The equation of state for a certain gas is , where and are constants distinct from zero. Ascertain w — States of Matter and Gaseous State Chemistry Question
Question
The equation of state for a certain gas is $P = \frac{RT}{V_m - b} - \frac{a}{V_m}$, where $a$ and $b$ are constants distinct from zero. Ascertain whether the gas has a critical point or not. Answer as 1 if the gas has critical point and as 2, if no critical point.
💡 Solution & Explanation
At a critical point, the first and second derivatives of pressure with respect to molar volume must be zero. $\frac{\partial P}{\partial V_m} = -\frac{RT}{(V_m - b)^2} + \frac{a}{V_m^2} = 0$ and $\frac{\partial^2 P}{\partial V_m^2} = \frac{2RT}{(V_m - b)^3} - \frac{2a}{V_m^3} = 0$. Dividing these two equations gives $V_m - b = V_m$, which implies $b = 0$. Since the problem states $b$ is distinctly non-zero, this contradiction means the conditions for a critical point can never be satisfied. Therefore, correct answer is 2.