“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)
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)
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
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
logP + γlogV = constant
log PVγ = constant
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.