Question

At a temperature T, let \(\beta\) and \(\kappa\) denote the volume expansivity and isothermal compressibility of a gas, respectively. Then \(\frac{\beta}{\kappa}\) is equal to

• A. \((\frac{\partial P}{\partial T})_V\)
• B. \((\frac{\partial P}{\partial V})_T\)
• C. \((\frac{\partial T}{\partial P})_V\)
• D. \((\frac{\partial T}{\partial V})_P\)

Answer 1

The Correct Option is A

To solve this problem, we need to understand the physical terms involved: volume expansivity (\(\beta\)) and isothermal compressibility (\(\kappa\)). These are fundamental thermodynamic properties of materials.

1. Volume Expansivity (\(\beta\)):

   It is a measure of how much the volume of a substance changes with temperature at constant pressure. Mathematically, it is defined as:

   \(\beta = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P\)
2. Isothermal Compressibility (\(\kappa\)):

   It measures the fractional change in volume of a substance as pressure changes at constant temperature. It is expressed as:

   \(\kappa = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T\)

The problem asks for the ratio \(\frac{\beta}{\kappa}\). Substituting the expressions for \(\beta\) and \(\kappa\), we get:

\(\frac{\beta}{\kappa} = \frac{\frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P}{-\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T}\)

After simplifying, the volumes (\(V\)) cancel out, leading to:

\(\frac{\beta}{\kappa} = -\left(\frac{\partial V}{\partial T}\right)_P \left(\frac{\partial P}{\partial V}\right)_T\)

Using the thermodynamic identity of Maxwell's relations, this ratio simplifies to:

\(\frac{\partial P}{\partial T}\)_V

This matches the first option, \(\left(\frac{\partial P}{\partial T}\right)_V\), which is the correct answer.

Therefore, the correct answer is:

\((\frac{\partial P}{\partial T})_V\)