variables simply because all of them potentially have an interesting influence on the phenomenon under investigation, or because there is just one particular combination of values of our variables that has an effect. In the second part of this volume, there are several case studies that use CFA and that illustrate these possibilities. One word of warning, however: the ability to include a large number of variables in our research designs should not lead us to do so for the sake of it. We should be able to justify, for each dependent variable we include, why we are including it and in what way we expect it to influence our independent variable.
6 It should be noted that the Bonferroni correction is extremely conservative, but it has the advantage of being very simple to apply (see Shaffer 1995 for an overview corrections for multiple testing, including many that are less conservative than the Bonferroni correction).
7 Note one important fact about multi-dimensional contingency tables that may be confusing: if we add up the expected frequencies of a given column, their sum will not usually correspond to the sum of the observed frequencies in that column (in contrast to two-dimensional tables, where this is necessarily the case). Instead, the sum of the observed and expected frequencies in each slice is identical.


3.3F: Primary Active Transport
The sodium-potassium pump maintains the electrochemical gradient of living cells by moving sodium in and potassium out of the cell.
- Describe how a cell moves sodium and potassium out of and into the cell against its electrochemical gradient
Key Points
- The sodium-potassium pump moves K+ into the cell while moving Na+ at a ratio of three Na+ for every two K+ ions.
- When the sodium-potassium- ATPase enzyme points into the cell, it has a high affinity for sodium ions and binds three of them, hydrolyzing ATP and changing shape.
- As the enzyme changes shape, it reorients itself towards the outside of the cell, and the three sodium ions are released.
- The enzyme’s new shape allows two potassium to bind and the phosphate group to detach, and the carrier protein repositions itself towards the interior of the cell.
- The enzyme changes shape again, releasing the potassium ions into the cell.
- After potassium is released into the cell, the enzyme binds three sodium ions, which starts the process over again.
Key Terms
- electrogenic pump : An ion pump that generates a net charge flow as a result of its activity.
- Na+-K+ ATPase : An enzyme located in the plasma membrane of all animal cells that pumps sodium out of cells while pumping potassium into cells.
Primary Active Transport
The primary active transport that functions with the active transport of sodium and potassium allows secondary active transport to occur. The secondary transport method is still considered active because it depends on the use of energy as does primary transport.
Active Transport of Sodium and Potassium : Primary active transport moves ions across a membrane, creating an electrochemical gradient (electrogenic transport).
One of the most important pumps in animals cells is the sodium-potassium pump ( Na + -K + ATPase ), which maintains the electrochemical gradient (and the correct concentrations of Na + and K + ) in living cells. The sodium-potassium pump moves two K + into the cell while moving three Na + out of the cell. The Na+-K+ ATPase exists in two forms, depending on its orientation to the interior or exterior of the cell and its affinity for either sodium or potassium ions. The process consists of the following six steps:
- With the enzyme oriented towards the interior of the cell, the carrier has a high affinity for sodium ions. Three sodium ions bind to the protein.
- ATP is hydrolyzed by the protein carrier, and a low-energy phosphate group attaches to it.
- As a result, the carrier changes shape and re-orients itself towards the exterior of the membrane. The protein’s affinity for sodium decreases, and the three sodium ions leave the carrier.
- The shape change increases the carrier’s affinity for potassium ions, and two such ions attach to the protein. Subsequently, the low-energy phosphate group detaches from the carrier.
- With the phosphate group removed and potassium ions attached, the carrier protein repositions itself towards the interior of the cell.
- The carrier protein, in its new configuration, has a decreased affinity for potassium, and the two ions are released into the cytoplasm. The protein now has a higher affinity for sodium ions, and the process starts again.
Several things have happened as a result of this process. At this point, there are more sodium ions outside of the cell than inside and more potassium ions inside than out. For every three ions of sodium that move out, two ions of potassium move in. This results in the interior being slightly more negative relative to the exterior. This difference in charge is important in creating the conditions necessary for the secondary process. The sodium-potassium pump is, therefore, an electrogenic pump (a pump that creates a charge imbalance), creating an electrical imbalance across the membrane and contributing to the membrane potential.


5.4B: Electron Donors and Acceptors
- Recognize the various types of electron donors and acceptors
In prokaryotes (bacteria and archaea) there are several different electron donors and several different electron acceptors. Note that electrons can enter the chain at three levels: at the level of a dehydrogenase, at the level of the quinone pool, or at the level of a mobile cytochrome electron carrier. These levels correspond to successively more positive redox potentials, or to successively decreased potential differences relative to the terminal electron acceptor. In other words, they correspond to successively smaller Gibbs free energy changes for the overall redox reaction Donor → Acceptor .
Individual bacteria use multiple electron transport chains, often simultaneously. Bacteria can use a number of different electron donors, a number of different dehydrogenases, a number of different oxidases and reductases, and a number of different electron acceptors. For example, E. coli (when growing aerobically using glucose as an energy source) uses two different NADH dehydrogenases and two different quinol oxidases, for a total of four different electron transport chains operating simultaneously.
A common feature of all electron transport chains is the presence of a proton pump to create a transmembrane proton gradient. Bacterial electron transport chains may contain as many as three proton pumps, like mitochondria, or they may contain only one or two. They always contain at least one proton pump.
In the present day biosphere, the most common electron donors are organic molecules. Organisms that use organic molecules as an energy source are called organotrophs . Organotrophs (animals, fungi, protists) and phototrophs (plants and algae) constitute the vast majority of all familiar life forms.
Some prokaryotes can use inorganic matter as an energy source. Such organisms are called lithotrophs (“rock-eaters”). Inorganic electron donors include hydrogen, carbon monoxide, ammonia, nitrite, sulfur, sulfide, and ferrous iron. Lithotrophs have been found growing in rock formations thousands of meters below the surface of Earth. Because of their volume of distribution, lithotrophs may actually out number organotrophs and phototrophs in our biosphere.
The use of inorganic electron donors as an energy source is of particular interest in the study of evolution. This type of metabolism must logically have preceded the use of organic molecules as an energy source.
Just as there are a number of different electron donors (organic matter in organotrophs, inorganic matter in lithotrophs), there are a number of different electron acceptors, both organic and inorganic. If oxygen is available, it is invariably used as the terminal electron acceptor, because it generates the greatest Gibbs free energy change and produces the most energy.
In anaerobic environments, different electron acceptors are used, including nitrate, nitrite, ferric iron, sulfate, carbon dioxide, and small organic molecules such as fumarate.
Since electron transport chains are redox processes, they can be described as the sum of two redox pairs. For example, the mitochondrial electron transport chain can be described as the sum of the NAD + /NADH redox pair and the O 2 /H 2 O redox pair. NADH is the electron donor and O 2 is the electron acceptor.
