Degree of Freedom and Equipartition Principle :
According to Law of equipartition of energy
(i) each translation and rotational degree of freedom in a molecule contributes `1/2 RT` to the thermal energy of one mole of a gas, and
(ii) each vibrational degree of freedom in a molecule contributes `RT` to the thermal energy of one mole of a gas.
The degree of freedom in a molecule are given by the number of coordinates required to locate all the mass points (atoms) in a molecule. If a molecule contains only one atom (as in a monatomic gas), it has three degree of freedom corresponding to translational motion in the three independent spatial directions `X`, `Y` and `Z`. lf a molecule contain `N` atoms, each atom contributes these three
degree of freedom, so the molecule has a total of `3N` degree of freedom. Since three coordinates (degree of freedom) are required to represent the translational motion of the molecule, the remaining (`3N - 3`) coordinates represent what are called the internal degree of freedom. If the molecule is linear, it has two rotational degrees of freedom; for a non-linear molecule, there are three rotational degree of freedom. The remaining degrees of freedom, that is `3N - 5` for linear and `3N- 6` for non-linear molecules are the vibrational degree of freedom. Table list the degrees of freedom for several molecular system.
In a monatomic molecule `E = 3RT//2` is in agreement with the simple model. For a diatomic molecule, there are three translational, two rotational (because the molecule is linear) and One vibrational degree of freedom making a total of six. The thermal energy per mole.
`barE = (1/2 RT)_(trans) + (1/2 RT)_(rot) + (1 RT)_(vib)`
and `barC_v = 3 R//2 + R + R = 7 R//2 cal deg^(-1) mol^(-1)`
Table shows that the observed of `barC_v` for diatomic deviate greatly from the predicted values. The fact that the observed values of `5 cal deg^(-1) mol^(-1)` (which is close to `5R//2` ) is most common for simple diatomic molecules shows that vibration degree of freedom are active only at very high temperature. The following graph shows.
Note: Variation of `C_v` with temperature highlights the fact that with increase in the temperature the vibration modes of motion also contribute to the heat capacity
Adiabatic exponent : Adiabatic exponent (`gamma`) for a mixture of gas with different heat capacity is defined as:
`gamma_(mix) = (n_1C_(P_1) + n_2C_(P_2)..........)/(n_1C_(v_1) +n_2C_(v_2)............)`
where `n_1, n_2 ........................` are moles of different gases.
(i) each translation and rotational degree of freedom in a molecule contributes `1/2 RT` to the thermal energy of one mole of a gas, and
(ii) each vibrational degree of freedom in a molecule contributes `RT` to the thermal energy of one mole of a gas.
The degree of freedom in a molecule are given by the number of coordinates required to locate all the mass points (atoms) in a molecule. If a molecule contains only one atom (as in a monatomic gas), it has three degree of freedom corresponding to translational motion in the three independent spatial directions `X`, `Y` and `Z`. lf a molecule contain `N` atoms, each atom contributes these three
degree of freedom, so the molecule has a total of `3N` degree of freedom. Since three coordinates (degree of freedom) are required to represent the translational motion of the molecule, the remaining (`3N - 3`) coordinates represent what are called the internal degree of freedom. If the molecule is linear, it has two rotational degrees of freedom; for a non-linear molecule, there are three rotational degree of freedom. The remaining degrees of freedom, that is `3N - 5` for linear and `3N- 6` for non-linear molecules are the vibrational degree of freedom. Table list the degrees of freedom for several molecular system.
In a monatomic molecule `E = 3RT//2` is in agreement with the simple model. For a diatomic molecule, there are three translational, two rotational (because the molecule is linear) and One vibrational degree of freedom making a total of six. The thermal energy per mole.
`barE = (1/2 RT)_(trans) + (1/2 RT)_(rot) + (1 RT)_(vib)`
and `barC_v = 3 R//2 + R + R = 7 R//2 cal deg^(-1) mol^(-1)`
Table shows that the observed of `barC_v` for diatomic deviate greatly from the predicted values. The fact that the observed values of `5 cal deg^(-1) mol^(-1)` (which is close to `5R//2` ) is most common for simple diatomic molecules shows that vibration degree of freedom are active only at very high temperature. The following graph shows.
Note: Variation of `C_v` with temperature highlights the fact that with increase in the temperature the vibration modes of motion also contribute to the heat capacity
Adiabatic exponent : Adiabatic exponent (`gamma`) for a mixture of gas with different heat capacity is defined as:
`gamma_(mix) = (n_1C_(P_1) + n_2C_(P_2)..........)/(n_1C_(v_1) +n_2C_(v_2)............)`
where `n_1, n_2 ........................` are moles of different gases.