N₂ gas behaves ideally at its Boyle temperature in the pressure range from 0 to 50 atm. The critical — States of Matter and Gaseous State Chemistry Question
Question
N₂ gas behaves ideally at its Boyle temperature in the pressure range from 0 to 50 atm. The critical temperature of N₂ gas is –177°C. Assume that the gas obeys van der Waal's equation in the entire range of pressure and temperature. Which of the following statement(s) must be incorrect regarding the molar volume of the gas at the given conditions?
💡 Solution & Explanation
Critical temp $T_c = 96 \text{ K}$. Boyle temp $T_B = \frac{27}{8} T_c = 3.375 \times 96 = 324 \text{ K} = 51^\circ\text{C}$. At $51^\circ\text{C}$ and $P \le 50 \text{ atm}$, it behaves ideally ($V_m = RT/P$). (a) At $T_c$, gas is non-ideal ($Z \approx 3/8$), so $V_m \neq 9.6 \text{ L}$. (b) At 127°C, gas is non-ideal, $V_m \neq 40 \text{ L}$ ideally calculated. (c) At 82.1 atm, pressure exceeds 50 atm threshold for ideal behavior at $T_B$. (d) At 51°C and 0.821 atm, it is ideal, so $V_m = 0.0821 \times 324 / 0.821 = 32.4 \text{ L}$, not 32.4 ml. All are mathematically incorrect. Therefore, correct answer is A,B,C,D.