An ideal gas mixture filled inside a balloon expands according to relation `PV^(1/2)` constant. The temperature of the gas inside the balloon is:

Options

A. Increasing
B. Constant
C. Decreasing
D. Cannot be said

Explanation

To solve the problem of finding the temperature of an ideal gas mixture inside a balloon that expands according to the relation \( PV^{1/2} = \text{constant} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Relation**:
The problem states that \( PV^{1/2} = \text{constant} \). This means that as the balloon expands, the product of pressure \( P \) and the square root of volume \( V \) remains constant.

2. **Use the Ideal Gas Law**:
The ideal gas law is given by:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

3. **Express Pressure in Terms of Volume and Temperature**:
From the ideal gas law, we can express pressure \( P \) as:
\[ P = \frac{nRT}{V} \]

4. **Substitute Pressure into the Given Relation**:
Substitute the expression for \( P \) into the relation \( PV^{1/2} = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)V^{1/2} = \text{constant} \]
Simplifying this gives:
\[ \frac{nRT}{V^{1/2}} = \text{constant} \]

5. **Rearranging the Equation**:
Rearranging the equation, we find:
\[ nRT = \text{constant} \cdot V^{1/2} \]
This implies:
\[ T = \frac{\text{constant} \cdot V^{1/2}}{nR} \]

6. **Analyze the Relationship**:
Since \( n \) and \( R \) are constants, we can see that \( T \) is directly proportional to \( V^{1/2} \):
\[ T \propto V^{1/2} \]

7. **Conclusion**:
If the volume \( V \) is increasing (as the balloon expands), then \( T \) must also increase. Therefore, the temperature of the gas inside the balloon increases as the balloon expands.

### Final Answer:
The temperature of the gas inside the balloon increases.