(Molar specific heat capacity at constant pressure) – (Molar specific heat capacity at constant volume) is equal to

(A) R

(B) P

(C) V

(D) T

The difference between the molar specific heat capacity at constant pressure and the molar specific heat capacity at constant volume is a fundamental concept in thermodynamics. This difference is represented by the equation: \( C_p - C_v = R \), where \( R \) is the ideal gas constant. Therefore, the correct answer to your question is (A) R.

Understanding Molar Specific Heat Capacities

To grasp why \( C_p - C_v = R \), let's break down the concepts of specific heat capacities at constant pressure and constant volume:

• Molar Specific Heat Capacity at Constant Pressure (C_p): This is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the pressure constant. When a substance is heated at constant pressure, it can expand, doing work on its surroundings.
• Molar Specific Heat Capacity at Constant Volume (C_v): This is the heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant. In this case, no work is done since the volume does not change.

The Relationship Between C_p and C_v

The difference between these two capacities arises from the work done during expansion. When heating at constant pressure, some of the energy goes into doing work against the external pressure, which is not the case at constant volume. This additional energy requirement at constant pressure is what leads to the relationship:

C_p - C_v = R

Why R?

The ideal gas constant \( R \) is a fundamental constant in the ideal gas law, which relates pressure, volume, temperature, and the number of moles of a gas. It serves as a bridge between the macroscopic properties of gases and their microscopic behavior. The value of \( R \) is approximately 8.314 J/(mol·K) and reflects the energy associated with the gas particles' motion and interactions.

Example for Clarity

Consider a gas in a piston. If you heat the gas while keeping the piston fixed (constant volume), all the heat goes into increasing the internal energy, which raises the temperature. However, if you allow the piston to move (constant pressure), part of the heat goes into doing work to push the piston out, meaning you need more heat to achieve the same temperature increase compared to the constant volume scenario.

This distinction is crucial in thermodynamics and helps in understanding various processes involving gases, such as in engines or refrigerators. By knowing the values of \( C_p \) and \( C_v \), you can predict how a gas will behave under different conditions.

In summary, the difference between the molar specific heat capacities at constant pressure and constant volume is equal to the ideal gas constant \( R \), making the answer (A) R. This relationship is a key concept in thermodynamics that illustrates how gases respond to heat under different constraints.