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Ideal Solutions

Draw the pres...

Question

Draw the pressure-temperature and volume-temperature diagrams of an isochoric process of $n$ moles of an ideal gas from pressure $P_{0}$ , volume $V_{0}$ to pressure $4 P_{0}$ , indicating the pressures and temperatures of the gas in the initial and the final states.

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Solution

The pressure of a gas is directly proportional to its temperature when the volume is constant. The ratio of pressure to temperature is constant when the volume is constant. A constant volume process is said to be isochoric.

From the equation of state for an ideal gas $P V = n R T$ , we get the following.

1. $P = \frac{n R T}{V}$ , i.e, $P \propto T (V = c o n s t a n t)$

At $P = P_{0}$ , and $V = V_{0}$ , we get $T = \frac{P_{0} V_{0}}{n R}$
And at $P = 4 P_{0}$ , and $V = V_{0}$ , we get $T = \frac{4 P_{0} V_{0}}{n R}$
The graph is a straight line (passing through the origin, when produced) shown in figure(i)

2. Since temperature increases from $\frac{P_{0} V_{0}}{n R}$ to $\frac{4 P_{0} V_{0}}{n R}$

Volume remains constant; so, the graph of volume-temperature will be as shown in the figure(ii).

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