A gas is heated at a constant pressure. The fraction of heat supplied used of external work is

Options

A. `(1)/(gamma)`
B. `(1 - (1)/(gamma))`
C. `gamma - 1`
D. `(1 - (1)/(gamma^(2)))`

Explanation

To solve the problem of finding the fraction of heat supplied that is used for external work when a gas is heated at constant pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Process**: The problem states that the gas is heated at constant pressure, which is an isobaric process.

2. **Write the Formula for Heat**: The heat supplied to the gas during an isobaric process can be expressed as:
\[ Q = nC_p\Delta T \]
where \( n \) is the number of moles, \( C_p \) is the molar heat capacity at constant pressure, and \( \Delta T \) is the change in temperature.

3. **Use the First Law of Thermodynamics**: According to the first law of thermodynamics:
\[ \Delta Q = \Delta U + \Delta W \]
where \( \Delta U \) is the change in internal energy and \( \Delta W \) is the work done by the system.

4. **Express Change in Internal Energy**: The change in internal energy for an ideal gas can be expressed as:
\[ \Delta U = nC_v\Delta T \]
where \( C_v \) is the molar heat capacity at constant volume.

5. **Calculate Work Done**: Rearranging the first law gives us:
\[ \Delta W = \Delta Q - \Delta U \]
Substituting the expressions for \( \Delta Q \) and \( \Delta U \):
\[ \Delta W = nC_p\Delta T - nC_v\Delta T \]
Factoring out \( n\Delta T \):
\[ \Delta W = n\Delta T (C_p - C_v) \]

6. **Relate \( C_p \) and \( C_v \)**: For ideal gases, the difference between the heat capacities is given by:
\[ C_p - C_v = R \]
where \( R \) is the universal gas constant.

7. **Substitute into Work Done**: Thus, we can express the work done as:
\[ \Delta W = n\Delta T R \]

8. **Calculate the Fraction of Heat Used for Work**: The fraction of heat supplied that is used for external work is given by:
\[ \text{Fraction} = \frac{\Delta W}{\Delta Q} = \frac{n\Delta T R}{nC_p\Delta T} \]
Simplifying this expression gives:
\[ \text{Fraction} = \frac{R}{C_p} \]

9. **Express in Terms of \( \gamma \)**: We know that:
\[ \gamma = \frac{C_p}{C_v} \]
and rearranging gives:
\[ C_p = \gamma C_v \]
Thus, we can express \( C_v \) in terms of \( C_p \):
\[ C_v = \frac{C_p}{\gamma} \]
Therefore, substituting \( C_p \) in terms of \( R \) and \( \gamma \):
\[ \text{Fraction} = \frac{R}{\gamma C_v} \]

10. **Final Expression**: After substituting and simplifying, we find:
\[ \text{Fraction} = \frac{\gamma - 1}{\gamma} \]
This can also be expressed as:
\[ 1 - \frac{1}{\gamma} \]

### Conclusion:
The fraction of heat supplied that is used for external work when a gas is heated at constant pressure is:
\[ \frac{\gamma - 1}{\gamma} \quad \text{or} \quad 1 - \frac{1}{\gamma} \]