A gas expands adiabatically at constant pressure such that `T propV^(-1//2)` The value of `gamma (C_(p,m)//C_(v,m))` of the gas will be :

Options

A. 1.3
B. 1.5
C. 1.7
D. 2

Explanation

To solve the problem, we need to find the value of gamma (γ), which is the ratio of the specific heat capacities at constant pressure (C_p) and constant volume (C_v) for a gas that expands adiabatically at constant pressure with the given relationship between temperature (T) and volume (V).

### Step-by-Step Solution:

1. **Understand the Given Relationship**:
We are given that the temperature (T) is directly proportional to \( V^{-1/2} \). This can be expressed mathematically as:
\[ T \propto V^{-1/2} \quad \Rightarrow \quad T = k \cdot V^{-1/2} \]
where \( k \) is a constant.

2. **Rearranging the Equation**:
We can rewrite this relationship as:
\[ T \cdot V^{1/2} = k \]
This indicates that the product of temperature and the square root of volume is constant.

3. **Using the Adiabatic Condition**:
For an adiabatic process, the relationship between temperature (T) and volume (V) is given by:
\[ T \cdot V^{\gamma - 1} = \text{constant} \]
We can compare this with our earlier expression \( T \cdot V^{1/2} = k \).

4. **Comparing the Exponents**:
From the adiabatic condition, we have:
\[ T \cdot V^{\gamma - 1} = \text{constant} \]
and from our earlier expression:
\[ T \cdot V^{1/2} = \text{constant} \]
By comparing the exponents of V, we can equate:
\[ \gamma - 1 = \frac{1}{2} \]

5. **Solving for γ**:
Rearranging the equation gives:
\[ \gamma = \frac{1}{2} + 1 = \frac{3}{2} \]

6. **Conclusion**:
Therefore, the value of gamma (γ) is:
\[ \gamma = \frac{3}{2} \quad \text{or} \quad 1.5 \]

### Final Answer:
The value of \( \gamma \) (C_p/C_v) of the gas is \( 1.5 \).