Concept of (Heat Capacity) CP and CV of Gas for JEE

Every time a body absorbs heat, the body's temperature rises as a result. The temperature also drops when heat is withdrawn. The temperature of an object can be used to calculate the total kinetic energy of the constituent parts that make up that object. As a result, when heat is absorbed by a substance, it is transformed into the kinetic energy of the particles, raising the substance's temperature. As a result, the relationship between temperature change and heat transmission is linear.

The body's temperature rises when heat is absorbed by it. Additionally, the temperature drops when heat is lost. The total kinetic energy of the particles that make up an object is measured by its temperature. In other words, when heat is absorbed by an object, it is converted into the kinetic energy of the particles, which raises the temperature. As a result, the relationship between temperature change and heat transmission is linear.

The specific heat capacity can be written mathematically as:

$Q=mc\Delta T$

where $Q=$ heat energy, m= mass, c= specific heat capacity, and $\Delta T$ is the change in temperature.

Specific Heat of Gas

The amount of energy needed to increase the temperature of one unit mass (1 g) of gas by one-degree celsius at constant pressure is known as the specific heat of a gas.

Definition of Molar Heat Capacity

The molar heat capacity (C) of a substance is the total amount of heat energy required to raise the temperature of 1 mole of that substance by 1 unit. The kind, quantity, and makeup of the substance in the system also have a big role.

$q=nC\Delta T$

where n is the amount in moles, and q is the amount of heat that is supplied or required to cause a change in temperature (T) in 1 mole of any given substance. The molar heat capacity of the given substance's body is represented by the constant C.

What is C_V?

The amount of heat energy that a substance absorbs or releases (per unit mass) in response to a change in temperature when no volumetric change occurs is known as C_V. In other words, it is the quantity of thermal energy that is transmitted from one system to another without causing any changes to the system's volume.

The quantity of internal energy that a system has is described by this term. A system's total potential and kinetic energy make up its internal energy. A system's ability to absorb or release energy into the environment depends on the temperature of the system. It is known as the heat capacity at constant volume if this change in internal energy takes place while the volume stays constant (C_V).

${{C}_{V}}=dU/dT$

where C_V is the heat capacity at a constant volume (or a constant temperature) and dU is the change in internal energy. The temperature change is represented by the symbol dT.

What is C_P?

The sign C_P stands for heat capacity at constant pressure. It is the amount of energy that a unit mass of a substance produces or absorbs when the substance's temperature varies while the pressure stays constant. In other words, it is the energy that is transferred when a system is continuously under pressure between the system and its surroundings.

This expression has to deal with the enthalpy of a certain system. An enthalpy is a unit of measurement for energy that is either received or released. It is described as the total internal energy plus a small amount of additional energy plus the product of pressure and volume. This is because the total amount of energy a system takes in or expels equals the sum of the energy it already possesses (internal energy) and the number of changes the system undergoes (PV). The enthalpy of the system will change as the system's temperature varies. The following can be provided as a consequence.

C_P = \[ \frac{dH}{dT}\]

where the temperature change is marked by the symbol dT, the enthalpy change is denoted by the symbol dH, and the heat capacity at constant pressure is given by the sign C_P.

Relationship Between CP and CV of an Ideal Gas

From the equation $q=nC\Delta T$, we can say

We have at constant pressure P

${{q}_{P}}=n{{C}_{P}}\Delta T$

This amount corresponds to the change in enthalpy, so

${{q}_{P}}=n{{C}_{P}}\Delta T=\Delta H$

Similarly, we have at constant volume V

${{q}_{V}}=n{{C}_{V}}\Delta T$

The change in internal energy is represented by this value, hence,

${{q}_{V}}=n{{C}_{V}}\Delta T=\Delta U$

For one mole (n=1) of an ideal gas, we are aware that

$\Delta H=\Delta U+\Delta (pV)=\Delta U+\Delta (RT)=\Delta U+R\Delta T$

Therefore, $\Delta H=\Delta U+R\Delta T$ 

Substituting the values of ∆H and ∆U from above in the former equation,

${{C}_{P}}\Delta T={{C}_{V}}\Delta T+R\Delta T$ 

C_P =C_V +R

C_P – C_V = R

Heat Capacity Ratio

In thermodynamics, the heat capacity ratio or ratio of heat capacities (C_P:C_V) is also known as the adiabatic index. It is the ratio of two heat capacities, C_p and C_V, and is given by:

The Heat Capacity at Constant Pressure (C_P) / Heat capacity at Constant Volume(C_V)

The heat capacity ratio is also known as the isentropic expansion factor that is also denoted for an ideal gas by $\gamma $ (gamma). Therefore, the ratio between C_P and C_V is the heat ratio, $\gamma $.

$\gamma =\frac{{{C}_{P}}}{{{C}_{V}}}$

Table: Gamma values for monoatomic, diatomic, and triatomic gases

Conclusion

The amount of heat needed to raise the temperature of a given quantity of a substance by one-degree Celsius is known as the substance's heat capacity. The thermodynamic characteristics of substances are specific heats. These characteristics give explanations for how a substance's or system's behaviour changes when the system's temperature changes. For ideal gases, the idea of specific heat holds valid, but when applied to real gases, it frequently needs to be changed. This is due to the fact that temperature and pressure often affect how gases behave.