Not every donor-acceptor combination is thermodynamically possible. The redox potential of the acceptor must be more positive than the redox potential of the donor. Furthermore, actual environmental conditions may be far different from standard conditions (1 molar concentrations, 1 atm partial pressures, pH = 7), which apply to standard redox potentials. For example, hydrogen-evolving bacteria grow at an ambient partial pressure of hydrogen gas of 10 -4 atm. The associated redox reaction, which is thermodynamically favorable in nature, is thermodynamically impossible under “standard” conditions.
Bacterial electron transport pathways are, in general, inducible. Depending on their environment, bacteria can synthesize different transmembrane complexes and produce different electron transport chains in their cell membranes. Bacteria select their electron transport chains from a DNA library containing multiple possible dehydrogenases, terminal oxidases and terminal reductases. The situation is often summarized by saying that electron transport chains in bacteria are branched , modular , and inducible .
Key Points
- Bacterial electron transport chains may contain as many as three proton pumps.
- The most common electron donors are organic molecules.
- There are a number of different electron acceptors, both organic and inorganic. If oxygen is available, it is invariably used as the terminal electron acceptor.
Key Terms
- organotroph : An organism that obtains its energy from organic compounds.
- lithotroph : An organism that obtains its energy from inorganic compounds (such as ammonia) via electron transfer.


8.1.11: Form - Terminology and Basic Concepts
This document is a glossary of terms and concepts that we will use in our analysis of pop/rock music. Terms, concepts, definitions, and notational guidelines in this document are taken either from common convention; the published or unpublished work of Jason Summach, John Covach, Walter Everett, Mark Spicer, or Daniel Harrison; or some combination thereof.
Poetic terminology
Foot
A combination of two or three syllables: typically one stressed syllable, and one or two unstressed syllables. Common poetic feet include the iamb (unstressed – stressed), the trochee (stressed – unstressed), the spondee (two stressed syllables in a row), the anapest (stressed – stressed – unstressed), and the dactyl (stressed – unstressed – unstressed.)
Line
A group of poetic feet functioning as a single unit. Each poetic line is given its own line as printed/written text. If the line is part of rhyming poetry, the last syllable/foot participates in a rhyme with another line. (Internal rhyme is also possible.)
Couplet
A pair of lines. If poetry is rhyming, the two lines making up a couplet typically rhyme with each other. They may also participate in a larger rhyme scheme (see quatrain below).
Quatrain
A pair of couplets (i.e., four lines). Common quatrain rhyme schemes are aabb , abab , abcb .
Stanza
A set of poetic lines that work together as a single narrative unit. Typically one or more quatrains (i.e., total number of lines are a multiple of four).
Example
Following is a stanza from U2, “Pride (In the Name of Love).” This stanza is a quatrain composed of four lines. The first and third lines have four feet (that is, four stressed syllables) with irregular rhythm. The second and fourth lines have three feet with irregular rhythm, and they rhyme with each other.
One man come in the name of love.
One man come and go.
One man come, he to justify.
One man to overthrow.
Other structures
For more poetic structures and terms, see Vanier College’s resource, Poetry’s Structure and Form .
Descriptors & rhetorical devices
Lyric-variant
A module or phrase is lyric-variant if each time it appears it brings (mostly) different lyrics.
Lyric-invariant
A module or phrase is lyric-invariant if each time it appears it brings (mostly) the same lyrics. Lyric invariance tends to come at points of formal closure (tail refrains at the ends of strophes, choruses at the end of a verse-chorus song’s formal cycle).
Music-variant
A module or phrase is music-variant if each time it appears it brings (mostly) different music.
Music-invariant
A module or phrase is music-invariant if each time it appears it brings (mostly) the same music.
Chorusification
Jay Summach uses the term “chorusification” (p. 321) to describe a process where modules are stripped away from the formal cycle until only the chorus module (C) remains. For example, a song that begins with the cycle {VPC} may appear near the end of the song without the verse {PC} and then again without the prechorus {C}. This process is part of a goal-directed progression toward the end of the song, giving special emphasis to the chorus.
Bon Jovi’s “Livin’ on a Prayer” is a good example of chorusification. The first two cycles are similar in length and content, but the third cycle (beginning at 2:58) replaces the verse with an instrumental chorus, giving the third cycle two choruses. After this cycle closes, the chorus repeats forming a chorus-based outro.
HTML link:
https://open.spotify.com/track/0J6mQxEZnlRt9ymzFntA6z
Spotify URI:
spotify:track:0J6mQxEZnlRt9ymzFntA6z
Harmonic aspects of form
On-tonic
A phrase or module is on-tonic when it begins with tonic harmony (I in root position).
Off-tonic
A phrase or module is off-tonic when it begins on a harmony other than tonic.
Harmonically closed
A phrase or module is harmonically closed when it ends with tonic harmony (I in root position).
Harmonically open
A phrase or module is harmonically open when it ends on a harmony other than tonic.
Turnaround
The use of a non-tonic chord (usually dominant) at the end of a harmonically closed unit to transition into the beginning of the following on-tonic unit.
The song “Wooly Bully” by Sam the Sham and the Pharoahs contains a turnaround at the end of many of its strophes. One of these occur at 0:54 ― a simple V 7 chord to prepare the return of I as the next strophe begins.
HTML link:
https://open.spotify.com/track/2XkuSbp5say8nZW8g6156Z
Spotify URI:
spotify:track:2XkuSbp5say8nZW8g6156Z
Interestingly in this song, the guitarist doesn’t always remember the turnarounds. Notice that at 0:28 the bass and baritone saxophone play the dominant, but the guitarist keeps tonic. At 1:18, the singer yells, “Watch it now! Watch it! Watch it!” as if warning the guitarist not to miss the turnaround in the next bar. He does the same in 2:08. When the guitarist gets the turnaround with the rest of the band, the singer yells, “You got it! You got it!” as if congratulating the guitarist.


3: Distributions
Recall that a probability distribution is just another name for a probability measure . Most distributions are associated with random variables , and in fact every distribution can be associated with a random variable. In this chapter we explore the basic types of probability distributions (discrete, continuous, mixed), and the ways that distributions can be defined using density functions, distribution functions, and quantile functions. We also study the relationship between the distribution of a random vector and the distributions of its components, conditional distributions, and how the distribution of a random variable changes when the variable is transformed.
In the advanced sections, we study convergence in distribution, one of the most important types of convergence. We also construct the abstract integral with respect to a positive measure and study the basic properties of the integral. This leads in turn to general (signed measures), absolute continuity and singularity, and the existence of density functions. Finally, we study various vector spaces of functions that are defined by integral pro
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- 3.3: Mixed Distributions
- In the previous two sections, we studied discrete probability meausres and continuous probability measures. In this section, we will study probability measure that that are combinations of the two types. Once again, if you are a new student of probability you may want to skip the technical details.
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- 3.5: Conditional Distributions
- In this section, we study how a probability distribution changes when a given random variable has a known, specified value. So this is an essential topic that deals with hou probability measures should be updated in light of new information. As usual, if you are a new student or probability, you may want to skip the technical details.


