A diatomic gas, having C_P = R and C_V =  R, is heated at constant pressure. The ratio dU ∶ dQ ∶ dW :

CONCEPT:

→The change in the internal energy is written as;

dU = nC_vdT

Here dU is the internal energy, C_v is the specific heat at constant volume, and dT is the change in temperature.

→The change in the heat energy is written as;

dQ = nC_pdT

Here dQ is the heat energy, C_p is the specific heat at constant pressure, and dT is the change in temperature.

→The change in work done is defined as;

dW = nRdT

Here we have R as the gas constant, and dT is the change in temperature.

CALCULATION:

Given: Specific heat at constant pressure 

and specific heat at a constant temperature 

The change in the internal energy is written as;

dU = nCvdT

⇒ dU =  dT    ----(1)

The change in the heat energy is written as;

dQ =  dT     ----(2)

The change in work done is defined as;

dW = nRdT     ----(3)

Now, on taking the ratio of (1), (2), and (3) we have;

dU ∶ dQ ∶ dW = 5 ∶ 7 ∶ 2

Hence, option 1) is the correct answer.