1. Use internal energy formula for a diatomic ideal gas

For one mole of a diatomic ideal gas (assuming no vibrational modes active), $C_V = \frac{5R}{2}$ So the change in internal energy is $\Delta U = n C_V \Delta T = \frac{5R}{2}(T_2-T_1)$ for $n=1$ mole.

Since internal energy depends only on temperature, we can compute it directly for each process.

2. Process $AB$

Given: $T_A=400\,K, \quad T_B=800\,K$

Therefore, $\Delta U_{AB} = \frac{5R}{2}(800-400)$ $= \frac{5R}{2}\cdot 400 = 1000R$

So option C which says $\Delta U_{AB}=-350R$ is false.

3. Process $BC$

Given: $T_B=800\,K, \quad T_C=600\,K$

Therefore, $\Delta U_{BC} = \frac{5R}{2}(600-800)$ $= \frac{5R}{2}(-200) = -500R$

So option D is true.

4. Process $CA$

Given: $T_C=600\,K, \quad T_A=400\,K$

Therefore, $\Delta U_{CA} = \frac{5R}{2}(400-600)$ $= \frac{5R}{2}(-200) = -500R$

So option B which says $700R$ is false.

5. Whole cyclic process $ABC$

In a complete cycle, the system returns to the initial state. Since internal energy is a state function, $\Delta U_{cycle}=0$

So option A which says $250R$ is false.

6. Final check of all options

• A: False, because $\Delta U_{cycle}=0$
• B: False, because $\Delta U_{CA}=-500R$
• C: False, because $\Delta U_{AB}=1000R$
• D: True, because $\Delta U_{BC}=-500R$

Hence the correct option is $\boxed{D}$