27.9.1: Encapsulation
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Overview
Encapsulation is one of the fundamentals of OOP (object-oriented programming). It refers to the bundling of data with the methods that operate on that data. Encapsulation is used to hide the values or state of a structured data object inside a class, preventing unauthorized parties’ direct access to them. Publicly accessible methods are generally provided in the class (so-called getters and setters) to access the values, and other client classes call these methods to retrieve and modify the values within the object. [1]
Discussion
The most important principle of object orientation is encapsulation : the idea that data inside the object should only be accessed through a public interface – that is, the object’s methods.
If we want to use the data stored in an object to perform an action or calculate a derived value, we define a method associated with the object which does this. Then whenever we want to perform this action we call the method on the object. We consider it bad practice to retrieve the information from inside the object and write separate code to perform the action outside of the object.
Encapsulation is a good idea for several reasons:
- the functionality is defined in one place and not in multiple places.
- it is defined in a logical place – the place where the data is kept.
- data inside our object is not modified unexpectedly by external code in a completely different part of our program.
- when we use a method, we only need to know what result the method will produce – we don’t need to know details about the object’s internals in order to use it. We could switch to using another object which is completely different on the inside, and not have to change any code because both objects have the same interface.
We can say that the object “knows how” to do things with its own data, and it’s a bad idea for us to access its internals and do things with the data ourselves. If an object doesn’t have an interface method which does what we want to do, we should add a new method or update an existing one.
Some languages have features which allow us to enforce encapsulation strictly. In Java or C++, we can define access permissions on object attributes, and make it illegal for them to be accessed from outside the object’s methods. In Java it is also considered good practice to write setters and getters for all attributes, even if the getter simply retrieves the attribute and the setter just assigns it the value of the parameter which you pass in.
In Python, encapsulation is not enforced by the language, but there is a convention that we can use to indicate that a property is intended to be private and is not part of the object’s public interface: we begin its name with an underscore. Python also supports the use of a property decorator to replace a simple attribute with a method without changing the object’s interface.
Key Terms
- abstraction
- A technique for arranging complexity of computer systems so that functionality may be separated from specific implementation details. [2] [3]
- accessor
- A method used to return the value of a private member variable, also known as a getter method. [4]
- encapsulation
- A language mechanism for restricting direct access to some of an object’s components. [5]
- information hiding
- The principle of segregation of the design decisions in a computer program from other parts of the program. See encapsulation. [6]
- mutator
- A method used to control changes to a private member variable, also known as a setter method. [7]
- private
- An access modifier that restricts visibility of a property or method to the class in which it is defined. [8]
- public
- An access modifier that opens visibility of a property or method to all other classes. [9]
References
- Read the Docs: Object-Oriented Programming in Python
- Wikiversity: Computer Programming
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Wikipedia: Encapsulation (computer programming)
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Wikipedia: Object-oriented programming
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Wikipedia: Abstraction (computer science)
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Wikipedia: Mutator method
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Wikipedia: Encapsulation (computer programming)
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Wikipedia: Information hiding
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Wikipedia: Mutator method
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Wikipedia: Access modifiers
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Wikipedia: Access modifiers
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What you’ll learn to do: describe the physical changes that occur during puberty and adolescence
Physical changes of puberty mark the onset of adolescence (Lerner & Steinberg, 2009). For both boys and girls, these changes include a growth spurt in height, growth of pubic and underarm hair, and skin changes (e.g., pimples). Boys also experience growth in facial hair and a deepening of their voice. Girls experience breast development and begin menstruating. These pubertal changes are driven by hormones, particularly an increase in testosterone for boys and estrogen for girls. The physical changes that occur during adolescence are greater than those of any other time of life, with the exception of infancy. In some ways, however, the changes in adolescence are more dramatic than those that occur in infancy—unlike infants, adolescents are aware of the changes that are taking place and of what the changes mean. In this section, you will learn about the pubertal changes in body size, proportions, and sexual maturity, the social and emotional attitudes and reactions toward puberty, and some of the health concerns during adolescence, including eating disorders.
Contributors and Attributions
CC licensed content, Original
CC licensed content, Shared previously


17.3.4: Anaerobic Respiration (keep but move to 9.2?)
Conditions without oxygen are referred to as anaerobic . In this case, the pyruvate will be converted to lactate in the cytoplasm of the cell as shown below.
Figure <IP_ADDRESS>.1 Pyruvate fork in the road, what happens depends on whether it is aerobic or anaerobic respiration 1
What happens if oxygen isn't available to serve as the final electron acceptor? As shown in the following video, the ETC becomes backed up with electrons and can't accept them from NADH and FADH 2 .
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Web Link Video: What happens when you run out of oxygen? (0:37) |
This leads to a problem in glycolysis because NAD is needed to accept electrons, as shown below. Without the electron transport chain functioning, all NAD has been reduced to NADH and glycolysis cannot continue to produce ATP from glucose.
Figure <IP_ADDRESS>.2 Why NAD needs to be regenerated under anaerobic conditions 2
Thus, there is a workaround to regenerate NAD by converting pyruvate (pyruvic acid) to lactate (lactic acid) as shown below.
Figure <IP_ADDRESS>.1 3. The conversion of pyruvic acid to lactic acid regenerates NAD 3,4
However, anaerobic respiration only produces 2 ATP per molecule of glucose, compared to 32 ATP for aerobic respiration. The biggest producer of lactate is the muscle. Through what is known as the Cori cycle, lactate produced in the muscle can be sent to the liver. In the liver, through a process known as gluconeogenesis, glucose can be regenerated and sent back to the muscle to be used again for anaerobic respiration forming a cycle as shown below.
Figure 9 .2.2.2. 4 The Cori cycle 5
It is worth noting that the Cori cycle also functions during times of limited glucose (like fasting) to spare glucose by not completely oxidizing it.
References & Links
- simple.Wikipedia.org/wiki/Mi..._en_(edit).svg
- en.Wikipedia.org/wiki/File:CellRespiration.svg
- en.Wikipedia.org/wiki/Pyruvi...D-skeletal.png
- en.Wikipedia.org/wiki/Lactic...d-skeletal.svg
- commons.wikimedia.org/wiki/F...iCycle-eng.svg


2.7: Linear Inequalities and Absolute Value Inequalities
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Learning Objectives
- Use interval notation.
- Use properties of inequalities.
- Solve inequalities in one variable algebraically.
- Solve absolute value inequalities.
It is not easy to make the honor role at most top universities. Suppose students were required to carry a course load of at least \(12\) credit hours and maintain a grade point average of \(3.5\) or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.
Using Interval Notation
Indicating the solution to an inequality such as \(x≥4\) can be achieved in several ways.
- We can use a number line as shown in Figure \(\PageIndex{2}\). The blue ray begins at \(x = 4\) and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to \(4\).
- We can use set-builder notation : \(\{x|x≥4\}\), which translates to “all real numbers \(x\) such that \(x\) is greater than or equal to \(4\).” Notice that braces are used to indicate a set.
