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(A) For an ideal gas at constant temperature, the equation of state is $PV = 	ext{constant}$.Taking the derivative with respect to $P$: $P \frac{dV}{

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(A) For an ideal gas at constant temperature, the equation of state is $PV = 	ext{constant}$.Taking the derivative with respect to $P$: $P \frac{dV}{dP} + V = 0$.This implies $P \frac{dV}{dP} = -V$, or $\frac{dV}{dP} = -\frac{V}{P}$.Substituting this into the expression for $\beta$:$\beta = - \left( \frac{dV}{dP} ight) / V = - \left( -\frac{V}{P} ight) / V = \frac{1}{P}$.Thus, $\beta \propto \frac{1}{P}$.This represents a rectangular hyperbola, where $\beta$ decreases as $P$ increases. Therefore, the correct graph is the one showing a rectangular hyperbola.

Which of the following graphs correctly represents the variation of $\beta = - \left( \frac{dV}{dP} \right)/V$ with $P$ for an ideal gas at constant temperature?

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