Relationships of variables Chemistry Dr. Breinan v3

In science, the type of mathematical relationship between two variables is extremely important. The two that you will study most often in chemistry are inverse and direct.

These two terms can have several different meanings, but we will agree at the start of the year to use them for two very specific relationships which are better called proportions. Read on for the explanation.

Direct:Let us consider two different variables, A and B. These could represent temperature, pressure, heat, mass, volume, etc.

The simplest definition of a “direct relationship” is that when one variable increases, the other increases. This also means that when one decreases, the other decreases. This is a very general idea and there are literally infinitely many relationships that would fall under this category. Any graph that slopes up and to the right would qualify under this simplistic definition. It is best to call these “positive” relationships (a graph of the data has positive slope) because…

when a chemist reports a “direct relationship”, she would generally be referring to a specific type of direct relationship more properly called a “direct proportion.” Thus when you or I say “relationship” we really mean “proportion!”

**The relationship between A and B is defined as “direct” if, for all the different trials in an experiment, A divided by B is always the same value, called a constant. In mathematical form,

a direct relationship is shown by A

Bk . The symbols “k” and “c” are often used for

‘constant’.If such a relationship exists, it is correct to say that “A and B are directly related” or “directly proportional.” (One more time: it is best to say a direct “proportion” but we will use “relationship” to mean the same.)

The graph of a direct relationship is very distinctive. If a graph is a straight line with a positive slope AND goes through the point (0,0), then the relationship is direct. Not all straight line graphs are direct because some slope the wrong way (negative slope, down and to the right) and some do not go through (0,0). (The general term for a straight line graph is “linear”)

One characteristic of a direct relationship is that as A increases in value, B also increases in value (recall that that is the most general definition). But, our definition is much more restrictive… it must satisfy other tests as well as explained above. Thus the fact that A increases when B increases may lead us to suspect that a direct relationship exists, but it is NOT PROOF! To prove it we have to show a straight line graph sloping up and to the right through (0,0) or we need to show that in multiple trials A/B (or B/A) always equals the same number. This best test is best shown by a table of values as follows:

Showing a direct relatiohship (PROOF from data!)If you obtain data for variables A and B, add a column which calculates B/A (or A/B). If this value is constant, the relationship is direct. If is not constant, it is not direct. See the following two examples:

trial A B B/A trial A B B/A1 0.20 0.81 4.0 1 0.30 1.2 4.02 0.80 3.25 4.1 2 0.80 8.1 103 2.75 11.15 4.05 3 2.75 76.0 27.64 3.32 13.40 4.04 4 3.32 140.2 42.2

B/A is constant. This is direct! B/A is not constant. This is not direct.

Relationships of variables Chemistry Dr. Breinan p. 2

Inverse:The simplest definition of an “inverse relationship” is that when one variable increases, the other decreases. This also means that when one decreases, the other increases. This is also a very general idea and there are literally infinitely many relationships that would fall under this category. Any graph that slopes down and to the right would qualify under this simplistic definition. It is best to just call this a “negative” relationship (for negative slope) because….

when a chemist reports an “inverse relationship”, she would generally be referring to a specific type of inverse relationship more properly called an “inverse proportion.” Thus when you or I say “relationship” we really mean “proportion!”

**The relationship between A and B is defined as “inverse” if, for all the different trials in an experiment, A multiplied by B is always the same value (constant). In mathematical form, an inverse relationship is shown by A•B = k.

If such a relationship exists, it is correct to say that “A and B are inversely related” or “inversely proportional” (One last time: it is best to say an inverse “proportion” but we will use “relationship” to mean the same.)

The graph of an inverse relationship is a function called a hyperbola. You will learn what this looks like later in the year. You should know that graphs of other relationships can appear similar to this type of graph even though they are not the same.

Similar to what we learned about a direct relationship, if A increases in value and B decreases in value (or vice versa) this is just one characteristic of an inverse relationship, but it is NOT PROOF! If we suspect that an inverse relationship exists, the table test is the best proof (in this case we are looking for A*B (or B*A) to equal a constant:

Proving an inverse relationship with a table:trial A B B*A trial A B B*A1 0.20 81.6 16 1 0.30 122 372 0.80 21.5 17 2 0.80 81 653 2.75 6.3 17 3 2.75 36 994 3.32 4.7 16 4 3.32 33 110

B*A is constant. This is inverse! B*A is not constant. This is not inverse.

