A diatomic gas done `80J` work when expanded isobarically. Find the heat given to the gas during this process?

Answer

To solve the problem of finding the heat given to a diatomic gas during an isobaric expansion where 80 J of work is done, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Process**:
The process is isobaric, meaning the pressure (P) remains constant during the expansion of the gas.

2. **Using the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
For an isobaric process, we can express the work done (W) as:
\[ W = P \Delta V \]
We can also relate the work done to the change in temperature (\(\Delta T\)) using:
\[ W = nR \Delta T \]

3. **Finding the Change in Temperature**:
Rearranging the equation for work:
\[ \Delta T = \frac{W}{nR} \]
Given that \(W = 80 \, \text{J}\), we can substitute this into the equation:
\[ \Delta T = \frac{80}{nR} \]

4. **Calculating Change in Internal Energy**:
The change in internal energy (\(\Delta U\)) for a diatomic gas can be expressed as:
\[ \Delta U = nC_V \Delta T \]
For a diatomic gas, \(C_V = \frac{5}{2}R\). Therefore:
\[ \Delta U = n \left(\frac{5}{2}R\right) \Delta T \]
Substituting \(\Delta T\) from the previous step:
\[ \Delta U = n \left(\frac{5}{2}R\right) \left(\frac{80}{nR}\right) \]
Simplifying this gives:
\[ \Delta U = \frac{5}{2} \cdot 80 = 200 \, \text{J} \]

5. **Applying the First Law of Thermodynamics**:
The first law of thermodynamics states:
\[ \Delta Q = \Delta U + W \]
Substituting the values we have:
\[ \Delta Q = 200 \, \text{J} + 80 \, \text{J} = 280 \, \text{J} \]

### Final Answer:
The heat given to the gas during this process is:
\[ \Delta Q = 280 \, \text{J} \]