If a gas (γ=4/3) is heated at constant pressure then what percentage of total heat supplied is used of external work ?

Options

A. 0.25
B. 0.5
C. 0.75
D. 0.8

Explanation

To solve the problem of determining what percentage of the total heat supplied to a gas (with γ = 4/3) at constant pressure is used for external work, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The adiabatic constant (γ) is given as 4/3.
- We are dealing with a gas heated at constant pressure.

2. **Identify Relevant Formulas**:
- The heat supplied at constant pressure (Q) is given by:
\[ Q = N C_P \Delta T \]
where \(N\) is the number of moles, \(C_P\) is the molar specific heat at constant pressure, and \(\Delta T\) is the change in temperature.
- The work done (W) during an isobaric process (constant pressure) is given by:
\[ W = N R \Delta T \]
where \(R\) is the gas constant.

3. **Relate Work Done to Heat Supplied**:
- We can find the ratio of work done to heat supplied:
\[ \frac{W}{Q} = \frac{N R \Delta T}{N C_P \Delta T} \]
- Canceling \(N\) and \(\Delta T\) gives:
\[ \frac{W}{Q} = \frac{R}{C_P} \]

4. **Use Mayer's Relation**:
- From Mayer's relation, we know:
\[ C_P - C_V = R \]
- We can express \(R\) in terms of \(C_P\) and \(C_V\):
\[ R = C_P - C_V \]
- Thus, we can substitute this into our ratio:
\[ \frac{W}{Q} = \frac{C_P - C_V}{C_P} \]

5. **Express in Terms of γ**:
- We know that:
\[ \frac{C_P}{C_V} = \gamma \]
- Therefore, we can express \(C_V\) as:
\[ C_V = \frac{C_P}{\gamma} \]
- Substituting this into our ratio gives:
\[ \frac{W}{Q} = \frac{C_P - \frac{C_P}{\gamma}}{C_P} = \frac{C_P(1 - \frac{1}{\gamma})}{C_P} = 1 - \frac{1}{\gamma} \]

6. **Calculate the Percentage**:
- Now substituting \(\gamma = \frac{4}{3}\):
\[ \frac{W}{Q} = 1 - \frac{1}{\frac{4}{3}} = 1 - \frac{3}{4} = \frac{1}{4} \]
- To find the percentage of work done:
\[ \text{Percentage} = \frac{W}{Q} \times 100 = \frac{1}{4} \times 100 = 25\% \]

### Final Answer:
The percentage of total heat supplied that is used for external work is **25%**.