Question

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of \( \Delta Q : \Delta U : \Delta W \) is:

• A. \(2:3:5\)
• B. \(5:3:2\)
• C. \(2:5:7\)
• D. \(7:5:2\)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
From the first law of thermodynamics, heat supplied (\(\Delta Q\)) is the sum of change in internal energy (\(\Delta U\)) and work done (\(\Delta W\)). At constant pressure, these quantities relate to the degrees of freedom of the gas molecules.
Step 2: Key Formula or Approach:
1. \(\Delta Q = n C_p \Delta T\)
2. \(\Delta U = n C_v \Delta T\)
3. \(\Delta W = P \Delta V = n R \Delta T\)
4. For diatomic gas: \(C_v = \frac{5}{2}R\), \(C_p = \frac{7}{2}R\) (degree of freedom \(f=5\)).
Step 3: Detailed Explanation:
Substitute the values for a diatomic gas into the definitions:
\[ \Delta Q = n \left( \frac{7}{2}R \right) \Delta T \]
\[ \Delta U = n \left( \frac{5}{2}R \right) \Delta T \]
\[ \Delta W = n R \Delta T \]
The ratio is:
\[ \Delta Q : \Delta U : \Delta W = \frac{7}{2} : \frac{5}{2} : 1 \]
Multiply by 2 to get integer values:
\[ \Delta Q : \Delta U : \Delta W = 7 : 5 : 2 \]
Step 4: Final Answer:
The ratio \(\Delta Q : \Delta U : \Delta W\) is 7 : 5 : 2.