#### Problem 55E

55. (a) The van der Waals equation for $n$ moles of a gas is

$$\left(P+\frac{n^{2} a}{V^{2}}\right)(V-n b)=n R T$$

where $P$ is the pressure, $V$ is the volume, and $T$ is the temperature of the gas. The constant $R$ is the universal gas constant and $a$ and $b$ are positive constants that are characteristic of a particular gas. If $T$ remains constant, use implicit differentiation to find $d V / d P$.

(b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of $V=10 \mathrm{~L}$ and a pressure of $P=2.5 \mathrm{~atm} .$ Use $a=3.592 \mathrm{~L}^{2}-\mathrm{atm} / \mathrm{mole}^{2}$ and $b=0.04267 \mathrm{~L} / \mathrm{mole}$