# Degree of Freedom and Specific Heats of Gas

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### 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 an 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

ET = ETranslational  + ERotational + EVibrational ……… (1)

• For translational motion, the molecule has three degrees of freedom (along 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

#### Ratio of Specific Heat Capacities of Monoatomic Gas:

• The monoatomic gases have only translational motion, hence has three translational degrees of freedom. The average energy of the molecule at temperature T is given by

By 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

#### Ratio of Specific Heat Capacities  of Diatomic Gas:

• 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 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

#### Ratio of Specific Heat Capacities  of Triatomic Gas:

• 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 nonrigid molecules there is additional vibrational motion.

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

By law of equipartition of energy we have

Thus the energy per mole of the gas is given by

#### Ratio of Specific Heat Capacities of Polyatomic Gas:

• 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.

#### Ratio of Specific Heat Capacities of Polyatomic Gasa 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

#### 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 an oscillation in one dimension is KBT. Thus the average energy of an oscillation is 3KBT.

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

Thus total energy

• Since there is negligible change in 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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