Partition function for ideal gas

Quantum Particle in a 3D Box

Consider a gas molecule in thermal equilibrium at temperature T. At any instant, the velocity components along x, y, and z directions are: v_x, v_y, v_z.

Kinetic Energy Components

Energy along x-direction:

E_x = (1/2) m v_x² = p_x² / (2m)

Where p_x is the momentum component along x-direction.

Quantum Condition

According to quantum mechanics, a moving particle behaves like a wave. For a particle confined in a box of length L_x:

p_x(2L_x) = n_x h

Where:

• h = Planck's constant
• n_x = positive integer
• ħ = h / (2π)

p_x = ħπ (n_x / L_x)

Energy in Each Direction

E_x = (ħ²π² / 2m) (n_x / L_x)²

E_y = (ħ²π² / 2m) (n_y / L_y)²

E_z = (ħ²π² / 2m) (n_z / L_z)²

Total Energy

E_i = E_x + E_y + E_z

E_i = (ħ²π² / 2m) [ (n_x² / L_x²) + (n_y² / L_y²) + (n_z² / L_z²) ]

Here, E_i represents the energy of the i-th state defined by quantum numbers n_x, n_y, n_z.

Partition Function

Z = Σ e^-βE_i

Z = Σ Σ Σ exp[-β (ħ²π² / 2m) ( n_x² / L_x² + n_y² / L_y² + n_z² / L_z² )]

Separated Form

Z = (Σ e^{-β (ħ²π² / 2m)(n_x² / L_x²)}) × (Σ e^{-β (ħ²π² / 2m)(n_y² / L_y²)}) × (Σ e^{-β (ħ²π² / 2m)(n_z² / L_z²)})