A diatomic gas, having $$C_P = \frac{7}{2}R$$ and $$C_V = \frac{5}{2}R$$, is heated at constant pressure. The ratio dU : dQ : dW

For a diatomic gas heated at constant pressure, we need to find the ratio $$dU : dQ : dW$$.

The change in internal energy at constant pressure is $$dU = nC_V\,dT = n \cdot \frac{5}{2}R\,dT$$.

The heat supplied at constant pressure is $$dQ = nC_P\,dT = n \cdot \frac{7}{2}R\,dT$$.

The work done by the gas at constant pressure is $$dW = dQ - dU = n(C_P - C_V)\,dT = nR\,dT$$.

Therefore the ratio is $$dU : dQ : dW = \frac{5}{2}R : \frac{7}{2}R : R = 5 : 7 : 2$$.

The correct answer is Option (3): $$5 : 7 : 2$$.