Question

If the ratio of molar heat capacities \(\frac{C_{p}}{C_{v}}=\gamma\) then the ratio of \(\frac{\Delta Q}{\Delta U}\) and \(\frac{\Delta Q}{\Delta W}\) in an isobaric process is

• A. \(2\gamma ,\frac{\gamma }{2}\)
• B. \(\gamma ,\frac{\gamma }{(\gamma -1)}\)
• C. \(\frac{\gamma }{(\gamma -1)}\)
• D. \(\frac{1}{\gamma },(\gamma -1)\)

Hint:

Write heat, internal energy change and work for an isobaric process in terms of Cp, Cv and the temperature change.

Answer 1

The Correct Option is B

Step 1: Use the First Law
$\Delta Q=\Delta U+\Delta W$. Divide by $\Delta Q$ and use $\dfrac{\Delta U}{\Delta Q}=\dfrac{1}{\gamma}$.

Step 2: Find the Work Fraction
$\dfrac{\Delta W}{\Delta Q}=1-\dfrac{1}{\gamma}=\dfrac{\gamma-1}{\gamma}$, so $\dfrac{\Delta Q}{\Delta W}=\dfrac{\gamma}{\gamma-1}$.

Step 3: Compare the options
The pair $\gamma,\dfrac{\gamma}{\gamma-1}$ is the second option.

Final Answer:
The ratios are gamma and gamma/(gamma - 1). \[ \boxed{\text{(B) }\gamma ,\dfrac{\gamma}{(\gamma-1)}} \]