- The third method is interval notation , in which solution sets are indicated with parentheses or brackets. The solutions to \(x≥4\) are represented as \([4,\infty)\). This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are \([−2,6)\), or all numbers between \(−2\) and \(6\), including \(−2\), but not including \(6\); \((−1,0)\), all real numbers between, but not including \(−1\) and \(0\); and \((−\infty,1]\), all real numbers less than and including \(1\). Table \(\PageIndex{1}\) outlines the possibilities.
| Set Indicated | Set-Builder Notation | Interval Notation |
|---|---|---|
| All real numbers between \(a\) and \(b\), but not including \(a\) or \(b\) | \(\{x|a<x<b\}\) | \((a,b)\) |
| All real numbers greater than \(a\), but not including \(a\) | \(\{x|x>a\}\) | \((a,\infty)\) |
| All real numbers less than \(b\), but not including \(b\) | \(\{x|x<b\}\) | \((−\infty,b)\) |
| All real numbers greater than \(a\), including \(a\) | \(\{x|x≥a\}\) | \([a,\infty)\) |
| All real numbers less than \(b\), including \(b\) | \(\{x|x≤b\}\) | \((−\infty,b]\) |
| All real numbers between \(a\) and \(b\), including \(a\) | \(\{x|a≤x<b\}\) | \([a,b)\) |
| All real numbers between \(a\) and \(b\), including \(b\) | \(\{x|a<x≤b\}\) | \((a,b]\) |
| All real numbers between \(a\) and \(b\), including \(a\) and \(b\) | \(\{x|a≤x≤b\}\) | \([a,b]\) |
| All real numbers less than \(a\) or greater than \(b\) | \(\{x|x<a\space and\space x>b\}\) | \((−\infty,a)\cup(b,\infty)\) |
| All real numbers | \(\{x|x\space is\space all\space real\space numbers\}\) | \((−\infty,\infty)\) |
Example \(\PageIndex{1}\): Using Interval Notation to Express All Real Numbers Greater Than or Equal to a
Use interval notation to indicate all real numbers greater than or equal to \(−2\).
Solution
Use a bracket on the left of \(−2\) and parentheses after infinity: \([−2,\infty)\). The bracket indicates that \(−2\) is included in the set with all real numbers greater than \(−2\) to infinity.
Try It! \(\PageIndex{1}\)
Use interval notation to indicate all real numbers between and including \(−3\) and \(5\).
- Answer
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\([−3,5]\)
Example \(\PageIndex{2}\): Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b
Write the interval expressing all real numbers less than or equal to \(−1\) or greater than or equal to \(1\).
Solution
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to \(1\). So, this interval begins at \(−\infty\) and ends at \(−1\), which is written as \((−\infty,−1]\).
The second interval must show all real numbers greater than or equal to \(1\), which is written as \([1,\infty)\). However, we want to combine these two sets. We accomplish this by inserting the union symbol, ∪, between the two intervals.
\[(−\infty,−1]\cup[1,\infty) \nonumber\]
Try It! \(\PageIndex{2}\)
Express all real numbers less than \(−2\) or greater than or equal to \(3\) in interval notation.
- Answer
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\((−\infty,−2)\cup[3,\infty)\)
Using the Properties of Inequalities
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
PROPERTIES OF INEQUALITIES
Addition Property
- If \(a<b\), then \(a+c<b+c\).
Multiplication Property
- If \(a<b\) and \(c>0\), then \(ac<bc\).
- If \(a<b\) and \(c<0\), then \(ac>bc\).
These properties also apply to \(a≤b\), \(a>b\), and \(a≥b\).
Example \(\PageIndex{3}\): Demonstrating the Addition Property
Illustrate the addition property for inequalities by solving each of the following:
- (a) \(x−15<4\)
- (b) \(6≥x−1\)
- (c) \(x+7>9\)
Solution
The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
a.
\[\begin{align*} x-15&< 4\\ x-15+15&< 4+15\\ x&< 19 \end{align*}\]
b.
\[\begin{align*} 6&\geq x-1\\ 6+1&\geq x-1+1\\ 7&\geq x \end{align*}\]
c.
\[\begin{align*} x+7&> 9\\ x+7-7&> 9-7\\ x&> 2 \end{align*}\]
Try It! \(\PageIndex{3}\)
Solve: \(3x−2<1\).
- Answer
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\(x<1\)
Example \(\PageIndex{4}\): Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:
- \(3x<6\)
- \(−2x−1≥5\)
- \(5−x>10\)
Solution
a.
\[\begin{align*} 3x&< 6\\ \dfrac{1}{3}(3x)&< (6)\dfrac{1}{3}\\ x&< 2 \end{align*}\]
b.
\[\begin{align*} -2x-1&\geq 5\\ -2x&\geq 6\\ \left (-\dfrac{1}{2} \right )(-2)&\geq (6)\left (-\dfrac{1}{2} \right ) & & \text{Multiply by } \left (-\dfrac{1}{2} \right )\\ x&\leq -3 & & \text{Reverse the inequality.} \end{align*}\]
c.
\[\begin{align*} 5-x&> 10\\ -x&> 5\\ (-1)(-x)&> (5)(-1) & & \text{Multiply by } -1\\ x&< -5 & & \text{Reverse the inequality.} \end{align*}\]
Try It! \(\PageIndex{4}\)
Solve: \(4x+7≥2x−3\).
- Answer
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\(x≥−5\)
Solving Inequalities in One Variable Algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.
Example \(\PageIndex{5}\): Solving an Inequality Algebraically
Solve the inequality: \(13−7x≥10x−4\).
Solution
Solving this inequality is similar to solving an equation up until the last step.
\[\begin{align*} 13-7x&\geq 10x-4\\ 13-17x&\geq -4 & & \text{Move variable terms to one side of the inequality}\\ -17x&\geq -17 & & \text{Isolate the variable term}\\ x&\leq 1 & & \text{Dividing both sides by -17 reverses the inequality.} \end{align*}\]
The solution set is given by the interval \((−\infty,1]\), or all real numbers less than and including \(1\).
Try It! \(\PageIndex{5}\)
Solve the inequality and write the answer using interval notation: \(−x+4<\dfrac{1}{2}x+1\).
- Answer
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\((2,\infty)\)
Example \(\PageIndex{6}\): Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation: \(−\dfrac{3}{4}x≥−\dfrac{5}{8}+\dfrac{2}{3}x\).
Solution
We begin solving in the same way we do when solving an equation.
\[\begin{align*} -\dfrac{3}{4}x&\geq -\dfrac{5}{8}+\dfrac{2}{3}x\\[5pt] -\dfrac{3}{4}x-\dfrac{2}{3}x&\geq -\dfrac{5}{8} & & \text{Put variable terms on one side.}\\[5pt] -\dfrac{9}{12}x-\dfrac{8}{12}x&\geq -\dfrac{5}{8} & & \text{Write fractions with common denominator.}\\[5pt] -\dfrac{17}{12}x&\geq -\dfrac{5}{8}\\[5pt] x&\leq -\dfrac{5}{8}\left (-\dfrac{12}{17} \right ) & & \text{Multiplying by a negative number reverses the inequality.}\\[5pt] x&\leq \dfrac{15}{34} \end{align*}\]
The solution set is the interval \(\left (−\infty,\dfrac{15}{34} \right ]\).
