$C_V$

Definition

$C_V$ is the heat capacity of a system at constant volume, which is a measure of how much energy is required to raise the temperature of a system by one unit while keeping the volume constant. It is an important concept in thermodynamics and is closely related to the equipartition of energy.

5 Must Know Facts For Your Next Test

1. $C_V$ is related to the number of degrees of freedom in a system, as described by the equipartition of energy principle.
2. For an ideal gas, $C_V$ is directly proportional to the number of degrees of freedom per particle.
3. The relationship between $C_V$ and $C_P$ is given by the equation $C_P - C_V = nR$, where $n$ is the number of moles of the substance and $R$ is the universal gas constant.
4. The value of $C_V$ can be used to determine the ratio of specific heats, $eta = C_P/C_V$, which is an important parameter in thermodynamics.
5. The equipartition of energy principle states that the average energy per degree of freedom in a system at thermal equilibrium is $\frac{1}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature.

Review Questions

• Explain the relationship between $C_V$ and the number of degrees of freedom in a system.
  • The heat capacity at constant volume, $C_V$, is directly related to the number of degrees of freedom in a system. According to the equipartition of energy principle, the average energy per degree of freedom in a system at thermal equilibrium is $\frac{1}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature. For an ideal gas, the number of degrees of freedom per particle is directly proportional to $C_V$, so a system with more degrees of freedom will have a higher $C_V$.
• Describe the relationship between $C_V$ and $C_P$ and explain the significance of the ratio $eta = C_P/C_V$.
  • The relationship between the heat capacity at constant volume, $C_V$, and the heat capacity at constant pressure, $C_P$, is given by the equation $C_P - C_V = nR$, where $n$ is the number of moles of the substance and $R$ is the universal gas constant. The ratio of these two heat capacities, $eta = C_P/C_V$, is an important parameter in thermodynamics as it determines the behavior of a system, such as the speed of sound in a gas and the adiabatic expansion or compression of a gas.
• Analyze how the equipartition of energy principle relates to the concept of $C_V$ and the average energy per degree of freedom in a system.
  • The equipartition of energy principle states that the average energy per degree of freedom in a system at thermal equilibrium is $\frac{1}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature. This principle is closely related to the concept of $C_V$, the heat capacity at constant volume, as $C_V$ is directly proportional to the number of degrees of freedom in a system. By understanding the equipartition of energy and its relationship to $C_V$, one can determine the average energy per degree of freedom in a system and how the system will respond to changes in temperature or volume.

Related terms

$C_P$: $C_P$ is the heat capacity of a system at constant pressure, which is a measure of how much energy is required to raise the temperature of a system by one unit while keeping the pressure constant.

Equipartition of Energy: The equipartition of energy is a principle in statistical mechanics that states that energy is equally distributed among all available degrees of freedom in a system at thermal equilibrium.

Specific Heat Capacity: Specific heat capacity is the amount of energy required to raise the temperature of a unit mass of a substance by one unit, and it is a characteristic property of the material.