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

Step 1: Understanding the Concept:
In an isobaric process (constant pressure), heat added to a gas is used both to increase the internal energy and to perform external work.
For an ideal gas, these quantities are related to the molar heat capacities $C_p$ and $C_v$ and the work done $W$.

Step 2: Key Formula or Approach:
For an isobaric process:
Work done, $W = P\Delta V = nR\Delta T$
Heat given, $Q = nC_p\Delta T$
Since $C_p = \frac{\gamma R}{\gamma - 1}$, we can write $Q$ in terms of $W$:
\[ Q = n \left( \frac{\gamma R}{\gamma - 1} \right) \Delta T = \frac{\gamma}{\gamma - 1} (nR\Delta T) = \frac{\gamma}{\gamma - 1} W \]

Step 3: Detailed Explanation:
We are given:
Work done $W = 100$ J
Ratio of specific heats for diatomic gas, $\gamma = 1.4$
Substituting these into the derived formula for $Q$:
\[ Q = \frac{1.4}{1.4 - 1} \times 100 \]
\[ Q = \frac{1.4}{0.4} \times 100 \]
\[ Q = \frac{14}{4} \times 100 \]
\[ Q = 3.5 \times 100 = 350 \text{ J} \]

Step 4: Final Answer:
The heat given to the gas during the isobaric expansion is 350 J.