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**Solids and liquids have only one specific heat, while gases have two specific heats: **

- In case of solids and liquids small change in temperature causes a negligible change in the volume and pressure, hence the external work performed is negligible. In such cases, all the heat supplied to the solid or liquid is used for raising the temperature. Thus there is only one value of specific heat for solids and liquids.
- When gases are heated small change in temperature causes considerable change in both the volume and the pressure. Due to which the specific heat of a gas can have any value between 0 and ∞. Therefore to fix the values of specific heat of a gas, either the volume or pressure is kept constant. Hence it is necessary to define two specific heats of gases. viz. Specific heat at constant pressure and specific heat at constant volume.

**Molar Specific Heat of Gas at Constant Volume:**

- The quantity of heat required to raise the temperature of one mole of a gas through 1K (or 1 °C) when the volume is kept constant is called molar specific heat at constant volume.
- It is denoted by C
_{V}. - Its S.I. unit is J K
^{-1}mol^{-1}.

**Molar Specific Heat of Gas at Constant Pressure:**

- The quantity of heat required to raise the temperature of one mole of a gas through 1K (or 1 °C) when pressure is kept constant is called molar specific heat at constant pressure.
- It is denoted by C
_{P}. - Its S.I. unit is J K
^{-1}mol^{-1}.

**Principal Specific Heat of Gas at Constant Volume:**

- The quantity of heat required to raise the temperature of unit mass of a gas through 1 K (or 1 °C) when its volume is kept constant, is called its principal specific heat at constant volume.
- It is denoted by c
_{V}. - Its S.I. unit is J K
^{-1}kg^{-1}.

**Principal Specific Heat of Gas at Constant Pressure:**

- The quantity of heat required to raise the temperature of unit mass of a gas through 1 K (or 1 °C) when its pressure is kept constant, is called its principal specific heat at constant pressure.
- It is denoted by c
_{P}. - Its S.I. unit is J K
^{-1}kg^{-1}.

#### Relation Between Molar Specific Heats and Principal Specific Heats:

- Molar specific heat at constant volume is Molecular mass times the principal specific heat at constant volume

C_{V} = M c_{V}

Note: Capital and small case used for C.

- Molar specific heat at constant pressure is Molecular mass times the principal specific heat at constant pressure

C_{P} = M c_{P}

Note: Capital and small case used for C.

**Explanation of C**_{P} ** >C**_{V} **:**

_{P}

_{V}

- When a gas is heated at constant volume there is no expansion of gas thus external work done is zero. Hence the heat given to the gas is completely used for the increase in the internal energy of the gas.
- Whereas when a gas is heated at constant pressure the heat given to the gas is used for two purposes i.e. a part of it is used for an increase in the internal energy of the gas and a part of it is used for doing external work.
- Now in both the cases of constant volume and constant pressure, the increase in internal energy is the same as the rise in temperature is the same for the same mass of the gas. Therefore, heat supplied at constant pressure is more than heat supplied at constant volume by an amount of heat which is used for doing external work.
- This explains why specific heat of a gas at constant pressure is greater than the specific heat at constant volume. i.e. C
_{P }> C_{V}_{.} - Further by Mayer’s relation C
_{P }– C_{V}= R . R is universal gas constant and is positive. Hence C_{P }> C_{V}_{.}

**Mayer’s Relation: **

- Let us consider a one le of a perfect gas enclosed in a cylinder fitted with a frictionless weightless airtight movable piston. Let P, V and T be the pressure, volume and absolute temperature of the gas respectively. Let the gas be heated at constant volume so that its temperature rises by dT. Let dQ
_{1 }be the heat given to the gas for this purpose. In this case, all the heat supplied to the gas is used for increasing the internal energy of the gas.

Now, dQ_{1} = 1 × C_{V} × dT = dE ………… (1)

Where C_{V} is the molar specific heat of the gas at constant volume.

- Now let the gas be heated at constant pressure so that its temperature rises by dT. Let dQ
_{2}be the heat given to the gas for this purpose.

Now, dQ_{2} = 1 × C_{P} × dT = C_{P} × dT ………… (2)

Where C= 1 × C_{P} × dT is the molar specific heat of the gas at constant pressure.

- In this case the heat supplied to the gas is used for two purposes i.e. a part of it is used for increase in the internal energy of the gas and a part of it is used for doing external work. Thus,

dQ_{2} = dE + dW ………… (3)

From equations (1) ,(2) and (3)

C_{P} × dT = C_{V} × dT + dW ………… (4)

- The gas is heated at a constant pressure so that it expands and in the process, the piston moves upward through a distance dx. If A is the area of cross-section of the cylinder (area of piston) and dV is the increase in the volume of the gas,

Then, dV = A dx. ……… (5)

Let dW be the work done by the gas during the increase in the volume.

Now, Work *=* Force × Displacement

But, Force = Pressure × Area

∴ Work = Pressure × Area × Displacement

dW = P × A × dx ……… (6)

From equations (5) and (6) we have

dW = P dV ……… (7)

From equations (7) and (4)

C_{P} × dT = C_{V} × dT + P .dV………… (8)

For one mole of a perfect gas

P V = R T

Where, R is the universal gas constant.

Differentiating both sides with respect to temperature.

P dV *=* R dT. ……………. (9)

From equations (8) and (9)

C_{P} × dT = C_{V} × dT + R.dT

∴ C_{P} = C_{V} + R

∴ C_{P} – C_{V} = R

This relation is known as Mayor’s relation between the two molar specific heats of a gas.

**Relation Between Principal Specific Heats of Gases:**

- Let C
_{P}– C_{V}= R be the molar heat capacities of the gas at constant pressure and constant volume. By Mayer’s relation

C_{P} – C_{V} = R ………..(1)

- Let c
_{P}and c_{V}be the principal heat capacities of the gas at constant pressure and constant volume. Let M be the molecular weight of the gas.

Then, C_{V} = M c_{V }and C_{P} = M c_{P}

Substituting these values in equation (1)

M c_{P } – M c_{V} = R

∴ M (c_{P } – c_{V}) = R

∴ (c_{P } – c_{V}) = R/M

∴ (c_{P } – c_{V}) = r

Where, r = gas constant for gas in consideration.

It is different for different gases.

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