Question

establish a relation between two molar heat capacities of an ideal gas.

Answer 1

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HomeChemistry ArticleHeat Capacity: Relationship between Cp and Cv
Heat Capacity: Relationship between Cp and Cv
November 25, 2016 0 Comments
When heat is absorbed by a body, temperature of the body increases. And when heat is lost, the temperature decreases. Temperature of any object is the measure of the total kinetic energy of the particles that make up that object. So when heat is absorbed by an object this heat gets translated into the kinetic energy of the particles and as a result the temperature increases. Thus, the change in temperature is proportional to the heat transferred. The formula q = n C ∆T represents the heat q required to bring about ∆T difference in temperature of one mole of any matter. The constant C here is called the molar heat capacity of the body. Thus, molar heat capacity of any substance is defined as the amount of heat energy required to change the temperature of 1 mole of that substance by 1 unit. It depends on the nature, size and composition of the system. In this article we will discuss two types of molar heat capacity – CP and CV and derive a relationship between them.

heat capacity

What are Heat Capacity C, CP and CV?

The molar heat capacity C, at constant pressure, is represented by CP.
At constant volume, the molar heat capacity C is represented by CV.
In the following section, we will find how CP and CV are related, for an ideal gas.

Relationship between CP and CV for an Ideal Gas

From the equation q = n C ∆T, we can say:

At constant pressure P, we have qP = n CP∆T

This value is equal to the change in enthalpy, that is, qP = n CP∆T = ∆H

Similarly, at constant volume V, we have qV = n CV∆T

This value is equal to the change internal energy, that is, qV = n CV∆T= ∆U

We know that for one mole (n=1) of ideal gas,

∆H = ∆U + ∆(pV )

= ∆U + ∆(RT )

= ∆U + R∆T

Therefore, ∆H = ∆U + R ∆T

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

CP∆T = CV∆T + R ∆T

Or CP = CV + R

Or CP – CV= R

Answer 2

The relation between the two molar heat capacities of an ideal gas is ${C_p} - {C_v} = R$

Here, $R$ is the gas contant.

Explanation:

• Molar heat capacity is the heat required to change the temperature of one mole gas by unit. It is given as

        $C = \frac{{\Delta Q}}{{n\Delta T}}$

• Here, $C$ is the heat capacity of the gas and $n$ is the number of moles. To change the temperature of the gas by amount $\Delta T$, $\Delta Q$ heat is required.
• There are two types of molar heat capacities-

1. When the change of temperature takes place and volume is constant, the heat required is called molar heat capacity at constant volume and is represented by ${C_v}$.
2. The other one is the molar heat capacity at constant pressure represented by ${C_p}$ which is the heat required when a change in temperature takes place and pressure is constant.

• Consider a cylinder with the piston and $n$ mole ideal gas filled in it as shown in the figure.
• At first, fix the pressure of gas and change its temperature by $\Delta T$ , for it the heat required is $\Delta Q$ then, $\Delta Q = n{C_p}\Delta T$
• Now, return the gas in the initial condition and again change its temperature by $\Delta T$ at fixed volume, then the heat required say $\Delta U$  and it is totally used to change the internal energy of gas and given as

       $\Delta U = n{C_v}\Delta T$

• From the first law of thermodynamics, we have,

       $\Delta Q = \Delta U + P\Delta V$

• From the gas equation,

        $P\Delta V = nR\Delta T$

• Therefore, the equation becomes

        $\Delta Q = \Delta U + nR\Delta T$

• Putting the values of $\Delta Q$ and $\Delta U$ in the above equation, we get,

       $\begin{gathered} n{C_p}\Delta T = n{C_v}\Delta T + nR\Delta T \hfill \\ {C_p} = {C_v} + R \hfill \\ {C_p} - {C_v} = R \hfill \\ \end{gathered}$

• Therefore, the molar heat capacities at constant pressure and constant volume differ by the value of the gas constant.

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