Adiabatic process and Work done in an adiabatic expansion

Adiabatic process

In Greek, adiabatic means ?nothing passes through?. The process in which pressure, volume and temperature of a system change in such a manner that during the change no heat enters or leaves the system is called adiabatic process. Thus in adiabatic process, the total heat of the system remains constant.

Let us consider a gas in a perfectly thermally insulated cylinder fitted with a piston. If the gas is compressed suddenly by moving the piston downward, heat is produced and hence the temperature of the gas will increase. Such a process is adiabatic compression.

If the gas is suddenly expanded by moving the piston outward, energy required to drive the piston is drawn from the internal energy of the gas, causing fall in temperature. This fall in temperature is not compensated by drawing heat from the surroundings. This is adiabatic expansion.

Both the compression and expansion should be sudden, so that there is no time for the exchange of heat. Hence, in an adiabatic process always there is change in temperature.

Expansion of steam in the cylinder of a steam engine, expansion of hot gases in internal combustion engine, bursting of a cycle tube or car tube, propagation of sound waves in a gas are adiabatic processes.

The adiabatic relation between P and V for a gas, is

PVγ = k, a constant  ??..(1)

where γ = specific heat capacity of the gas at constant pressure / specific heat capacity of the gas at constant volume

From standard gas equation,

PV = RT

P=RT/V

substituting the value P in (1)

(RT/V ) V^γ = constant

RT/ V^γ-1 = constant

In an adiabatic process Q =constant

∴ ∆Q = 0

∴ specific heat capacity C = ∆Q/m∆T

C=0

Work done in an adiabatic expansion

Consider one mole of an ideal gas enclosed in a cylinder with perfectly non conducting walls and fitted with a perfectly frictionless, non conducting piston.

Let P₁, V₁ and T₁ be the initial pressure, volume and temperature of the gas. If A is the area of cross section of the piston, then force exerted by the gas on the piston is

F = P ? A, where P is pressure of the gas at any instant during expansion. If we assume that pressure of the gas remains constant during an infinitesimally small outward displacement dx of the piston,

then work done dW = F ? dx = P ? A dx dW = P dV

Total work done by the gas in adiabatic expansion from volume V₁ to V₂ is

W = ∫_v1^v2 PdV

But PV^γ = constant (k) for adiabatic process

where γ = C_p/C_v

W=∫_v1^v2kV^{- γ}dV = k[V^{- γ} /I- γ] _v1^v2

W = 1/(1- γ)[kV₂^1-γ ? kV₁^1-γ]

but, P₂V₂γ = P₁V₁γ = k   ??.(2)

Substituting the value of k in (1)

W = 1/(1- γ)[P₂V₂ ? P₁V₁] 

If T2 is the final temperature of the gas in adiabatic expansion, then

P₁V₁ = RT₁,

 P₂V₂ = RT₂

Substituting in (3)

W = 1/(1- γ)[RT₂ ? RT₂] 

W= R/(1- γ)[ T₂ ?T₂] 

This is the equation for the work done during adiabatic process.