WBJEE 2024 Physics & Chemistry Paper

The compressibility factor $(Z)$ is a measure of how much a real gas deviates from ideal gas behavior. For an ideal gas, the compressibility factor is 1 . For real gases, $Z$ can be less than or greater than 1 depending on the conditions, such as pressure and temperature.The van der Waals equation for real gases is given by: $\left(P + \;\frac{a}{V_m^2}\right) (V_m - b) = R T$ Where:$P$ is the pressure$V_m$ is the molar volumeR is the universal gas constantT is the temperature$a$ and $b$ are van der Waals constants for the gas At high pressures, the term $\;\frac{a}{V_m^2}$ becomes negligible, and the van der Waals equation can $b$ approximated as:$P (V_m - b) = R T$Rearranging for $V_m$, we get: $V_m = \;\frac{R\,T}{P} + b$ The compressibility factor $Z$ is defined as:$Z = \;\frac{P\,V_m}{R\,T}$ Substituting the expression for $V_m$ into the definition of Z , we get:$Z = \;\frac{P \left( \frac{R\,T}{P} + b \right)}{R\,T}$ Simplifying this expression gives us:$Z = 1 + \;\frac{P\,b}{R\,T}$ Thus, the compressibility factor for a van der Waals gas at high pressure is:Option B: $1 + \frac{Pb}{RT}$