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Chapter 5: Problem 131

Draw a qualitative graph to show how the first property varies with the second in each of the following (assume 1 mole of an ideal gas and \(T\) in kelvin). a. \(P V\) versus \(V\) with constant \(T\) b. \(P\) versus \(T\) with constant \(V\) c. \(T\) versus \(V\) with constant \(P\) d. \(P\) versus \(V\) with constant \(T\) e. \(P\) versus 1\(/ V\) with constant \(T\) f. \(P V / T\) versus \(P\)

Short Answer

Expert verified
a. Since \(T\) is constant, the graph of \(P V\) versus \(V\) is a straight line with a positive slope. b. With constant \(V\), the graph of \(P\) versus \(T\) is a straight line with a positive slope. c. Keeping \(P\) constant, the graph of \(T\) versus \(V\) is a straight line with a positive slope. d. With constant \(T\), the graph of \(P\) versus \(V\) is a hyperbola. e. Plotting \(P\) versus 1\(/V\) with constant \(T\) results in a straight line with a positive slope. f. The graph of \(P V/T\) versus \(P\) is a horizontal straight line, as the ratio remains constant.

Step by step solution

01

a. \(P V\) versus \(V\) with constant \(T\)

Since the temperature \(T\) is constant in this case, we can use the ideal gas law, \(P V = n R T\), where \(n\) is the number of moles (set to 1 in the problem), and \(R\) is the ideal gas constant. Since \(n R T\) is constant, as \(P V\) increases, \(V\) must also increase in proportion. Thus, the graph is a straight line with a positive slope.
02

b. \(P\) versus \(T\) with constant \(V\)

Here, we have constant volume \(V\), and we will again use the ideal gas law. By rearranging the equation, we get \(P = \frac{n R T}{V}\). Since \(n R / V\) is constant, as \(T\) increases, the pressure \(P\) increases in proportion. The graph is again a straight line with a positive slope.
03

c. \(T\) versus \(V\) with constant \(P\)

With constant pressure \(P\), we can write the ideal gas law as \(T = \frac{P V}{n R}\). Since the pressure and number of moles are both constant, as the volume \(V\) increases, the temperature \(T\) must also increase in proportion. The graph is a straight line with a positive slope.
04

d. \(P\) versus \(V\) with constant \(T\)

This case is the same as case (a), but we are now asked to plot the graph of \(P\) versus \(V\). Using the ideal gas law, we get \(P = \frac{n R T}{V}\). Since \(n R T\) is constant, as the volume \(V\) increases, the pressure \(P\) decreases in inverse proportion. The graph is a hyperbola.
05

e. \(P\) versus 1\(/ V\) with constant \(T\)

In this case, we want to plot \(P\) versus \(1/V\) while the temperature is constant. From the ideal gas law, \(P = \frac{n R T}{V}\). This is a proportionality relation between \(P\) and 1\(/V\). As the value of 1\(/V\) increases, the pressure \(P\) also increases. The graph is a straight line with a positive slope.
06

f. \(P V / T\) versus \(P\)

We are asked to plot the ratio \(P V / T\) versus pressure \(P\). From the ideal gas law, we have \(P V = n R T\), and thus, \(P V/T = n R\). Since \(n\) (number of moles) and \(R\) (ideal gas constant) are constant, the ratio \(P V/T\) remains constant regardless of the value of pressure \(P\). The graph is a horizontal straight line.

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