For a specific hypothetical gas, the critical temperature () is and the critical pressure () is . As — States of Matter and Gaseous State Chemistry Question
Question
For a specific hypothetical gas, the critical temperature ($T_c$) is $47^\circ\text{C}$ and the critical pressure ($P_c$) is $2.0\text{ atm}$. Assuming it perfectly follows the van der Waals equation where the critical compressibility factor $Z_c = 0.375$, what is its critical volume ($V_c$) in $\text{L mol}^{-1}$? (Use $R = 0.0821\text{ L atm K}^{-1}\text{ mol}^{-1}$ and round to the nearest integer).
💡 Solution & Explanation
First, convert $T_c$ to Kelvin: $T_c = 47 + 273 = 320\text{ K}$. By definition, $Z_c = P_cV_c / RT_c$. We are given $Z_c = 0.375$. Substituting the values: $0.375 = (2.0 \times V_c) / (0.0821 \times 320)$. Solving for $V_c$: $V_c = 0.375 \times 26.272 / 2.0 = 9.852 / 2.0 = 4.926\text{ L mol}^{-1}$. Rounding to the nearest integer yields 5.