An ideal gas with adiabatic exponent `gamma` is heated at constant pressure. It absorbs Q amount of heat. Fraction of heat absorbed in increasing the temperature is

Options

A. `gamma`
B. `1/gamma`
C. `1-1/gamma`
D. `2gamma`

Answer

To solve the problem, we need to find the fraction of heat absorbed by an ideal gas at constant pressure that is used to increase its temperature. Let's break down the steps:

### Step 1: Understand the process
We know that the gas is being heated at constant pressure. In this case, the heat absorbed (Q) can be expressed in terms of the change in temperature (ΔT) and the heat capacity at constant pressure (Cₚ).

### Step 2: Write the equation for heat absorbed
The heat absorbed by the gas at constant pressure is given by:
\[ Q = n C_p \Delta T \]
where:
- \( n \) is the number of moles of the gas,
- \( C_p \) is the molar heat capacity at constant pressure,
- \( \Delta T \) is the change in temperature.

### Step 3: Apply the first law of thermodynamics
According to the first law of thermodynamics, the heat absorbed by the system is equal to the change in internal energy (ΔU) plus the work done (W) by the system:
\[ Q = \Delta U + W \]

### Step 4: Determine the work done
At constant pressure, the work done by the gas can be expressed as:
\[ W = P \Delta V \]
Using the ideal gas law, we can relate this to the change in temperature:
\[ W = n R \Delta T \]
where \( R \) is the universal gas constant.

### Step 5: Relate internal energy change to temperature change
The change in internal energy (ΔU) for an ideal gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where \( C_v \) is the molar heat capacity at constant volume.

### Step 6: Substitute into the first law equation
Now, substituting the expressions for ΔU and W into the first law equation:
\[ Q = n C_v \Delta T + n R \Delta T \]

### Step 7: Factor out common terms
Factoring out \( n \Delta T \):
\[ Q = n \Delta T (C_v + R) \]

### Step 8: Use the relationship between Cp, Cv, and R
We know that:
\[ C_p = C_v + R \]
Thus, we can rewrite the equation as:
\[ Q = n \Delta T C_p \]

### Step 9: Find the fraction of heat absorbed in increasing temperature
Now, we need to find the fraction of heat absorbed that goes into increasing the temperature:
\[ F = \frac{\Delta U}{Q} = \frac{n C_v \Delta T}{n C_p \Delta T} \]

### Step 10: Simplify the fraction
Cancelling \( n \Delta T \) from the numerator and denominator gives:
\[ F = \frac{C_v}{C_p} \]

### Step 11: Relate Cv and Cp to gamma
We know that:
\[ \gamma = \frac{C_p}{C_v} \]
Thus, we can express \( F \) as:
\[ F = \frac{1}{\gamma} \]

### Conclusion
The fraction of heat absorbed in increasing the temperature is:
\[ F = \frac{1}{\gamma} \]