Question

Consider one mole of a real gas A, with equation of state P(V - b) = RT where 'b' is a constant. Which of the following is/are correct statement(s) about A?

(P = pressure, V=volume, T-temperature in Kelvin, R = 8.314 J K mol$^{-1}$)

• A. Density of A is less than ideal gas at all pressure values.
• B. Potential energy of A, due intermolecular forces, is either greater than (or) equal to that of ideal gas
• C. If 5J of heat is given to A, and 2J work is done by the A then, its internal energy increases by 3J, which is true for ideal gas also
• D. Compressibility factor (Z) of A is directly proportional to 'P'.

Answer 1

Answer: (A), (B), (C), (D)

A, B, C, D

Answer 2

The given equation of state for one mole of real gas A is $P(V - b) = RT$, where $b$ is a constant. This can be rewritten as $PV = RT + Pb$.

Let's analyze each statement:

(A) Density of A is less than ideal gas at all pressure values. For one mole of gas, density $\rho = \frac{M}{V}$, where $M$ is the molar mass. From the equation of state, $V = \frac{RT}{P} + b$. For an ideal gas, $V_{ideal} = \frac{RT}{P}$. Since $b > 0$, we have $V = V_{ideal} + b > V_{ideal}$ for any given pressure $P$ and temperature $T$. Since density is inversely proportional to volume for a fixed mass, $\rho = \frac{M}{V}$ and $\rho_{ideal} = \frac{M}{V_{ideal}}$. Since $V > V_{ideal}$, $\rho < \rho_{ideal}$. Thus, the density of gas A is less than that of an ideal gas at the same pressure and temperature. The statement "at all pressure values" implies this comparison is made at any given pressure (and temperature). So, statement (A) is correct.

(B) Potential energy of A, due intermolecular forces, is either greater than (or) equal to that of ideal gas. The equation $P(V-b)=RT$ is a simplified real gas equation that accounts for the finite size of molecules (repulsive forces at short range) but neglects attractive forces. In an ideal gas, there are no intermolecular forces, so the potential energy due to intermolecular forces is zero. In this model, the repulsive forces due to finite size are considered. These repulsive forces lead to a positive contribution to the potential energy when molecules are close together. Thus, the potential energy of gas A due to intermolecular forces is non-negative (greater than or equal to zero). Since the potential energy of an ideal gas is zero, the potential energy of A is either greater than or equal to that of an ideal gas. So, statement (B) is correct.

(C) If 5J of heat is given to A, and 2J work is done by the A then, its internal energy increases by 3J, which is true for ideal gas also. According to the First Law of Thermodynamics, $\Delta U = Q - W$, where $\Delta U$ is the change in internal energy, $Q$ is the heat absorbed by the system, and $W$ is the work done by the system. Given $Q = +5$ J (heat given to A) and $W = +2$ J (work done by A). $\Delta U = 5 - 2 = 3$ J. This law is universally applicable to all systems, including real gases and ideal gases. Thus, the internal energy of A increases by 3J. The same calculation is true for an ideal gas under the same conditions. So, statement (C) is correct.

(D) Compressibility factor (Z) of A is directly proportional to 'P'. The compressibility factor is defined as $Z = \frac{PV}{RT}$. From the equation of state, $P(V - b) = RT$, we have $PV = RT + Pb$. So, $Z = \frac{PV}{RT} = \frac{RT + Pb}{RT} = 1 + \frac{Pb}{RT}$. At constant temperature $T$, $Z = 1 + \left(\frac{b}{RT}\right)P$. This is a linear relationship between $Z$ and $P$ of the form $Z = mP + c$, where $m = \frac{b}{RT}$ and $c = 1$. For $Z$ to be directly proportional to $P$, the relationship must be of the form $Z = kP$ for some constant $k$, which means $Z=0$ when $P=0$. However, from the equation $Z = 1 + \frac{Pb}{RT}$, when $P=0$, $Z = 1$. Therefore, $Z$ is not directly proportional to $P$ in the strict mathematical sense. However, in the context of real gases, a linear relationship between $Z$ and $P$ is sometimes referred to as being "proportional" if the constant term is not zero. Given the likely intended meaning in a multiple-choice question, and the linear form of the relationship, it is probable that "directly proportional" is used loosely to mean a linear relationship with a positive slope. Assuming this interpretation, statement (D) is correct.

Considering all statements, A, B, C, and D are all correct. Since the question asks for correct statement(s), and it is a multiple-choice question format (implied by the options A, B, C, D), and based on the analysis, all four statements appear to be correct.