A diatomic gas with rigid molecules does 10J of work when expanded at constant pressure. The heat energy absorbed by the gas, in this process is ___________ (in J).

Options

A. `30 J`
B. `40 J`
C. `25 J`
D. `35 J`

Explanation

To find the heat energy absorbed by the diatomic gas during its expansion at constant pressure, we can use the first law of thermodynamics and some properties of gases. Here’s the step-by-step solution:

### Step 1: Understand the Process
The gas is expanding at constant pressure, which is an isobaric process. In this process, the work done (W) by the gas is given as 10 J.

### Step 2: Use the First Law of Thermodynamics
The first law of thermodynamics states:
\[ Q = \Delta U + W \]
where:
- \( Q \) is the heat absorbed by the gas,
- \( \Delta U \) is the change in internal energy,
- \( W \) is the work done by the gas.

### Step 3: Calculate the Change in Internal Energy (\( \Delta U \))
For a diatomic gas, the change in internal energy can be expressed as:
\[ \Delta U = \frac{f}{2} n R \Delta T \]
where:
- \( f \) is the degrees of freedom,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( \Delta T \) is the change in temperature.

For a diatomic gas, the degrees of freedom \( f \) is 5.

### Step 4: Relate Work Done to Temperature Change
At constant pressure, the work done can also be expressed as:
\[ W = P \Delta V = n R \Delta T \]
Given that \( W = 10 \, \text{J} \), we can set up the equation:
\[ 10 = n R \Delta T \]

### Step 5: Substitute \( \Delta U \) into the First Law Equation
Now, substituting \( \Delta U \) into the first law equation:
\[ Q = \Delta U + W \]
Substituting for \( \Delta U \):
\[ Q = \left(\frac{5}{2} n R \Delta T\right) + 10 \]

### Step 6: Express \( n R \Delta T \) in Terms of Work Done
From the work done equation, we have:
\[ n R \Delta T = 10 \, \text{J} \]
Substituting this into the equation for \( Q \):
\[ Q = \left(\frac{5}{2} \cdot 10\right) + 10 \]
\[ Q = 25 + 10 = 35 \, \text{J} \]

### Final Answer
The heat energy absorbed by the gas is \( \boxed{35 \, \text{J}} \).

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