In a process, the molar heat capacity of a diatomic gas is `(10)/(3) R`. When heat Q is supplied to the gas, find the work done by the gas

Options

A. `(3Q)/(4)`
B. `(Q)/(4)`
C. `(Q)/(3)`
D. `(2Q)/(3)`

Explanation

To solve the problem step by step, we will use the first law of thermodynamics and the relationships between heat, internal energy, and work done.

### Step 1: Identify Given Information
- Molar heat capacity \( C = \frac{10}{3} R \)
- The gas is diatomic, which means it has 5 degrees of freedom.

### Step 2: Write the Formula for Change in Internal Energy
The change in internal energy \( \Delta U \) for a gas can be calculated using the formula:
\[ \Delta U = N C_v \Delta T \]
For a diatomic gas, the molar heat capacity at constant volume \( C_v \) is given by:
\[ C_v = \frac{5}{2} R \]
Thus, we can express \( \Delta U \) as:
\[ \Delta U = N \left(\frac{5}{2} R\right) \Delta T \]

### Step 3: Relate Heat Supplied to the Gas
The heat supplied to the gas \( Q \) is related to the molar heat capacity \( C \) and the change in temperature \( \Delta T \) as follows:
\[ Q = N C \Delta T \]
Substituting the value of \( C \):
\[ Q = N \left(\frac{10}{3} R\right) \Delta T \]

### Step 4: Express \( \Delta T \) in Terms of \( Q \)
From the equation for \( Q \):
\[ \Delta T = \frac{Q}{N \left(\frac{10}{3} R\right)} = \frac{3Q}{10NR} \]

### Step 5: Substitute \( \Delta T \) into the Internal Energy Equation
Now substitute \( \Delta T \) back into the equation for \( \Delta U \):
\[ \Delta U = N \left(\frac{5}{2} R\right) \left(\frac{3Q}{10NR}\right) \]
This simplifies to:
\[ \Delta U = \frac{5 \cdot 3Q}{2 \cdot 10} = \frac{15Q}{20} = \frac{3Q}{4} \]

### Step 6: Apply the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ Q = W + \Delta U \]
Rearranging gives us:
\[ W = Q - \Delta U \]
Substituting the value of \( \Delta U \):
\[ W = Q - \frac{3Q}{4} = \frac{1Q}{4} = \frac{Q}{4} \]

### Final Answer
The work done by the gas is:
\[ W = \frac{Q}{4} \]