Question

The figure shows isotherms of a real gas compared to an ideal gas. Which of the following figures correctly represents the behavior of Hydrogen gas at 25°C?

• A. (a)
• B. (b)
• C. (c)
• D. (d)

Answer 1

Answer: (A)

(a)

Answer 2

The compressibility factor $Z = \frac{PV}{nRT}$ describes the deviation of a real gas from ideal behavior ($Z=1$). For a van der Waals gas, $Z = \frac{V}{V-nb} - \frac{a}{RTV}$. The term $\frac{a}{RTV}$ accounts for attractive forces, while $\frac{V}{V-nb}$ accounts for repulsive forces. The Boyle temperature ($T_B$) is where a gas behaves most ideally. For a van der Waals gas, $T_B = \frac{a}{Rb}$. For Hydrogen ($H_2$), $a = 0.244 \, \text{L}^2\text{atm/mol}^2$ and $b = 0.0266 \, \text{L/mol}$. Thus, $T_B = \frac{0.244}{0.0821 \times 0.0266} \approx 112 \, \text{K}$ (-161°C). At temperatures $T > T_B$, $Z$ is always greater than 1, meaning the real gas pressure is always greater than the ideal gas pressure. Since 25°C (298 K) is much greater than $T_B$ (112 K), Hydrogen gas will behave such that its isotherm is always above the ideal gas isotherm.