The ratio of work done by an ideal diatomic gas to the heat supplied by the gas in an isobatic process is

Options

A. `5/7`
B. `3/5`
C. `2/7`
D. `5/3`

Answer

To solve the problem of finding the ratio of work done by an ideal diatomic gas to the heat supplied by the gas in an isobaric process, we can follow these steps:

### Step 1: Understand the First Law of Thermodynamics
The first law of thermodynamics states:
\[ W = Q - \Delta U \]
where:
- \( W \) is the work done by the gas,
- \( Q \) is the heat supplied to the gas,
- \( \Delta U \) is the change in internal energy.

### Step 2: Express Work Done and Heat Supplied
For an ideal gas, the work done \( W \) in an isobaric process can be expressed as:
\[ W = nC_p\Delta T - nC_v\Delta T \]
where:
- \( n \) is the number of moles,
- \( C_p \) is the specific heat at constant pressure,
- \( C_v \) is the specific heat at constant volume,
- \( \Delta T \) is the change in temperature.

### Step 3: Substitute Values for Diatomic Gas
For a diatomic ideal gas:
- \( C_p = \frac{7}{2}R \)
- \( C_v = \frac{5}{2}R \)

Substituting these values into the equation for work done:
\[ W = n\left(\frac{7}{2}R\Delta T - \frac{5}{2}R\Delta T\right) \]
\[ W = nR\Delta T \]

### Step 4: Calculate Heat Supplied
The heat supplied \( Q \) in an isobaric process is given by:
\[ Q = nC_p\Delta T \]
Substituting the value of \( C_p \):
\[ Q = n\left(\frac{7}{2}R\right)\Delta T \]
\[ Q = \frac{7}{2}nR\Delta T \]

### Step 5: Find the Ratio of Work Done to Heat Supplied
Now, we need to find the ratio \( \frac{W}{Q} \):
\[ \frac{W}{Q} = \frac{nR\Delta T}{\frac{7}{2}nR\Delta T} \]
The \( nR\Delta T \) terms cancel out:
\[ \frac{W}{Q} = \frac{1}{\frac{7}{2}} = \frac{2}{7} \]

### Final Answer
Thus, the ratio of work done by the ideal diatomic gas to the heat supplied by the gas in an isobaric process is:
\[ \frac{W}{Q} = \frac{2}{7} \]