Question

Write the expression for the internal energy (U) of n moles of a diatomic gas at temperature T​

Answer 1

The internal energy (U) for a diatomic gas at temperature T is 5/2 NkT.

• A diatomic gas such as H₂, O₂, etc. have their internal energies as the sum of translational kinetic energies and rotational kinetic energies.
• The diatomic molecules have 3 degrees of freedom for translational motion hence the internal energy for translational kinetic energy is 3/2 NkT.

• The diatomic molecules have 2 degrees of freedom for rotational motion hence the internal energy for rotational kinetic energy is 2/2 NkT.
• Therefore U = 3/2NkT + 2/2NkT = 5/2 NkT.

Answer 2

To write:

Expression for internal energy (U) for n moles of diatomic gas at temperature T .

Calculation:

The general expression for internal energy for any ideal gas is :

$\therefore \: \Delta U = n \times C_{V} \times \Delta T$

• ∆U is change in internal energy.
• n is moles
• C_(V) is Specific Heat capacity at constant volume
• ∆T is change in temperature

Now, at T = 0 Kelvin (absolute zero) , internal energy will be zero.

So, at T = T Kelvin, we can say:

$\implies\: U - 0 = n \times C_{V} \times (T - 0)$

$\implies\: U = n \times C_{V} \times T$

For a diatomic gas , C_(V) = 5R/2 ( R is Universal Gas Constant):

$\implies\: U = n \times \dfrac{5R}{2} \times T$

$\implies\: U = \dfrac{5nRT}{2}$

So, final answer is:

$\boxed{ \bf\: U = \dfrac{5nRT}{2}}$