An ideal gas mixture filled inside a balloon expands according to the relation `PV^(2//3)=` constant. What will be the temperature inside the balloon

Options

A. increasing
B. decreasing
C. constant
D. Cannot be defined

Answer

To solve the problem, we need to analyze the relationship given in the question: \( PV^{\frac{2}{3}} = \text{constant} \).

### Step 1: Understand the relationship
The equation \( PV^{\frac{2}{3}} = \text{constant} \) indicates that the product of pressure \( P \) and the volume \( V \) raised to the power of \( \frac{2}{3} \) remains constant during the expansion of the gas mixture.

**Hint:** Recognize that this is a specific form of a thermodynamic process involving an ideal gas.

### Step 2: Use the Ideal Gas Law
We know from the Ideal Gas Law that:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

**Hint:** Relate the given equation to the Ideal Gas Law to express pressure in terms of temperature and volume.

### Step 3: Express Pressure in Terms of Temperature and Volume
From the Ideal Gas Law, we can express pressure \( P \) as:
\[ P = \frac{nRT}{V} \]

**Hint:** Substitute this expression for \( P \) into the original equation.

### Step 4: Substitute into the Original Equation
Substituting \( P \) into the equation \( PV^{\frac{2}{3}} = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)V^{\frac{2}{3}} = \text{constant} \]

### Step 5: Simplify the Equation
This simplifies to:
\[ \frac{nRT}{V^{\frac{1}{3}}} = \text{constant} \]
Rearranging gives:
\[ nRT = \text{constant} \cdot V^{\frac{1}{3}} \]

**Hint:** Recognize that since \( n \) and \( R \) are constants, the temperature \( T \) must be related to the volume \( V \).

### Step 6: Relate Temperature and Volume
From the equation \( nRT = \text{constant} \cdot V^{\frac{1}{3}} \), we can express \( T \):
\[ T = \frac{\text{constant} \cdot V^{\frac{1}{3}}}{nR} \]
This shows that \( T \) is directly proportional to \( V^{\frac{1}{3}} \).

**Hint:** Consider how changes in volume affect temperature.

### Step 7: Determine the Relationship
Since \( V \) is increasing as the gas expands, it follows that \( T \) must also increase because they are directly proportional.

### Conclusion
Therefore, the temperature inside the balloon will **increase** as the gas expands.

**Final Answer:** The temperature inside the balloon will increase.