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Question

The relation between internal energy U, pressure P and volume V of a gas in an adiabatic process is given as
U=a+bPV
where a and b are constants. What is the value of adiabatic constant (γ)?
  1. ab
  2. b+1b
  3. a+1a
  4. ba


Solution

The correct option is B b+1b
From first law of thermodynamics,
dQ=dU+dW ...(i)
we can write small amount of work done as dW=PdV
For an adiabatic process heat supplied is zero i.e dQ=0
From Eq(i),
dQ=dU+PdV
0=dU+PdV
substituting value of U
 d(a+bPV)+PdV=0 ...(ii)
since a & b are constants, da=0, db=0
Applying chain rule for finding d(bPV) and putting in Eq (ii)
 bPdV+bVdP+PdV=0
or (b+1)PdV+bVdP=0
Dividing both sides by PV,
(b+1)dVV+bdPP=0 ...(iii)
Integrating Eq.(iii) both sides,
(b+1)logV+blogP=constant
logVb+1+logPb=constant
Applying logm+logn=log(mn)
log(Vb+1Pb)=constant
Vb+1Pb=constant
Taking power 1b both sides,
PVb+1b=constant1b
PVb+1b=constant ...(iv)
Comparing Eq. (iv) with PVγ=constant
γ=b+1b

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