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Question

# Prove for an ideal gas that (a) the P = constant lines on a T-v diagram are straight lines and (b) the high-pressure lines are steeper than the low-pressure lines.

Solution

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To solve this problem we first need to determine the $\textbf{relation}$ for the temperature $T$ change with respect to specific volume $v$, $\partial T/\partial v$. To do that we will use the $\textbf{ideal gas equation}.$

\begin{align} \nonumber T &= \frac{p v}{R} \\ \frac{\partial T}{\partial v}&=\frac{p}{R} \end{align}

A) From the equation (1) we can see that the temperature change with respect to specific volume $\textbf{depends on the pressure}$ $p$ and the specific gas constant $R$. If we assume the pressure to be constant then the $\partial T/\partial v$ will also be constant.

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