A change in Pressure & Volume of gas in which temperature also changes is called an adiabatic process“.
In such a change no heat is allowed to enter into or escape from gas.

Essential Conditions for a perfect Adiabatic Change :-

1. The walls of the container must be perfectly non conducting in order to prevent any exchange of heat between system & surroundings.

2. The process of compression & expansion should be sudden so that there is no time for the exchange of heat.

Adiabatic Equation of a perfect gas :-
Let us consider one gm molecule of gas thermally insulated from surrounding. Let it suffer a very small adiabatic expansion doing external work at cost of its internal energy.
If the volume of gas increases by an infinitesimal amount dV against an external pressure P , the external work done by gas in its expansion will be

dW = PdV  ……………..(i)

Since in a perfect gas,the molecules do not attract one another , the internal energy depends only on its tempreture.
Hence the decrease in internal energy of the gas,suffering a fall dT in its temperature , is equal to heat drawn from it i.e.

dU = 1× Cv ×dT   …………(ii)
where,

Cv = specific heat for one gm. mol of gas at constant volume.

Using first law of thermodynamics

dQ = dU +dW

or           dQ = CvdT + PdV
Since , In adiabatic change, dQ =0

Therefore, we have

CvdT +P.dV = 0    ……… (iii)

For one gram molecule of perfect gas,

PV = RT, which on differentiation gives ,

P.dV + V.dP = RdT
or,         dT = P.dV + V.dP / R
Put dT in eq. (iii)
Therefore,

Cv( P.dV +V.dP) /(R) + P.dV = 0

or,    Cv(P.dV + V.dP) + R(P.dV) = 0
But from Mayer’s formulae

Cp – Cv = R
Therefore,

Cv(P.dV + V.dP) + (Cp – Cv)P.dV = 0

or            Cv.V.dP + Cp.P.dV = 0
Now divide by CvPV,

dP/P +CpdV/CvV =0

But ,     Cp/Cv = γ
Hence,      dP /P + γdV/V = 0
Integrating ,

logP + γlogV = constant

log PVγ = constant
or
PVγ = constant    ………(iv)

This equation connecting Pressure & Volume & is known as Poisson’s law.
There are two other forms of above relation.

Relation between T & V:-
Putting P= RT/V in eq. (iv).

RT/V.Vγ = constant
or ,            TVγ-1 = constant

Relation between T & P :-
Putting V = RT/ P in eq. (iv)
we get ,

P(RT/P)γ = constant
or ,   TP(1-γ)/γ = constant

Thus, we’ve discussed Adiabatic process & Adiabatic equation for a perfect gas, Any query ask us in comment box.