Question

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K respectively. Choose the correct statement:

• A. The change in internal energy in the process AB is -350 R.
• B. The change in internal energy in the process BC is -500 R.
• C. The change in internal energy in whole cyclic process is 250 R.
• D. The change in internal energy in the process CA is 700 R.

Answer 1

The correct option is B The change in internal energy in the process BC is -500 R.

a) At constant volume the change in internal energy $\left(dU\right)$ = $C_{v}dT$. For diatomic ideal gas the value of $C_v=\frac{5R}{2}$
Hence,
$dU=\frac{5R}{2}\times (800-400)$
$=1000R$....(i)
Hence (a) is wrong .

b) Since $C\to A$ is isobaric process. Using first law of thermodynamics
$dU=dq+dw$
$=C_{p}dT-P(V_2-V_1)$
$=C_{p}dT-nR(T_2-T_1)\because PV=nRT$
$=(C_p-nR)dT$
$=C_{v}dT$ (using $C_p-C_v=nR$)
$=\frac{5R}{2}(400-600)$
$=-500R$.....(ii)
Since the given process is a cyclic process and $U$ is a state function. Hence $d{U}_{cycle}=0$
$d{U}_{A\to B}+d{U}_{B\to C}+d{U}_{C\to A}=0$
Using equation (i) and (ii)
$d{U}_{B\to C}=-500R$