A gas expands adiabatically at constant pressure such that its temperature `Tprop(1)/(sqrt(V))` , the value of `C_(P)//C_(V)` of gas is

Options

A. 1.3
B. 1.5
C. 1.67
D. 2

Answer

To solve the problem, we need to analyze the relationship between the temperature, volume, and the specific heat capacities of the gas during an adiabatic expansion at constant pressure.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that during the adiabatic expansion of a gas at constant pressure, the temperature \( T \) is directly proportional to \( \frac{1}{\sqrt{V}} \). This can be expressed mathematically as:
\[ T \propto \frac{1}{\sqrt{V}} \implies T = k \cdot \frac{1}{\sqrt{V}} \]
where \( k \) is a constant.

2. **Using the Adiabatic Condition**:
For an adiabatic process, we know that:
\[ PV^{\gamma} = \text{constant} \]
and
\[ TV^{\gamma - 1} = \text{constant} \]
Here, \( \gamma = \frac{C_p}{C_v} \).

3. **Substituting the Temperature Relation**:
From the relationship \( T = k \cdot \frac{1}{\sqrt{V}} \), we can substitute \( T \) into the adiabatic condition:
\[ \left(k \cdot \frac{1}{\sqrt{V}}\right) V^{\gamma - 1} = \text{constant} \]
Simplifying this gives:
\[ k \cdot V^{-\frac{1}{2}} \cdot V^{\gamma - 1} = \text{constant} \]
\[ k \cdot V^{\gamma - \frac{3}{2}} = \text{constant} \]

4. **Analyzing the Exponent**:
For the above equation to hold true for all volumes \( V \), the exponent of \( V \) must be zero:
\[ \gamma - \frac{3}{2} = 0 \implies \gamma = \frac{3}{2} \]

5. **Finding the Value of \( \frac{C_p}{C_v} \)**:
Since \( \gamma = \frac{C_p}{C_v} \), we conclude that:
\[ \frac{C_p}{C_v} = \frac{3}{2} \]

### Final Answer:
The value of \( \frac{C_p}{C_v} \) of the gas is \( \frac{3}{2} \).

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