An ideal diatomic gas is heated at constant pressure such that it performs a work W=2.0 J . Find the amount of heat supplied.

Answer

To find the amount of heat supplied to an ideal diatomic gas that is heated at constant pressure and performs a work of W = 2.0 J, we can follow these steps:

### Step 1: Understand the Process
The gas is undergoing an isobaric process (constant pressure). In this process, the work done by the gas can be expressed as:
\[ W = P \Delta V \]
where \( P \) is the pressure and \( \Delta V \) is the change in volume.

### Step 2: Relate Work Done to Temperature Change
For an ideal gas, we can relate the work done to the change in temperature. Since the pressure is constant, we can use the ideal gas law:
\[ PV = nRT \]
From this, we can derive that:
\[ W = P \Delta V = nR \Delta T \]
Given that \( W = 2.0 \, \text{J} \), we have:
\[ nR \Delta T = 2.0 \, \text{J} \]

### Step 3: Use the Heat Capacity at Constant Pressure
The amount of heat supplied at constant pressure can be expressed as:
\[ Q = n C_p \Delta T \]
where \( C_p \) is the molar specific heat at constant pressure.

### Step 4: Determine the Value of \( C_p \) for a Diatomic Gas
For a diatomic gas, the value of \( C_p \) is:
\[ C_p = \frac{7}{2} R \]

### Step 5: Substitute and Solve for Heat Supplied
Now, substituting \( C_p \) into the heat equation:
\[ Q = n \left(\frac{7}{2} R\right) \Delta T \]
We already established that \( nR \Delta T = 2.0 \, \text{J} \). Therefore, we can substitute this into the equation for \( Q \):
\[ Q = n \left(\frac{7}{2} R\right) \Delta T = \frac{7}{2} (nR \Delta T) \]
\[ Q = \frac{7}{2} \times 2.0 \, \text{J} \]
\[ Q = 7.0 \, \text{J} \]

### Final Answer
The amount of heat supplied to the gas is:
\[ Q = 7.0 \, \text{J} \]

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