A diatomic gas `(gamma = 1.4)` does 400J of work when it is expanded isobarically. The heat given to the gas in th process is ________J.

Explanation

To find the heat given to a diatomic gas during an isobaric expansion, we can use the following steps:

### Step 1: Understand the isobaric process
In an isobaric process, the pressure remains constant while the gas expands. The work done by the gas during this expansion can be expressed as:
\[ W = P \Delta V \]
However, we also know that for an isobaric process, the work done can be related to the change in temperature and the number of moles of the gas.

### Step 2: Use the relationship for work done
The work done by the gas is given as:
\[ W = n R \Delta T \]
where \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( \Delta T \) is the change in temperature.

### Step 3: Use the heat equation for an isobaric process
The heat added to the gas during an isobaric process is given by:
\[ Q = n C_p \Delta T \]
where \( C_p \) is the specific heat capacity at constant pressure.

### Step 4: Relate \( C_p \) to \( \gamma \)
For a diatomic gas, the relationship between \( C_p \) and \( \gamma \) (the heat capacity ratio) is:
\[ C_p = \frac{\gamma}{\gamma - 1} R \]
Given that \( \gamma = 1.4 \), we can substitute this into the equation.

### Step 5: Substitute \( C_p \) into the heat equation
Substituting the expression for \( C_p \) into the heat equation, we have:
\[ Q = n \left( \frac{\gamma}{\gamma - 1} R \right) \Delta T \]

### Step 6: Relate \( Q \) to \( W \)
From the work done equation, we can express \( n R \Delta T \) in terms of \( W \):
\[ n R \Delta T = W \]
Thus, we can substitute this into the heat equation:
\[ Q = \frac{\gamma}{\gamma - 1} W \]

### Step 7: Substitute the values
Now we can substitute \( W = 400 \, J \) and \( \gamma = 1.4 \):
\[ Q = \frac{1.4}{1.4 - 1} \times 400 \]
\[ Q = \frac{1.4}{0.4} \times 400 \]
\[ Q = 3.5 \times 400 \]
\[ Q = 1400 \, J \]

### Final Answer
The heat given to the gas in this process is **1400 J**.

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