Work Done in an Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat transfer occurs between the system and surroundings. This can happen in two ways:

• The process occurs so rapidly that heat has no time to transfer (e.g., sound wave propagation, sudden compression of gas)
• The system is perfectly thermally insulated (e.g., gas in a perfectly insulated cylinder)

Key condition: ΔQ = 0

From the First Law of Thermodynamics: ΔQ = ΔU + W 0 = ΔU + W Therefore: W = −ΔU

Work done equals the decrease in internal energy of the system.

For an adiabatic process, the relation between pressure and volume is: PV^γ = constant

Integrating this, the work done by the gas is:

W = (P₁V₁ − P₂V₂) / (γ − 1)

Alternatively, using temperature: W = nR(T₁ − T₂) / (γ − 1)

Where:

• P₁, V₁ = initial pressure and volume
• P₂, V₂ = final pressure and volume
• T₁, T₂ = initial and final temperatures
• n = number of moles
• R = universal gas constant (8.314 J/mol·K)
• γ = Cp/Cv (1.4 for diatomic gases like air, O₂, N₂)

Adiabatic Expansion (gas expands, V increases):

• Temperature drops (T₂ < T₁)
• Gas does positive work on surroundings (W > 0)
• Internal energy decreases
• Example: Air rushing out of a punctured tyre feels cold

Adiabatic Compression (gas compressed, V decreases):

• Temperature rises (T₂ > T₁)
• Work done ON the gas (W < 0 for gas)
• Internal energy increases
• Example: Diesel engine — air compressed so strongly it ignites fuel without a spark plug