Algebraic Proof of direct and inverse relationships:The mathematical definitions given above do constitute proof of a direct or inverse relationship. However, be aware that algebra lets you change the appearance of an equation without changing the relationship of variables! For example, a direct relationship between A

and B can be written in any of the following ways: A

Bk , , or A = B∙k

These all represent the same direct relationship. Similarly, an inverse relationship can be

written as any of these: A∙B = k, , or . In each case, the first expression is most

useful because the constant is solved for and you can use the descriptions given above (variables divided to give a constant are directly related; variables multiplied to give a constant are inversely related.) Solving for the constant in an equation is a reliable way of determining inverse or direct relationships.

Relationships of variables worksheet Chemistry Dr. Breinan

The worksheet on the following page will help you practice all these skills.

Relationships of variables Chemistry Dr. Breinan p. 4

P1. For an object moving at constant speed (v), the relationship of distance (d) traveled and time (t)

traveled is: . (a) Is the relationship between distance and time inverse or direct?

(b) This relationship can be written as d = v∙t. Is the relationship between d and t now inverse or direct?

P2. An electrical circuit in your home operates at constant voltage (V). The resistance of the circuit (R) is related to the current (I) by V = I ∙ R. (a) Is the relationship between current and resistance

inverse or direct? (b) if the equation is written as , is the relationship of I and R now inverse or

direct?

P3. Determine whether the relationship of fluid pressure (P) and the height of a column of fluid (h) is inverse or direct at constant weight density (wd). P = wd ∙ h.

Distinguishing direct from inverse by experiment (PROOF from data!)If you obtain data for variables A and B, you can determine whether they have a direct, inverse, or other type of relationship by completing a table like the one below and seeing if you get a constant (the same value each time) in one of the columns. Once again if A/B is constant, the relationship is direct. If A∙B is a constant the relationship is inverse. If neither column is constant, the relationship is neither inverse nor direct. Remember, when working with real data, the “constant” may vary a little bit due to experimental errors. This table shows a direct relationship because the A/B column is essentially constant.

trial A B A/B A•B1 0.15 0.89 .17 .132 1.34 8.0 .17 113 52.7 316 .167 167004 332 1940 .171 6.44x105

P4. (a) Is the value of B increasing or decreasing when A increases? (b) Based only on your answer to (a), could the relationship of A and B be direct? inverse? (c) Fill out the table and state whether the relationship is inverse, direct, or other.

trial A B A/B A•B1 4.5 0.892 0.34 11.913 2.7 1.454 8.8 0.45

P5. Here’s a few more!For each table below answer the following:(a) Look at the table. Is the value of B increasing or decreasing when A increases?

P1a) direct (b) direct (same relationship, just written differently) P2a) inverse (b) inverse (same relationship, just written differently) P3) direct (solve for wd because it is the constant) P4a) B decreases. (b) It could be inverse, but it cannot be direct (c) The relationship is inverse because A∙B always equals 4.0.

Relationships of variables worksheet Chemistry Dr. Breinan

(b) Based only on your answer to (a), could A and B be related directly? (yes or no?)(c) Based only on your answer to (a), could A and B be related inversely? (yes or no?)(d) Fill out the table and state whether the relationship is truly inverse, direct, or other.(e) In the right hand margin area, make a quick sketch of what the graph would look like and state

whether the relationship is positive, negative, or other.

Ex.1

trial A B A/B A•B a)1 0.15 0.892 1.34 8.0 b)3 52.7 3164 332 1940 c)

d)

Ex.2

trial A B A/B A•B a)1 1.2 1.92 2.7 2.5 b)3 9.5 3.64 12.8 3.9 c)

d)

Ex.3

trial A B A/B A•B a)1 0.35 6.32 3.7 145 b)3 14.5 8924 28.8 45 c)5 51.4 2.3

d)