A sample of an ideal diatomic gas is heated at constant pressure. If an amount of 100 J of heat is supplied to the gas, find the work done by the gas.

Answer

To solve the problem of finding the work done by an ideal diatomic gas when heated at constant pressure, we can follow these steps:

### Step 1: Understand the relationship between heat, work, and temperature change.
When a gas is heated at constant pressure, the heat supplied (q) can be expressed using the formula:
\[ q = n C_p \Delta T \]
where:
- \( n \) = number of moles of the gas,
- \( C_p \) = molar specific heat at constant pressure,
- \( \Delta T \) = change in temperature.

### Step 2: Identify the specific heat for a diatomic gas.
For a diatomic gas, the value of \( C_p \) is given by:
\[ C_p = \frac{7}{2} R \]
where \( R \) is the universal gas constant.

### Step 3: Substitute the known values into the heat equation.
Given that \( q = 100 \, \text{J} \), we can write:
\[ 100 = n \left(\frac{7}{2} R\right) \Delta T \]

### Step 4: Rearrange the equation to find \( n R \Delta T \).
From the equation, we can express \( n R \Delta T \):
\[ n R \Delta T = \frac{100 \times 2}{7} = \frac{200}{7} \, \text{J} \]

### Step 5: Use the ideal gas law to find the work done.
For an isobaric process (constant pressure), the work done (W) can be expressed as:
\[ W = P \Delta V \]
Using the ideal gas law \( PV = nRT \), we can express \( P \Delta V \) in terms of \( n R \Delta T \):
\[ W = n R \Delta T \]

### Step 6: Substitute the value of \( n R \Delta T \) into the work equation.
From our previous calculation:
\[ W = n R \Delta T = \frac{200}{7} \, \text{J} \]

### Step 7: Calculate the final value of work done.
Now, we can compute:
\[ W = \frac{200}{7} \approx 28.57 \, \text{J} \]

### Final Answer:
The work done by the gas is approximately \( 28.57 \, \text{J} \).

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