Question 10

The van der Waals equation is $(P + \frac{a}{V_m^2})(V_m - b) = RT$. Expanding: $PV_m - Pb + \frac{a}{V_m} - \frac{ab}{V_m^2} = RT$. Multiply by $\frac{V_m^2}{P}$ and rearrange: $V_m^3 - (b + \frac{RT}{P})V_m^2 + \frac{a}{P}V_m - \frac{ab}{P} = 0$. Coefficient of $V_m^2$ ($c_2$) $= -(b + \frac{RT}{P})$. Coefficient of $V_m$ ($c_1$) $= \frac{a}{P}$. Ratio $\frac{c_2}{c_1} = \frac{-(b + \frac{RT}{P})}{\frac{a}{P}} = -\frac{Pb + RT}{a}$. Substitute values: $P = 300$, $b = 0.060$, $R = 0.082$, $T = 300$, $a = 6.0$. Ratio $= -\frac{(300 \times 0.060) + (0.082 \times 300)}{6.0} = -\frac{18 + 24.6}{6.0} = -\frac{42.6}{6.0} = -7.1$.