Define two molar specific heats of gas , and deduce the relation between them.

In thermodynamics, the molar specific heats of a gas are crucial for understanding how gases respond to heat transfer. There are two primary types of molar specific heats: the molar specific heat at constant volume, denoted as \( C_v \), and the molar specific heat at constant pressure, denoted as \( C_p \). Let’s break down what these terms mean and how they relate to each other.

Understanding Molar Specific Heats

The molar specific heat at constant volume, \( C_v \), is defined as the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius (or one Kelvin) while keeping the volume constant. This means that no work is done by the gas since it cannot expand against an external pressure. Mathematically, it can be expressed as:

C_v = \left( \frac{\partial Q}{\partial T} \right)_V

On the other hand, the molar specific heat at constant pressure, \( C_p \), is the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius (or one Kelvin) while keeping the pressure constant. In this case, the gas can expand, and thus work is done by the gas. This can be expressed as:

C_p = \left( \frac{\partial Q}{\partial T} \right)_P

Relationship Between \( C_p \) and \( C_v \)

To deduce the relationship between \( C_p \) and \( C_v \), we can use the first law of thermodynamics and the ideal gas law. The first law states that the change in internal energy (\( \Delta U \)) of a system is equal to the heat added to the system (\( Q \)) minus the work done by the system (\( W \)). For an ideal gas, the change in internal energy can be expressed as:

\( \Delta U = n C_v \Delta T \)

Where \( n \) is the number of moles and \( \Delta T \) is the change in temperature. When heat is added at constant pressure, the heat added can be expressed as:

\( Q = n C_p \Delta T \)

Now, when a gas expands at constant pressure, the work done by the gas is given by:

\( W = P \Delta V \)

Using the ideal gas law, we know that \( P \Delta V = n R \Delta T \), where \( R \) is the ideal gas constant. Therefore, we can rewrite the first law of thermodynamics for a process at constant pressure:

\( Q = \Delta U + W \)

Substituting the expressions for \( Q \), \( \Delta U \), and \( W \) gives us:

\( n C_p \Delta T = n C_v \Delta T + n R \Delta T \)

Dividing through by \( n \Delta T \) (assuming \( \Delta T \) is not zero), we arrive at the relationship:

\( C_p = C_v + R \)

Key Takeaways

• Molar specific heat at constant volume (\( C_v \)): Heat required to raise temperature with constant volume.
• Molar specific heat at constant pressure (\( C_p \)): Heat required to raise temperature with constant pressure.
• Relationship: \( C_p = C_v + R \), where \( R \) is the ideal gas constant.

This relationship highlights that \( C_p \) is always greater than \( C_v \) for ideal gases because when a gas expands at constant pressure, it does additional work, requiring more heat input compared to when it is held at constant volume. Understanding these concepts is fundamental in thermodynamics and helps in analyzing various processes involving gases.