Law of Equipartition of energy

According to this law, in equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to (1/2)k_BT.

1. Each translational degree of freedom contributes (1/2) k_B
2. Each rotational degree of freedom contributes (1/2) k_B
3. Each vibrational degree of freedom contributes 2x (1/2)k_B

Specific Heat Capacity for monoatomic gases

• Monoatomic gases will only have translational degree of freedom.
• Maximum they can have is three translational degrees of freedom.
• Each degree of freedom will contribute (1/2) k_B
• Therefore 3 degrees of freedom will contribute (3/2) k_B
• By using law of equipartition of energy, the total internal energy of 1 mole of gas U=(3/2) k_BTxN_A=(3/2) RT
• Specific heat capacity at constant volume C_V= dU/dT=(3/2) R(i)
• For an ideal gas C_P-C_V=R, By using equation(i)C_P=(5/2)R
• Ratio of specific heatsγ=C_P/C_V=(5/3)

Specific Heat of Diatomic gases (rigid)

• A rigid diatomic gas means they will have translational as well as rotational degree of freedom but not vibrational.
• They are rigid oscillator.
• A rigid diatomic molecule will have 3 translational degrees of freedom and 2 rotational degrees of freedom. Total 5 degrees of freedom.
• By law of equipartition of energy, each degree of freedom will contribute (1/2) k_B
• Therefore 5 degree of freedom will contribute (5/2) k_B
• Therefore the total internal energy of 1 mole of gas, U=(5/2) k_BTxN_A=(5/2)RT
• Specific heat capacity at constant volume(C_V) =dU/dT=(5/2)R
• Specific heat capacity at constant pressure of a rigid diatomic is given as C_P=(7/2)R
• Ratio of specific heats γ=C_P/C_V=(7/5)

Specific Heat of Diatomic gases (non-rigid)

• A no-rigid diatomic gas has translational, rotational as well as vibrational degrees of freedom.
• There will be 3 translational degrees of freedom and 2 rotational degrees of freedom and 1 vibrational degree of freedom.
• Total contribution by translational= (1/2) k_BT, rotational=2x (1/2)k_BT and vibrational =k_B
• Total Internal energy for 1 mole =(5/2)k_BT+k_BT = (7/2)k_BT= (7/2)RT.
• C_V=dU/dT = (7/2) R.
• C_P=C_V+R= (9/2) R.
• γ= C_P/C_V =(9/7)

Specific Heat Capacity for polyatomic gases

• Polyatomic gases will have 3 translational degree of freedom, 3 rotational degrees of freedom and ‘f’ number of vibrational modes.
• Total internal energy of 1 mole of gas =(3x(1/2)k_BT + 3x(1/2)k_BT+ fk_BT)x N_A

((3/2) + (3/2) +f)RT = (3+f) RT.

• C_V=dU/dT = (3+f)R
• C_P=C_V+R=(4+f)R
• γ= C_P/C_V = (4+f)/(3+f)

Specific Heat Capacity for solids

• Consider there are N atoms in a solid. Each atom can oscillate about its mean position.
• Therefore vibrational degree of freedom = k_BT
• In one-dimensional average energy=k_BT, in three-dimensional average energy =3K_BT
• Therefore total internal energy (U) of 1 mole of solid = 3K_BTxN_A= 3RT
• At constant pressure, ΔQ = ΔU + PΔV change in volume is very less in solids .Therefore ΔV = 0.
• =>ΔQ = ΔU
• C_V=(dU/dT)_v
• C_P=(dQ/dT)_Vas ΔQ = ΔU, Therefore C_V=dU/dT=3R
• Therefore C_P=C_V=3R

Specific Heat Capacity of water

• Consider water as solid,so it will have ‘N’ number of atoms.
• Therefore for each atom average energy =3k_BT
• No of molecules in H₂O= 3 atoms.
• Total internal energy U=3k_BTx3xN_A =9RT.
• C_V=C_P=9R.

Conclusion on Specific heat

• According to classical mechanics, the specific heat which is calculated based on degree of freedom should be independent of temperature.
• However T->0,degree of freedom becomes inefficient.
• This shows classical mechanics is not enough; as a result quantum mechanics came into play.
• According to quantum mechanics minimum non-zero energy is required for degree of freedom to come into play.
• Specific heats of all substances approach zero as T->0.