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)`

Explanation

To determine which gas has the largest velocity of sound 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:
- \( V \) is the velocity of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

Since the conditions of temperature and pressure are the same for all gases, we can simplify our analysis to focus on the ratio \( \frac{\gamma}{M} \). The velocity of sound is directly proportional to the square root of this ratio.

### Step 1: Identify the gases and their properties
We will analyze the following gases:
1. Hydrogen (H₂)
2. Nitrogen (N₂)
3. Helium (He)
4. Carbon Dioxide (CO₂)

### Step 2: Determine the values of \( \gamma \) and \( M \) for each gas
- **Hydrogen (H₂)**:
- Degrees of freedom: 5 (monatomic)
- \( \gamma = \frac{5}{3} \)
- Molar mass \( M = 2 \, \text{g/mol} \)

- **Nitrogen (N₂)**:
- Degrees of freedom: 5 (diatomic)
- \( \gamma = \frac{7}{5} \)
- Molar mass \( M = 28 \, \text{g/mol} \)

- **Helium (He)**:
- Degrees of freedom: 3 (monatomic)
- \( \gamma = \frac{5}{3} \)
- Molar mass \( M = 4 \, \text{g/mol} \)

- **Carbon Dioxide (CO₂)**:
- Degrees of freedom: 6 (linear molecule)
- \( \gamma = \frac{7}{5} \)
- Molar mass \( M = 44 \, \text{g/mol} \)

### Step 3: Calculate \( \frac{\gamma}{M} \) for each gas
- **Hydrogen**:
\[ \frac{\gamma}{M} = \frac{5/3}{2} = \frac{5}{6} \]

- **Nitrogen**:
\[ \frac{\gamma}{M} = \frac{7/5}{28} = \frac{7}{140} \]

- **Helium**:
\[ \frac{\gamma}{M} = \frac{5/3}{4} = \frac{5}{12} \]

- **Carbon Dioxide**:
\[ \frac{\gamma}{M} = \frac{7/5}{44} = \frac{7}{220} \]

### Step 4: Compare the values of \( \frac{\gamma}{M} \)
Now we will compare the calculated ratios:
- Hydrogen: \( \frac{5}{6} \)
- Nitrogen: \( \frac{7}{140} \)
- Helium: \( \frac{5}{12} \)
- Carbon Dioxide: \( \frac{7}{220} \)

### Step 5: Determine the largest value
Among these values, \( \frac{5}{6} \) (for Hydrogen) is the largest.

### Conclusion
Thus, the velocity of sound will be largest in Hydrogen gas.

### Final Answer
The correct answer is Hydrogen (H₂).

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