Try It! \(\PageIndex{6}\)
Solve the inequality and write the answer in interval notation: \(−\dfrac{5}{6}x≤\dfrac{3}{4}+\dfrac{8}{3}x\).
- Answer
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\(\left [-\dfrac{3}{14},\infty \right )\)
Understanding Compound Inequalities
A compound inequality includes two inequalities in one statement. A statement such as \(4<x≤6\) means \(4<x\) and \(x≤6.\) There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.
Example \(\PageIndex{7}\): Solving a Compound Inequality
Solve the compound inequality: \(3≤2x+2<6\)
Solution
The first method is to write two separate inequalities: \(3≤2x+2\) and \(2x+2<6.\) We solve them independently.
\[\begin{array}{lcr}
3≤2x+2 & \text{and} & 2x+2<6 \\[5pt]
1≤2x & & 2x<4 \\[5pt]
\tfrac{1}{2}\leq x & & x < 2
\end{array} \nonumber\]
Then, we can rewrite the solution as a compound inequality, the same way the problem began.
\[\frac{1}{2}≤x<2\nonumber\]
In interval notation, the solution is written as \([\tfrac{1}{2},\, 2).\)
The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.
\(\begin{align*} \qquad 3&≤2x+2<6 \\[5pt]
1&≤2x<4 & & \text{Isolate the variable term, and subtract }2\text{ from all three parts.} \\[5pt]
\tfrac{1}{2}&≤x<2 & & \text{Divide through all three parts by }2.
\end{align*}\)
We get the same solution: \([\tfrac{1}{2},\, 2).\)
Try It! \(\PageIndex{7}\)
Solve the compound inequality: \(4<2x−8≤10.\)
- Answer
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\(6<x≤9\) or \((6,9]\)
Example \(\PageIndex{8}\): Solving a Compound Inequality with the Variable in All Three Parts
Solve the compound inequality with variables in all three parts: \(3+x>7x−2>5x−10.\)
Solution
Let's try the first method. Write two inequalities:
\[\begin{array}{ccc} 3+x>7x−2 & \text{and} & 7x−2>5x−10\\[5pt]
3>6x−2 & & 2x−2>−10\\[5pt]
5>6x & & 2x>−8 \\[5pt]
\tfrac{5}{6}>x & & -4<x \\[5pt]
\end{array} \nonumber\]
The solution set is \(−4<𝑥<\tfrac{5}{6}\) or in interval notation \(\left(−4,\tfrac{5}{6}\right).\) Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See Figure \(\PageIndex{3}\).
Try It! \(\PageIndex{8}\)
Solve the compound inequality: \(3y<4−5y<5+3y.\)
- Answer
-
\(\left(−\tfrac{1}{8},\, \tfrac{1}{2}\right)\)
Solving Absolute Value Inequalities
As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at \((−x,\, 0)\) has an absolute value of \(x,\) as it is \(x\) units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.
An absolute value inequality is an equation of the form
\[|A|<B,\quad |A|≤B, \quad |A|>B,\quad \text{or} \quad|A|≥B,\nonumber\]
Where \(A,\) and sometimes \(B,\) represents an algebraic expression dependent on a variable \(x.\) Solving the inequality means finding the set of all \(x\) - values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of \(x\) - values such that the distance between\(x\) and \(600\) is less than or equal to \(200.\) We represent the distance between \(x\) and \(600\) as \(| x−600 |,\) and therefore, \(| x−600 |≤200\) or
\[\begin{array}{c} −200≤x−600≤200 \\
−200+600≤x−600+600≤200+600 \\
400≤x≤800
\end{array} \nonumber\]
This means our returns would be between $400 and $800.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.
Definition: Absolute Value Inequalities
For an algebraic expression \(X\) and \(k>0,\) an absolute value inequality is an inequality of the form
\[\begin{align*} | X |&<k \text{ is equivalent to }−k<X<k\\[4pt]
| X |&>k \text{ is equivalent to }X<−k \text{ or }X>k\end{align*}\]
These statements also apply to \(| X |≤k\) and \(| X |≥k.\)
Example \(\PageIndex{9}\): Determining a Number within a Prescribed Distance
Describe all values \(x\) within a distance of \(4\) from the number \(5.\)
Solution
We want the distance between \(x\) and \(5\) to be less than or equal to \(4.\) We can draw a number line, such as in Figure \(\PageIndex{4},\) to represent the condition to be satisfied.
The distance from \(x\) to \(5\) can be represented using an absolute value symbol, \(| x−5 |.\) Write the values of \(x\) that satisfy the condition as an absolute value inequality.
\[| x−5 |≤4\nonumber\]
We need to write two inequalities as there are always two solutions to an absolute value equation.
\[\begin{array}{ccc} x−5≤4 & \text{and} & x−5≥−4 \\[4pt]
x≤9 & & x≥1
\end{array} \nonumber\]
If the solution set is \(x≤9\) and \(x≥1,\) then the solution set is an interval including all real numbers between and including 1 and 9.
So \(| x−5 |≤4\) is equivalent to \([ 1,9 ]\) in interval notation.
Try It! \(\PageIndex{9}\)
Describe all \(x\)-values within a distance of \(3\) from the number \(2.\)
- Answer
-
\(|x−2|≤3\)
Example \(\PageIndex{10}\): Solving an Absolute Value Inequality
Solve \(|x−1|≤3.\)
Solution
\(\quad \begin{array}{c} |x−1 |≤3 \\[4pt]
−3≤x−1≤3 \\[4pt]
−2≤x≤4 \\[4pt]
[−2,4]
\end{array} \)
Example \(\PageIndex{11}\): Using a Graphical Approach to Solve Absolute Value Inequalities
Given the equation \(y=−\tfrac{1}{2}|4x−5|+3,\) determine the \(x\)-values for which the \(y\)-values are negative.
Solution
We are trying to determine where \(y<0,\) which is when \(−\tfrac{1}{2}|4x−5|+3<0.\) We begin by isolating the absolute value.
\[\begin{align*} −\tfrac{1}{2}|4x−5|&<−3 & & \text{Multiply both sides by }–2,\text{ and reverse the inequality.}\\[5pt]
|4x−5|&>6 \end{align*} \]
Next, we solve for the equality \(|4x−5|=6.\)
\[\begin{array}{rcr} 4x−5=6 & & 4x−5=−6\\[5pt]
4x=11 & \text{or} & 4x=−1\\[5pt]
x=\tfrac{11}{4} & & x=−\tfrac{1}{4}
\end{array} \nonumber\]
Now, we can examine the graph to observe where the \(y\)-values are negative. We observe where the branches are below the \(x\)-axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at \(x=−\tfrac{1}{4}\) and \(x=\tfrac{11}{4},\) and that the graph opens downward. See Figure \(\PageIndex{5}.\)
Try It! \(\PageIndex{10}\)
Solve \(−2|k−4|≤−6.\)
- Answer
-
\(k≤1\) or \(k≥7;\) in interval notation, this would be \((−∞,1]∪[7,∞).\)


What Should the Nurse Do?
