Ideal Gas Expansivity Calculation

The question asks for the isothermal expansivity of an ideal gas. Expansivity typically refers to the coefficient of volume expansion, $\beta$, which measures the relative change in volume per unit change in temperature at constant pressure. The standard definition is $\beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P$. While the term ""isothermal expansivity"" can be ambiguous, the provided options suggest we should calculate $\beta$ for an ideal gas.

Ideal Gas Law Foundation

The behavior of an ideal gas is described by the ideal gas law: $PV = nRT$ where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is temperature.

Deriving the Expansivity Coefficient

To find the expansivity $\beta$, we first express volume $V$ as a function of temperature $T$ at constant pressure $P$: $V = \frac{nRT}{P}$ Next, we find the partial derivative of volume with respect to temperature, holding pressure constant: $\left( \frac{\partial V}{\partial T} \right)_P = \frac{nR}{P}$ Now, substitute this into the definition of $\beta$: $\beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P = \frac{1}{V} \left( \frac{nR}{P} \right) = \frac{nR}{VP}$ This expression directly matches option 2.

Simplifying the Result

We can also simplify the expression $\frac{nR}{VP}$ using the ideal gas law. From $PV = nRT$, we can rearrange to get $nR = \frac{PV}{T}$. Substituting this into our expression for $\beta$: $\beta = \frac{PV/T}{VP} = \frac{1}{T}$ Thus, the isobaric volume expansivity of an ideal gas is $\frac{1}{T}$. The expression $\frac{nR}{VP}$ is equivalent to $\frac{1}{T}$.

Conclusion

The calculated expression for the expansivity coefficient $\beta$ of an ideal gas is $\frac{nR}{VP}$.

Correct Option: 2.

$nR/(VP)$