In this article, we shall study the concept of specific heat capacities or Specific heats of gases.

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

In the case of solids and liquids, a 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 a considerable change in both the volume and the pressure. Due to which the specific heat of 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 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 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 a unit mass of 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 a unit mass of 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

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

C_P = M c_P

Explanation of C_P >C_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 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 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 the universal gas constant and is positive. Hence C_P > C_V.

Mayer’s Relation:

Let us consider one mole 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 a constant volume so that its temperature rises by dT.  Let dQ₁ 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, dQ₁ =   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 dQ2 be the heat given to the gas for this purpose.

Now, dQ₂ =   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 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. Thus,

dQ₂ = dE   +   dW    …………  (3)

From equations (1) ,(2) and (3)

C_P ×   dT = C_V ×   dT +   dW …………  (4)

The gas is heated at 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 the 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:

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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In this article, we shall study the concept of degree of freedom of gas molecules. Degree of freedom for different types of gases. We shall also find the ratio of specific heats for different types of gases.

Degree of Freedom of a Gas Molecule:

A molecule free to move in space needs three coordinates to specify its location. If a molecule is constrained to move along a line it requires one co-ordinate to locate it. Thus it has one degree of freedom for motion in a line. If a molecule is constrained to move in a plane it requires two coordinates to locate it. Thus it has two degrees of freedom for motion in a plane. If a molecule is free to move in a space it requires three coordinates to locate it. Thus it has three degrees of freedom for motion in a space.

Law of Equipartition of Energy:

Statement: In equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having average energy equal to

Explanation: A molecule has three types of kinetic energy a) Translational kinetic energy, b) Rotational kinetic energy, and c) Vibrational kinetic energy. Thus total energy of a molecule is given by

E_T = E_{Translational} + E_Rotational + E_Vibrational ……… (1)

For translational motion, the molecule has three degrees of freedom (along the x-axis, along the y-axis and along the z-axis). Hence,

For rotational motion, it has two degrees of freedom along its centre of mass (clockwise and anticlockwise).

For vibrational motion only one degree of freedom (to and fro).

Where k is the force constant and y is the vibrational coordinate.

Thus the total energy of the molecule is

It is to be noted that in the vibrational mode the energy has two components potential and kinetic and by the law of equipartition energy, each part is equal to

Thus the total vibrational component of the energy is

Specific Heats of Gases:

In the 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 a considerable change in both the volume and the pressure. Due to which the specific heat of 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 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 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.

Relation Between C_P and C_V :

C_P – C_V = R

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

The ratio C_P/C_V is known as ratio of specific heats and is denoted by the letter ‘γ’

Ratio of Specific Heats for Different Gases:

Monoatomic Gases:

Helium and argon are monoatomic gases. The monoatomic gases have only translational motion, hence they have three translational degrees of freedom. The average energy of the molecule at temperature T is given by

By the law of equipartition of energy we have

Thus the energy per mole of the gas is given by

Thus the ratio of specific heat capacities of monoatomic gas is 1.67

Diatomic Gases:

Dinitrogen, dioxygen, and dihydrogen are diatomic gases. The diatomic gases have translational motion (three translational degrees of freedom) as well as rotational motion(rotational degree of freedom). The average energy of the molecule at temperature T is given by

By the law of equipartition of energy we have

Thus the energy per mole of the gas is given by

Thus the ratio of specific heat capacities of diatomic gas is 1.4

Triatomic Gas:

Trioxygen (ozone) and carbondioxide are triatomic gases. The triatomic gases have translational motion, rotational motion as well as vibrational motion, hence has three translational degrees of freedom and two rotational degrees of freedom. For non-rigid molecules, there is an additional vibrational motion.

The average energy of the molecule at temperature T is given by

By the law of equipartition of energy we have

Thus the energy per mole of the gas is given by

Polyatomic Gases:

A polyatomic molecule has 3 translational, 3 rotational and certain number (say f) of vibrational modes. By the law of equipartition of energy, one mole of such gas has

Thus the energy per mole of the gas is given by

This is an expression for the ratio of specific heats of polyatomic gases. Where f is the degree of freedom of vibration.

Polyatomic Gas having ‘f’ degree of Freedom:

By the law of equipartition of energy for ‘f’ degree of freedom we have

Thus the energy per mole of the gas is given by

The ratio of specific heats 1+ 2/f

The Expression for the Molar Specific Heat Capacity of Solid:

Consider a solid of N atoms, each vibrating about mean position. These atoms don’t have translational or rotational modes. The average energy of oscillation in one dimension is K_BT. Thus the average energy of oscillation is 3K_BT.

For one mole of Solid N = Avogadro’s number = N_A.

Thus total energy 

Since there is negligible change in the volume of a solid on heating, solids have only one specific heat.

∴ C = 3R

Molar Specific Heat Capacity of Water:

A water molecule has three atoms (2hydrogens and 1 oxygen). If we treat water like solid, the total energy of water molecules is three times the average energy of an atom of solid. Thus for Water

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