Question

The Boyle temperature (\(T_B\)) is defined as the temperature at which the properties of a real gas coincide with those of an ideal gas in the low pressure limit. The graph that shows the pressure dependence of the compression factor (Z) for a real gas at \(T_B\) is

• A.
• B.
• C.
• D.

Hint:

At the Boyle temperature, the real gas behaves like an ideal gas at low pressure, and the compressibility factor \( Z \) approaches 1.

Answer 1

The Correct Option is B

$\mathbf{Z-P}$ Behavior at Boyle Temperature ($\mathbf{T_B}$)

The Boyle temperature ($\mathbf{T_B}$) is the temperature at which a real gas exhibits ideal gas behavior over an appreciable range of pressure, especially in the low-to-moderate pressure region.

Ideal Gas Behavior ($\mathbf{Z=1}$): For an ideal gas, the compressibility factor $Z = \frac{PV}{nRT}$ is equal to 1 at all temperatures and pressures.

Definition of $\mathbf{T_B}$: At $T_B$, the effects of intermolecular attractive forces (which tend to make $Z<1$) and repulsive forces (which tend to make $Z>1$) effectively balance out. This is formally defined as the temperature where the second virial coefficient ($B_2(T)$) is zero.

Low Pressure Limit: Because the attractive and repulsive forces balance, the gas behaves ideally (i.e., $Z \approx 1$) over a wider pressure range compared to any other temperature. Therefore, the $\mathbf{Z}$ vs. $\mathbf{P}$ curve starts at $Z=1$ at $P=0$ and remains close to $Z=1$ for low pressure .

High Pressure Limit: As pressure increases significantly, the repulsive forces (due to the finite volume of the gas molecules, represented by the Van der Waals constant $b$) eventually dominate. This causes the molar volume of the real gas to be greater than that of an ideal gas, leading to $\mathbf{Z > 1}$. Consequently, the curve must bend upwards.

Graph (B) starts at $Z=1$ and remains horizontal or nearly horizontal for an initial pressure range before gradually increasing to $Z>1$ at high pressures, which correctly depicts the behavior of a real gas at its Boyle temperature.