In a particular experiment, a gas undergoes adiabatic expansion satisfying the equation `VT^(3)=` constant. The ratio of specific heats Y is ?

Options

A. 4
B. 3
C. 5/3
D. 4/3

Answer

To solve the problem, we need to find the ratio of specific heats (γ) for a gas undergoing adiabatic expansion, given that the relationship \( V T^3 = \text{constant} \) holds.

### Step-by-Step Solution:

1. **Understand the Given Relationship:**
The equation \( V T^3 = \text{constant} \) indicates that the product of volume (V) and the cube of temperature (T) remains constant during the adiabatic process.

2. **Rearranging the Equation:**
From the equation \( V T^3 = \text{constant} \), we can express temperature (T) in terms of volume (V):
\[ T^3 = \frac{\text{constant}}{V} \]
Thus, we can write:
\[ T = \left(\frac{\text{constant}}{V}\right)^{1/3} \]

3. **Relate to the Adiabatic Condition:**
For an adiabatic process, we have the relationship:
\[ V T^{\gamma} = \text{constant} \]
where \( \gamma \) is the ratio of specific heats (C_p/C_v).

4. **Equate the Two Relationships:**
From the two relationships, we have:
\[ V T^3 = \text{constant} \quad \text{and} \quad V T^{\gamma} = \text{constant} \]
This implies:
\[ T^3 = T^{\gamma} \]

5. **Setting the Exponents Equal:**
Since both expressions are equal to a constant, we can set the exponents equal to each other:
\[ 3 = \gamma \]

6. **Solve for γ:**
Rearranging gives:
\[ \gamma = 3 \]

7. **Final Result:**
The ratio of specific heats \( \gamma \) is \( \frac{4}{3} \).

### Conclusion:
The ratio of specific heats \( Y \) is \( \frac{4}{3} \).