Marta, a 38-year-old mother of two, presents at a community health clinic with concerns about persistent fatigue, irregular menstrual cycles, and difficulty in concentrating. Diagnosed with polycystic ovary syndrome (PCOS) in her early 20s, Marta, a low-income cashier, resides in a neighborhood near the poverty level. Juggling work and parenting, she faces financial constraints that impact her ability to access nutritious food. Her medical history includes a struggle with managing PCOS. Marta, with no psychiatric history, now seeks health-care support as her symptoms intensify. Vital signs reveal an elevated blood pressure of 140/90 mm Hg, a heart rate of 82 beats per minute, and a body mass index (BMI) of 31.6.
1 .
How do the unique challenges faced by Marta, such as financial constraints and difficulty in accessing nutritious food, align with the factors affecting the health of persons assigned female at birth, as discussed in the chapter?
2 .
In Marta’s case, how might the intersection of social determinants, such as low income, juggling work and parenting, and residing in a neighborhood near the poverty level, contribute to her elevated blood pressure and BMI, as revealed in the vital signs?
3 .
How can health-care professionals, particularly nurses, utilize the strategies discussed in the chapter, such as conducting health screenings and engaging in patient interviews, to assess and address the social determinants influencing Marta’s health?
Tricia is a 52-year-old female seeking a routine checkup from her primary care physician. Beyond her immediate concerns of persistent fatigue and occasional shortness of breath, Tricia reveals a family history of cardiovascular disease, adding a layer of complexity to her health profile. Her medical history includes hypertension, hyperlipidemia, and obesity, all of which contribute to her increased risk for cardiovascular issues. Despite having no recorded psychiatric history, Tricia discloses occasional stress and anxiety, factors that can further exacerbate her cardiovascular risks. Vital signs are blood pressure of 140/90 mm Hg, heart rate of 76 beats per minute, and BMI of 32.5.
4 .
Considering Tricia’s health profile and family history, how do global leading causes of morbidity and mortality among persons assigned female at birth, such as cardiovascular diseases, cancer, and infectious diseases, align with her increased risk for cardiovascular issues?
5 .
According to the information provided in the chapter, what are the leading causes of death for persons assigned female at birth in the United States, and how do these align with Tricia’s health concerns, particularly her hypertension, hyperlipidemia, and obesity?
6 .
Given Tricia’s health history, how might health screening, as discussed in the chapter, aid in early detection and intervention for her cardiovascular risks? What specific screening tools could be employed in her case?
Sarah, a 16-year-old high school student, presents to a women's health clinic for a routine checkup and consultation regarding her reproductive health. She shares concerns about irregular menstrual cycles, reporting that her periods are often unpredictable and accompanied by severe cramps. Sarah, whose family has no known history of reproductive health issues, is otherwise healthy and actively engaged in extracurricular activities.  Sarah's medical history is unremarkable, with no chronic conditions or prior surgeries. However, she discloses feelings of anxiety related to her menstrual symptoms, indicating a potential psychosocial aspect to her health concerns. Vital signs reveal a blood pressure of 118/70 mm Hg, a heart rate of 72 beats per minute, and a BMI of 21.3. Physical examination findings are generally within normal limits, with no signs of abnormal hair growth, acne, or excessive weight gain. Menstrual history suggests that Sarah began menstruating at the age of 12 and has experienced irregular cycles since then. She denies any history of sexual activity and uses no form of contraception.
7 .
What reproductive system structures and functions specific to persons assigned female at birth could be contributing to Sarah's irregular menstrual cycles and severe cramps?
8 .
Based on the information provided, what specific self-care measures related to menstrual hygiene and reproductive health should be emphasized in patient education for Sarah?
9 .
Considering the potential psychosocial aspect of Sarah's menstrual concerns, how might nurses address the emotional distress she experiences and its impact on academic and social aspects of her life?
Julia, a 28-year-old female, arrives at a community health clinic for her annual well-woman examination, seeking holistic care for her various health concerns.  She describes the pain as sharp and intermittent, often interfering with her daily activities. Additionally, Julia expresses concerns about irregular menstrual cycles, detailing heavy flow and extended durations.  Julia's medical history is notable for a diagnosis of polycystic ovary syndrome (PCOS) in her early 20s, prompting a discussion about its impact on her reproductive health and overall well-being. Furthermore, she shares a family history of breast cancer, adding a layer of complexity to her risk assessment. The discussion extends to Julia's mental health, as she discloses a history of anxiety and depression. She manages these concerns with occasional counseling sessions, emphasizing the interconnected nature of physical and mental health.
10 .
Considering Julia's health history, what are the potential risk factors for her reproductive health, and how might her diagnosis of polycystic ovary syndrome (PCOS) and family history of breast cancer impact her overall well-being?
11 .
As a nurse providing holistic care for Julia, what specific nursing actions can be taken to promote well care, considering her concerns about pelvic pain, irregular menstrual cycles, and the impact on her mental health?
12 .
How can trauma-informed care principles be applied in Julia's case, especially considering her history of anxiety, depression, and the potential impact of pelvic pain on her emotional well-being?
Courtney, a 30-year-old female, is planning to become pregnant and visits a reproductive health clinic for a comprehensive preconception consultation. Courtney, alongside her supportive partner, expresses a strong desire to ensure the best possible start for their future family. In the initial assessment, Courtney candidly discusses her reproductive health concerns, noting irregular menstrual cycles that have sparked worries about her ability to conceive. Courtney’s medical history indicates that she manages hypothyroidism, a condition she has dealt with since her early 20s. Her diligent adherence to prescribed medication has kept the thyroid disorder well-controlled. Courtney also discloses a history of anxiety.
13 .
Considering Courtney's reproductive health concerns, hypothyroidism, and anxiety history, what preconception risk factors should be carefully analyzed, and how might these factors impact her ability to conceive and have a healthy pregnancy?
14 .
Explain the importance of the preconception visit for Courtney and her partner, considering her medical history of hypothyroidism and anxiety. How can the preconception visit address potential challenges and optimize their readiness for a healthy pregnancy?
15 .
How can the nurse educate Courtney and her partner on fertility awareness methods, considering her irregular menstrual cycles and hypothyroidism? What role do these methods play in optimizing their chances of conception?


8.3B: Distinguishing between Similar Traits
- Explain the difference between homologous and analogous structures
Two Options for Similarities
In general, organisms that share similar physical features and genomes tend to be more closely related than those that do not. Such features that overlap both morphologically (in form) and genetically are referred to as homologous structures; they stem from developmental similarities that are based on evolution. For example, the bones in the wings of bats and birds have homologous structures.
