**Adiabatic Process**

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*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)

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)

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

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

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

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

Cp – Cv = R

Therefore,

Cv(P.dV + V.dP) + (Cp – Cv)P.dV = 0

or Cv.V.dP + Cp.P.dV = 0

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

dP/P +CpdV/CvV =0

But , Cp/Cv = γ

Hence, dP /P + γdV/V = 0

Integrating ,

logP + γlogV = constant

log PVγ = constant

or

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 ,

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

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.