Question

A diatomic gas ($\gamma = 1.4$) does 100 J of work when it is expanded isobarically. Then the heat given to the gas is _________ J.

Hint:

For any isobaric process, the ratio of heat added to work done is $\frac{Q}{W} = \frac{C_p}{R} = \frac{\gamma}{\gamma - 1}$. For diatomic gases, this ratio is $3.5$.

Answer 1

Correct Answer: 350

To find the heat given to a diatomic gas that undergoes isobaric expansion, we use the first law of thermodynamics, which is expressed as: ΔQ = ΔU + W where: ΔQ is the heat added to the system, ΔU is the change in internal energy, and W is the work done by the system. Given, W = 100 J and the specific heat ratio γ = 1.4 for a diatomic gas, the relationship between the molar heat capacities is: C_p = γC_v For an isobaric process, C_p is used, and ΔU for a diatomic gas is: ΔU = nC_vΔT We know: C_v = R/(γ-1) For diatomic gases, C_v = (5R/2) and C_p = (7R/2), hence: ΔU = (5R/2)ΔT and W (At constant pressure) = nRΔT For the work done, 100 J = nRΔT Therefore, we rewrite ΔU: ΔU = (5/2)(100 J)/(RΔT)/nR = (5/2) × 100 Now back to: ΔQ = (5/2) × 100 + 100 = 250 + 100 = 350 J Conclusively, the heat given to the gas is 350 J, which fits perfectly within the expected range of 350 to 350 J.