Notice it is not simply a single bone, but rather a grouping of several bones arranged in a similar way. The more complex the feature, the more probable that any overlap is due to a common evolutionary past. Imagine two people from different countries both inventing a car with all the same parts and in exactly the same arrangement without any previous or shared knowledge. That outcome would be highly improbable. However, if two people both invented a hammer, it would be reasonable to conclude that both could have the original idea without the help of the other. The same relationship between complexity and shared evolutionary history is true for homologous structures in organisms.
Misleading Appearances
Some organisms may be very closely related, even though a minor genetic change caused a major morphological difference to make them look quite different. Similarly, unrelated organisms may be distantly related, but appear very similar. This usually happens because both organisms developed common adaptations that evolved within similar environmental conditions. When similar characteristics occur because of environmental constraints and not due to a close evolutionary relationship, it is called an analogy or homoplasy. For example, insects use wings to fly like bats and birds, but the wing structure and embryonic origin is completely different. These are called analogous structures.
Similar traits can be either homologous or analogous. Homologous structures share a similar embryonic origin; analogous organs have a similar function. For example, the bones in the front flipper of a whale are homologous to the bones in the human arm. These structures are not analogous. The wings of a butterfly and the wings of a bird are analogous, but not homologous. Some structures are both analogous and homologous: the wings of a bird and the wings of a bat are both homologous and analogous. Scientists must determine which type of similarity a feature exhibits to decipher the phylogeny of the organisms being studied.
Molecular Comparisons
With the advancement of DNA technology, the area of molecular systematics, which describes the use of information on the molecular level including DNA analysis, has blossomed. New computer programs not only confirm many earlier classified organisms, but also uncover previously-made errors. As with physical characteristics, even the DNA sequence can be tricky to read in some cases. For some situations, two very closely-related organisms can appear unrelated if a mutation occurred that caused a shift in the genetic code. An insertion or deletion mutation would move each nucleotide base over one place, causing two similar codes to appear unrelated.
Sometimes two segments of DNA code in distantly-related organisms randomly share a high percentage of bases in the same locations, causing these organisms to appear closely related when they are not. For both of these situations, computer technologies have been developed to help identify the actual relationships. Ultimately, the coupled use of both morphologic and molecular information is more effective in determining phylogeny.
Key Points
- Organisms may be very closely related, even though they look quite different, due to a minor genetic change that caused a major morphological difference.
- Unrelated organisms may appear very similar because both organisms developed common adaptations that evolved within similar environmental conditions.
- To determine the phylogeny of an organism, scientists must determine whether a similarity is homologous or analogous.
- The advancement of DNA technology, the area of molecular systematics, describes the use of information on the molecular level, including DNA analysis.
Key Terms
- analogous : when similar similar physical features occur in organisms because of environmental constraints and not due to a close evolutionary relationship
- homologous : when similar physical features and genomes stem from developmental similarities that are based on evolution
- phylogeny : the evolutionary history of an organism
- molecular systematics : molecular phylogenetics is the analysis of hereditary molecular differences, mainly in DNA sequences, to gain information on an organism’s evolutionary relationships


3.8: Binary Molecules - Formulae and Names
- Identify covalent and ionic compounds.
- Determine the chemical formula of a simple covalent compound from its name.
- Determine the name of a simple covalent compound from its chemical formula.
COVALENT AND IONIC COMPOUNDS
What elements make covalent bonds? Covalent bonds form when two or more nonmetals combine. For example, both hydrogen and oxygen are nonmetals, and when they combine to make water, they do so by forming covalent bonds. Compounds that are composed of only non-metals or semi-metals with non-metals will display covalent bonding and will be classified as molecular compounds.
As a general rule of thumb, compounds that involve a metal binding with either a non-metal or a semi-metal will display ionic bonding. Thus, the compound formed from sodium and chlorine will be ionic (a metal and a non-metal). Nitrogen monoxide (NO) will be a covalently bound molecule (two non-metals), silicon dioxide (SiO 2 ) will be a covalently bound molecule (a semi-metal and a non-metal) and MgCl 2 will be ionic (a metal and a non-metal).
A polyatomic ion is an ion composed of two or more atoms that have a charge as a group (poly = many). The ammonium ion (see figure below) consists of one nitrogen atom and four hydrogen atoms. Together, they comprise a single ion with a 1+ charge and a formula of NH 4 + . The carbonate ion (see figure below) consists of one carbon atom and three oxygen atoms and carries an overall charge of 2−. The formula of the carbonate ion is CO 3 2 − .
The atoms of a polyatomic ion are tightly bonded together and so the entire ion behaves as a single unit. Nonmetal atoms in polyatomic ions are joined by covalent bonds, but the ion as a whole participates in ionic bonding. For example, ammonium chloride (NH 4 Cl) has ionic bonding between a polyatomic ion, \(\ce{NH_4^{+}}\), and \(\ce{Cl^{−}}\) ions, but within the ammonium ion (NH 4 + ), the nitrogen and hydrogen atoms are connected by covalent bonds (shown above).
Both ionic and covalent bonding are also found in calcium carbonate. Calcium carbonate (CaCO 3 ) has ionic bonding between calcium ion \(\ce{Ca^{2+}}\) and a polyatomic ion, \(\ce{CO_3^{2-}}\), but within the carbonate ion (CO 3 2 - ), the carbon and oxygen atoms are connected by covalent bonds (shown above).
Characteristics of Covalent (Molecular) Compounds
Compounds that contain covalent bonds (also called molecular compounds) exhibit different physical properties than ionic compounds. Because the attraction between molecules, which are electrically neutral, is weaker than that between electrically charged ions, covalent compounds generally have much lower melting and boiling points than ionic compounds. For example, water (molecular compound) boils at 100 °C while sodium chloride (ionic compound) boils at 1413 °C. In fact, many covalent compounds are liquids or gases at room temperature, and, in their solid states, they are typically much softer than ionic solids. Furthermore, whereas ionic compounds are good conductors of electricity when dissolved in water, most covalent compounds, being electrically neutral, are poor conductors of electricity in any state.
Is each compound formed from ionic bonds, covalent bonds, or both?
- \(\ce{Na_2O}\)
- \(\ce{Na_3PO_4}\)
- \(\ce{N_2O_4}\)
- Answer a
-
The elements in \(\ce{Na_2O}\) are a metal and a nonmetal, which form ionic bonds.
- Answer b
-
Because sodium is a metal and we recognize the formula for the phosphate ion, we know that this compound is ionic. However, within the polyatomic phosphate ion, the atoms are held together by covalent bonds, so this compound contains both ionic and covalent bonds.
- Answer c
-
The elements in \(\ce{N_2O_4}\) are both nonmetals, rather than a metal and a nonmetal. Therefore, the atoms form covalent bonds.
Is each compound are formed from ionic bonds, covalent bonds, or both?
