Why is Constant= PV^((Cp/Cv)) For Adiabatic Process?

Why Is Constant:

Someone asked me this question on Quora, and I am giving the answer of it in this post.

·        Not only Adiabatic but all type of process such as Isochoric process, Isobaric Process, Isothermal process, etc. can be represented by one single equation which is PVk=C, where k represents the slope of process on P-V diagram & C is a constant. Different types of thermodynamics system and transformation of ideal gas equation

·        Value of k decides types of process. And these values are obtained by series of an experiment. Let’s see the proof.

First just take k=0, so if k is slope process on P-V diagram, then the graph of the process will be the straight line. And if you put k=0 in above equation then you will get P=C. So the process will be Isobaric.

As above value of k=∞ graph of a process will be vertical on a P-V diagram which is for an isochoric process. So as K=1 for an isothermal process. And so as a value of k=1.4 or k= (Cp÷ Cv) represents Adiabatic Process.Real gas equation

Here as per shown in a graph you also can decide types of processes as per their slopes on P-V diagram.

DERIVATION OF CONDITION FOR 

ADIABATIC BY IDEAL GAS EQUATION: 

PV =mRT (ideal gas equation)

Differentiate above equation

V(dP) + P(dV) = mR (dT)  -------------- (1)

Now, dQ = dU + dW (First law of thermodynamics) ---------------- (2)We

For adiabatic process, as we know, dQ = 0 (there is no heat transfer)

& dU = m Cv dT & dW = P dV

So from equation (2)

dW = - m Cv dT         Þ    P dV÷Cv=  - m  dT                   --------------- (3)

Now from equation (1) and (3)

(P (dV))/Cv  = -  (V(dP) + P(dV))/R

Take R = Cp - Cv, and by rearranging above equation, you will get

dP/P+ ɣ dV/V=0

Now by integrating above equation, you can get

lnP+ ɣ lnV = lnC    ( Here C is constant )

 C=P V^(ɣ  ) ( Here ɣ = Cp/Cv  = ratio of specific heats )

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