Proof of Mayer's Relation or Formula : (Cp – Cv = R) |- Thermodynamics - Class 11 Physics

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Proof of Mayer's Relation

We know that:

Ideal Gas Equation:

$$PV=nRT$$

First Law of Thermodynamics:

$$\Delta Q=\Delta U+W$$

Step 1: Constant Volume Process

At constant volume,

$$\Delta V=0$$

Therefore, the work done is

$$W=P\Delta V=0$$

Substituting into the first law of thermodynamics:

$$\Delta Q=\Delta U$$

For an ideal gas,

$$\Delta U=nC_v\Delta T$$

Hence,

$$\Delta Q=nC_v\Delta T$$

Step 2: Constant Pressure Process

At constant pressure, the work done is

$$W=P\Delta V$$

Using the first law of thermodynamics,

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

For a temperature change $$\Delta T$$,

$$\Delta Q=nC_p\Delta T$$

and

$$\Delta U=nC_v\Delta T$$

Substituting these values:

$$nC_p\Delta T=nC_v\Delta T+P\Delta V$$

Step 3: Using the Ideal Gas Equation

From the ideal gas equation,

$$PV=nRT$$

Differentiating for constant pressure:

$$P\Delta V=nR\Delta T$$

Step 4: Substitution and Simplification

Substituting $$P\Delta V=nR\Delta T$$ into the previous equation:

$$nC_p\Delta T=nC_v\Delta T+nR\Delta T$$

Dividing both sides by $$n\Delta T$$:

$$C_p=C_v+R$$

Therefore,

$$\boxed{C_p-C_v=R}$$

Mayer's Relation

$$\boxed{C_p-C_v=R}$$

This relation is known as Mayer's Relation for an ideal gas.

Why is the Change in Internal Energy the Same in Both Cases?

For an ideal gas, internal energy depends only on temperature and not on pressure or volume.

Therefore, for the same temperature change $\Delta T$,

$$\Delta U=nC_v\Delta T$$

Whether the process occurs at constant volume or constant pressure, the change in internal energy remains the same.

The difference lies in the distribution of supplied heat:

  • Constant Volume: All supplied heat increases the internal energy of the gas.
  • Constant Pressure: Part of the supplied heat increases internal energy, while the remaining part is used to perform external work.

Key Formulae

Ideal Gas Equation:

$PV=nRT$

First Law of Thermodynamics:

$\Delta Q=\Delta U+W$

Work Done:

$W=P\Delta V$

Change in Internal Energy:

$\Delta U=nC_v\Delta T$

Heat Supplied at Constant Volume:

$Q_v=nC_v\Delta T$

Heat Supplied at Constant Pressure:

$Q_p=nC_p\Delta T$

Mayer's Relation:

$\boxed{C_p-C_v=R}$

Frequently Asked Questions (FAQ)

1. What is Mayer's Relation?

Mayer's Relation is the relationship between molar specific heat at constant pressure and molar specific heat at constant volume for an ideal gas.

$\boxed{C_p-C_v=R}$

2. For which gases is Mayer's Relation valid?

Mayer's Relation is valid only for ideal gases. Real gases obey it approximately under ordinary conditions.

3. Why is $C_p$ greater than $C_v$?

At constant pressure, a gas expands while being heated and performs external work. Therefore, more heat is required to raise its temperature by the same amount.

Hence,

$C_p>C_v$

4. What is the value of $$R$$ in Mayer's Relation?

$R=8.314\ \text{J mol}^{-1}\text{K}^{-1}$

5. Why does internal energy remain the same for both constant pressure and constant volume processes?

For an ideal gas, internal energy depends only on temperature.

$\Delta U=nC_v\Delta T$

6. What happens to the supplied heat during a constant-volume process?

Since no work is done, the entire supplied heat increases the internal energy of the gas.

$Q_v=\Delta U$

7. What happens to the supplied heat during a constant-pressure process?

A part of the supplied heat increases internal energy and the remaining part is used to perform work.

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

8. Can $C_p$ and $C_v$ be equal?

No. For an ideal gas,

$C_p-C_v=R$

Since $R$ is positive, $C_p>C_v$

Quiz (MCQs)

1. Mayer's Relation for an ideal gas is

A) $C_p+C_v=R$

B) $C_p-C_v=R$

C) $C_v-C_p=R$

D) $C_pC_v=R$

Answer: B) $C_p-C_v=R$

2. The SI unit of the universal gas constant $$R$$ is

A) J

B) J K-1

C) J mol-1 K-1

D) N m-2

Answer: C) J mol-1 K-1

3. During a constant-volume process, the work done is

A) Positive

B) Negative

C) Zero

D) Maximum

Answer: C) Zero

$W=P\Delta V=0$

4. For an ideal gas, internal energy depends only on

A) Pressure

B) Volume

C) Temperature

D) Density

Answer: C) Temperature

5. Which quantity is always greater for an ideal gas?

A) $C_v$

B) $C_p$

C) Both are equal

D) Cannot be determined

Answer: B) $C_p$

6. At constant pressure, the heat supplied is

A) $nC_v\Delta T$

B) $nC_p\Delta T$

C) $P\Delta V$

D) Zero

Answer: B) $nC_p\Delta T$

7. The work done during a constant-pressure process is

A) $P\Delta V$

B) $V\Delta P$

C) $PV$

D) Zero

Answer: A) $P\Delta V$

8. According to the first law of thermodynamics,

A) $Q=W-\Delta U$

B) $Q=\Delta U+W$

C) $Q=\Delta U-W$

D) $Q=W$

Answer: B) $Q=\Delta U+W$

9. The value of the gas constant $R$ is approximately

A) 1.38

B) 6.67

C) 8.314

D) 9.8

Answer: C) 8.314

10. Mayer's Relation was derived using

A) Boyle's Law only

B) Charles' Law only

C) First Law of Thermodynamics and Ideal Gas Equation

D) Newton's Law

Answer: C) First Law of Thermodynamics and Ideal Gas Equation

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