Under simuliar conditions of temperature and pressure, in which of the following gases the velocity of sound will be largest?

Options

A. `H_(2)`
B. `N_(2)`
C. `He`
D. `CO_(2)`

Answer

To determine in which gas the velocity of sound will be largest under similar conditions of temperature and pressure, we can use the formula for the velocity of sound in a gas:

\[ v = \sqrt{\frac{\gamma RT}{M}} \]

Where:
- \( v \) = velocity of sound
- \( \gamma \) = adiabatic index (ratio of specific heats)
- \( R \) = universal gas constant
- \( T \) = absolute temperature
- \( M \) = molar mass of the gas

Since the conditions of temperature and pressure are the same for all gases, we can simplify our analysis by focusing on the ratio \( \frac{\gamma}{M} \). The gas with the highest value of \( \frac{\gamma}{M} \) will have the largest velocity of sound.

### Step 1: Identify the values of \( \gamma \) and \( M \) for each gas.
- **Hydrogen (H₂)**:
- \( \gamma \) = 7/5 (approximately 1.4)
- Molar mass \( M \) = 2 g/mol
- \( \frac{\gamma}{M} = \frac{1.4}{2} = 0.7 \)

- **Helium (He)**:
- \( \gamma \) = 5/3 (approximately 1.67)
- Molar mass \( M \) = 4 g/mol
- \( \frac{\gamma}{M} = \frac{1.67}{4} = 0.4175 \)

- **Carbon Dioxide (CO₂)**:
- \( \gamma \) = 7/5 (approximately 1.4)
- Molar mass \( M \) = 44 g/mol
- \( \frac{\gamma}{M} = \frac{1.4}{44} \approx 0.0318 \)

- **Nitrogen (N₂)**:
- \( \gamma \) = 7/5 (approximately 1.4)
- Molar mass \( M \) = 28 g/mol
- \( \frac{\gamma}{M} = \frac{1.4}{28} \approx 0.05 \)

### Step 2: Compare the values of \( \frac{\gamma}{M} \).
- Hydrogen: \( 0.7 \)
- Helium: \( 0.4175 \)
- Carbon Dioxide: \( 0.0318 \)
- Nitrogen: \( 0.05 \)

### Step 3: Determine which gas has the largest \( \frac{\gamma}{M} \).
From the calculated values, we see that hydrogen has the highest value of \( \frac{\gamma}{M} \) at \( 0.7 \).

### Conclusion:
Since hydrogen has the largest \( \frac{\gamma}{M} \), the velocity of sound will be largest in hydrogen gas under similar conditions of temperature and pressure.

**Final Answer:** The velocity of sound will be largest in hydrogen (H₂).

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