A diatomic gas (𝛾 = 1.4) does 100 J of work in an isobaric expansion. The heat given to the gas is:

Correct Answer: The correct option is 350 J.

Step-by-Step Explanation:

1. Understand the Given Information:
We are given:
- Type of gas: Diatomic gas, which has an adiabatic index (ratio of specific heats) $\gamma =1.4$.
- Work done by the gas during expansion ($W$): $W=100\text{ J}$.
- Process: Isobaric expansion (constant pressure process).

2. Formulas for an Isobaric Process:
For an isobaric process with $n$ moles of an ideal gas:
- Work done by the gas is given by:

$W=P\Delta V=nR\Delta T$

- Heat given to the gas ($Q$) at constant pressure is given by:

$Q=n{C}_p\Delta T$

3. Express Specific Heat Capacity ($C_p$) in Terms of $\gamma$:
The molar heat capacity at constant pressure $C_p$ for an ideal gas is related to the adiabatic index $\gamma$ by the relation:

$C_p=\frac{\gamma R}{\gamma -1}$

4. Relate Heat Supplied ($Q$) to Work Done ($W$):
Substitute the expression for $C_p$ into the formula for $Q$:

$Q=n\left(\frac{\gamma R}{\gamma -1}\right)\Delta T$

Since $W=nR\Delta T$, we can substitute $W$ into the equation for $Q$:

$Q=\left(\frac{\gamma }{\gamma -1}\right)W$

5. Calculate the Heat Given to the Gas:
Now, substitute the given values $\gamma =1.4$ and $W=100\text{ J}$:

$Q=\left(\frac{1.4}{1.4-1}\right)\times 100$

$Q=\left(\frac{1.4}{0.4}\right)\times 100$

$Q=3.5\times 100=350\text{ J}$

Conclusion:
The heat given to the gas during the isobaric expansion is 350 J.