Adiabatic Process & Adiabatic Equation for a perfect gas

Adiabatic Process

“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 Process:

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 the 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 was done by the 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 temperature.
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 the above relation.


The 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 the adiabatic process & Adiabatic equation for a perfect gas. If still any query left, ask us in the comment box.

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