Question

Which of the following is true in the case of an adiabatic process where $\gamma =C_p/C_v$?

• A. $P^{1-\gamma }{T}^{\gamma }=constant$
• B. $P^{\gamma }{T}^{1-\gamma }=constant$
• C. $P{T}^{\gamma }=constant$
• D. $P^{\gamma }T=constant$

Answer 1

The correct option is A $P^{1-\gamma }{T}^{\gamma }=constant$

In an adiabatic process heat supplied is zero. Hence internal energy change equals work done.

Hence, $W=\Delta U$

$-PdV=\frac{nR}{\gamma -1}dT=\frac{PdV+VdP}{\gamma -1}$

$-\gamma PdV=VdP$

$-\gamma \frac{dV}{V}=\frac{dP}{P}$

$ln\left(P\right)+\gamma ln\left(V\right)=C$

$P{V}^{\gamma }=constant$

Putting $V=\frac{nRT}{P}$

$P^{1-\gamma }{T}^{\gamma }=constant$

Answer is option A.