Specific Heat Relation \(C_p - C_v\) Explained

The question asks for the correct thermodynamic relation between the specific heat at constant pressure (\(C_p\)) and the specific heat at constant volume (\(C_v\)). These two quantities represent the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin) under different conditions.

Understanding the Terms

• \(C_p\): Specific heat capacity at constant pressure.
• \(C_v\): Specific heat capacity at constant volume.
• \(v\): Specific volume, which is the volume per unit mass of the substance.
• \(T\): Absolute temperature in Kelvin.
• \(\beta\): Coefficient of volume expansion. It measures how the volume of a substance changes with temperature at constant pressure. It is defined as \(\beta = \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_p\).
• \(K\): Isothermal Bulk Modulus. It measures the resistance of a substance to uniform compression. It is defined as \ K = -v \left( \frac{\partial p}{\partial v} \right)_T\), where \(p\) is pressure.

The Thermodynamic Relation

For any substance, the difference between specific heat capacities at constant pressure and constant volume is related to its thermal expansion and compressibility. The energy required to heat a substance at constant pressure is always greater than at constant volume because, at constant pressure, the substance expands and does work on its surroundings. This difference is quantified by the following thermodynamic relation:

\(C_p - C_v = \frac{vT\beta^2}{K}\)

This equation shows that the difference \(C_p - C_v\) is positive, as expected, since \(v\), \(T\), \(\beta^2\), and \(K\) are all positive quantities for most substances under normal conditions. The terms \(\beta\) and \(K\) specifically relate the energy changes to the mechanical work done due to volume changes caused by temperature variations.

Evaluating the Options

• Option 1: \(\rm C_p-C_v=\frac{vT\beta^2}{K}\) - This matches the derived thermodynamic relation.
• Option 2: \(\rm C_v-C_p=\frac{vT\beta^2}{K}\) - This suggests \(C_v > C_p\), which is incorrect.
• Option 3: \(\rm C_p-C_v=\frac{vT}{K\beta^2}\) - The inverse relationship of \(\beta^2\) and \(K\) is incorrect.
• Option 4: \(\rm C_p-C_v=\frac{pT\beta^2}{K}\) - The pressure \(p\) should not be in the numerator instead of specific volume \(v\) in this standard form of the relation.

Therefore, the correct relation is given in Option 1.