Statement-1 : A balloon filled with `CO_(2)` gas acts as a coverging lens for a sound wave
Statement-2 : Sound waves travel faster in air than in `CO_(2)`

Options

A. Statement-1 is true, Statement-2 is true , Statement-2 is correct explanation for Statement-16
B. Statement-1 is true,Statement-2 is true , Statement-2 is NOT a correct explanation for Statement-16
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

To solve the problem, we need to analyze both statements provided in the question.

### Step 1: Analyze Statement 1
**Statement 1**: A balloon filled with `CO₂` gas acts as a converging lens for a sound wave.

- To determine if this statement is true, we need to understand how sound waves behave when they travel through different media.
- Sound waves travel at different speeds in different gases. The speed of sound in a medium is influenced by its density and the adiabatic index (γ).
- In this case, we have a balloon filled with carbon dioxide (CO₂) gas.

### Step 2: Speed of Sound in CO₂ and Air
- The speed of sound in a gas can be expressed as:
\[ v = \sqrt{\frac{\gamma R T}{M}} \]
where \( v \) is the speed of sound, \( \gamma \) is the adiabatic index, \( R \) is the gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

- For CO₂, we have:
- \( \gamma_{CO₂} \approx 1.28 \)
- Molar mass of CO₂ \( M_{CO₂} = 44 \, g/mol \)

- For air, we have:
- \( \gamma_{air} \approx 1.67 \)
- Molar mass of air \( M_{air} \approx 28.8 \, g/mol \)

### Step 3: Compare Speeds
- To compare the speeds of sound in air and CO₂, we can set up a ratio:
\[ \frac{v_{CO₂}}{v_{air}} = \frac{\gamma_{CO₂} \cdot R \cdot T / M_{CO₂}}{\gamma_{air} \cdot R \cdot T / M_{air}} = \frac{\gamma_{CO₂} \cdot M_{air}}{\gamma_{air} \cdot M_{CO₂}} \]

- Plugging in the values:
\[ \frac{v_{CO₂}}{v_{air}} = \frac{1.28 \cdot 28.8}{1.67 \cdot 44} \approx 0.7 \]
- This indicates that the speed of sound in CO₂ is less than in air.

### Step 4: Conclusion for Statement 1
- Since sound waves travel slower in CO₂ (denser medium) than in air (rarer medium), when sound waves pass from air into CO₂, they bend towards the normal.
- This bending behavior is similar to how light behaves when passing through a converging lens.
- Therefore, **Statement 1 is true**.

### Step 5: Analyze Statement 2
**Statement 2**: Sound waves travel faster in air than in `CO₂`.

- From our previous calculations, we established that the speed of sound in air is greater than in CO₂.
- Thus, **Statement 2 is also true**.

### Final Conclusion
Both statements are true, and Statement 2 provides a correct explanation for Statement 1. Therefore, the correct conclusion is that both statements are true.

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