The pitch of an organ pipe is highest when the pipe is filled with

Options

A. Air
B. Hydrogen
C. Oxygen
D. Carbon dioxide

Answer

To determine the gas that will produce the highest pitch in an organ pipe, we need to analyze the relationship between the speed of sound in a gas, its molecular mass, and the pitch (which is related to frequency).

### Step-by-Step Solution:

1. **Understanding Pitch and Frequency**:
- The pitch of a sound is directly related to its frequency. Higher frequency corresponds to a higher pitch.

2. **Speed of Sound in Gases**:
- The speed of sound \( v \) in a gas can be expressed using the formula:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( \gamma \) = adiabatic index (ratio of specific heats)
- \( R \) = universal gas constant
- \( T \) = absolute temperature
- \( M \) = molar mass of the gas

3. **Analyzing the Formula**:
- From the formula, we can see that the speed of sound \( v \) is directly proportional to \( \sqrt{\frac{\gamma}{M}} \). To achieve a higher pitch (higher frequency), we need to maximize the speed of sound.

4. **Comparing Gases**:
- We have four gases to compare: air, hydrogen, oxygen, and carbon dioxide.
- We need to evaluate \( \frac{\gamma}{M} \) for each gas:
- **Hydrogen (H₂)**:
- \( \gamma \approx \frac{7}{5} \)
- \( M \approx 2 \, \text{g/mol} \)
- \( \frac{\gamma}{M} \approx \frac{7/5}{2} = \frac{7}{10} \)
- **Oxygen (O₂)**:
- \( \gamma \approx \frac{7}{5} \)
- \( M \approx 32 \, \text{g/mol} \)
- \( \frac{\gamma}{M} \approx \frac{7/5}{32} = \frac{7}{160} \)
- **Carbon Dioxide (CO₂)**:
- \( \gamma \approx \frac{4}{3} \)
- \( M \approx 44 \, \text{g/mol} \)
- \( \frac{\gamma}{M} \approx \frac{4/3}{44} = \frac{4}{132} \approx \frac{1}{33} \)
- **Air** (approximately diatomic):
- \( \gamma \approx \frac{7}{5} \)
- \( M \approx 29 \, \text{g/mol} \)
- \( \frac{\gamma}{M} \approx \frac{7/5}{29} = \frac{7}{145} \)

5. **Conclusion**:
- Comparing the values of \( \frac{\gamma}{M} \):
- Hydrogen has the highest value of \( \frac{\gamma}{M} \).
- Therefore, the speed of sound is highest in hydrogen, leading to the highest pitch when the organ pipe is filled with hydrogen.

### Final Answer:
The pitch of an organ pipe is highest when the pipe is filled with **hydrogen**.

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