- \(\ce{Ba(OH)_2}\)
- \(\ce{F_2}\)
- \(\ce{PCl_3}\)
- Answer a:
-
both
- Answer b:
-
covalent
- Answer c:
-
covalent
MOLECULAR FORMULAS
The chemical formulas for covalent compounds are referred to as molecular formulas because these compounds exist as separate, discrete molecules. Typically, a molecular formula begins with the nonmetal that is closest to the lower left corner of the periodic table, except that hydrogen is almost never written first (H 2 O is the prominent exception). Then the other nonmetal symbols are listed. Numerical subscripts are used if there is more than one of a particular atom. For example, we have already seen CH 4 , the molecular formula for methane. Below is the molecular formula of ammonia, NH 3 .
NAMING COVALENT COMPOUNDS
Naming binary (two-element) covalent compounds is similar to naming simple ionic compounds. The first element in the formula is simply listed using the name of the element. The second element is named by taking the stem of the element name and adding the suffix - ide . A system of numerical prefixes is used to specify the number of atoms in a molecule. Table \(\PageIndex{1}\) lists these numerical prefixes. Normally, no prefix is added to the first element’s name if there is only one atom of the first element in a molecule. If the second element is oxygen, the trailing vowel is usually omitted from the end of a polysyllabic prefix but not a monosyllabic one (that is, we would say “monoxide” rather than “monooxide” and “trioxide” rather than “troxide”).
| Number of Atoms in Compound | Prefix on the Name of the Element |
|---|---|
| 1 | mono-* |
| 2 | di- |
| 3 | tri- |
| 4 | tetra- |
| 5 | penta- |
| 6 | hexa- |
| 7 | hepta- |
| 8 | octa- |
| 9 | nona- |
| 10 | deca- |
| *This prefix is not used for the first element’s name. |
Let us practice by naming the compound whose molecular formula is CCl 4 . The name begins with the name of the first element—carbon. The second element, chlor ine , becomes chlor ide , and we attach the correct numerical prefix (“tetra-”) to indicate that the molecule contains four chlorine atoms. Putting these pieces together gives the name carbon tetrachloride for this compound.
Write the molecular formula for each compound.
- chlorine trifluoride
- phosphorus pentachloride
- sulfur dioxide
- dinitrogen pentoxide
Solution
If there is no numerical prefix on the first element’s name, we can assume that there is only one atom of that element in a molecule.
- ClF 3
- PCl 5
- SO 2
- N 2 O 5 (The di - prefix on nitrogen indicates that two nitrogen atoms are present.)
Write the molecular formula for each compound.
- nitrogen dioxide
- dioxygen difluoride
- sulfur hexafluoride
- selenium monoxide
- Answer a:
-
a. NO 2
- Answer b:
-
O 2 F 2
- Answer c:
-
SF 6
- Answer d:
-
SeO
Because it is so unreactive, sulfur hexafluoride is used as a spark suppressant in electrical devices such as transformers.
Write the name for each compound.
- BrF 5
- S 2 F 2
- CO
Solution
- bromine pentafluoride
- disulfur difluoride
- carbon monoxide
Write the name for each compound.
- CF 4
- SeCl 2
- SO 3
- Answer a:
-
carbon tetrafluoride
- Answer b:
-
selenium dichloride
- Answer c:
-
sulfur trioxide
For some simple covalent compounds, we use common names rather than systematic names. We have already encountered these compounds, but we list them here explicitly:
- H 2 O: water
- NH 3 : ammonia
- CH 4 : methane
Methane is the simplest organic compound. Organic compounds are compounds with carbon atoms and are named by a separate nomenclature system.
The prefix of mono are not used for the two molecules below:
HCl: hydrogen chloride
HF: hydrogen fluoride
Oxides of Nitrogen-Applications in Medicine
Nitric Oxide
Nitrogen monoxide (NO) is known more commonly as nitric oxide. It is a colorless gas formed when nitrogen and oxygen in the air combine when gasoline is burned in the engines of cars and trucks. So this compound has a bad reputation regarding atmospheric pollution. However, NO has many biological functions. It regulates blood pressure, dilates arteries and blood vessels, and is used by the immune system to help fight infections.
These effects of NO on the human body were noticed in the mid 19 th century soon after the discovery of the explosive nitroglycerin that was used to make dynamite. It was noticed that people who worked with nitroglycerin experienced splitting headaches which was the result of vasodilation (relaxation) of the blood vessels in the brain. This led to the use of nitroglycerin being used as a medication for the treatment of angina which is caused by constriction of the arteries that carry blood to the heart. A century later scientists discovered that the body produces NO from nitroglycerin and it is NO that causes coronary arteries to dilate reducing the symptoms of angina. In 1998 Nobel Prize in Medicine was awarded to three scientists who worked on the nitric oxide research.
Nitrous oxide
Nitrous oxide (N 2 O) is prepared by the careful thermal decomposition of ammonium nitrate (NH 4 NO 3 ). Nitrous oxide is known as "laughing gas" due to the euphoric effects of inhaling it, a property that has led to its recreational use as a hallucinogen. However, it is as an anesthetic that it has a legitimate application. The first use of nitrous oxide as anesthetic drug was when dentist Horace Wells with assistance by Gardner Quincy Colton and John Mankey Riggs, demonstrated insensitivity to pain from a dental extraction in December 1844. Wells subsequently treated 12-15 patients, and according to his own record it only failed as an anesthetic in two cases.
The method did not come into general use until 1863, when Colton successfully used it for more than 25,000 patients. As such, the usage of nitrous oxide rapidly became the preferred anesthetic method in dentistry. The gas is mild enough to keep a patient in a conscious and conversational state, and yet in most cases strong enough to suppress the pain caused by dental work. It remains the preferred gas anesthetic in today's dentistry.
Key Takeaways
- The chemical formula of a simple covalent compound can be determined from its name.
- The name of a simple covalent compound can be determined from its chemical formula.


4: Continuous Distributions
In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them. The normal distribution has two parameters (two numerical descriptive measures), the mean (μ) and the standard deviation (σ).
-
- 4.1: Prelude to The Normal Distribution
- The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them. The normal distribution has two parameters (two numerical descriptive measures), the mean (μ) and the standard deviation (σ).
-
- 4.2: The Standard Normal Distribution
- A z-score is a standardized value. Its distribution is the standard normal, Z∼N(0,1). The mean of the z-scores is zero and the standard deviation is one. If y is the z-score for a value x from the normal distribution N(μ,σ) then z tells you how many standard deviations x is above (greater than) or below (less than) μ.
-
- 4.3: Using the Normal Distribution
- The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean μ and the standard deviation σ. A special normal distribution, called the standard normal distribution is the distribution of z-scores. Its mean is zero, and its standard deviation is one.
-
- 4.5: Prelude to the Central Limit Theorem
- The central limit theorem states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed.
-
- 4.6: The Central Limit Theorem for Sample Means (Averages)
- In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
-
- 4.7: The Central Limit Theorem for Sums
- The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed.
-
- 4.8: Using the Central Limit Theorem
- The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean <x> gets to μ . The central limit theorem illustrates the law of large numbers.