Using the fact that hydrogen gas consister of diatomic molecules with `M = 2 kg//K - mol`. Find the speed of sound in hydrogen at `27^(@) C` .

Explanation

To find the speed of sound in hydrogen gas at 27°C, we can use the formula for the speed of sound in a gas, which is given by:

\[ v = \sqrt{\frac{\gamma R T}{M}} \]

where:
- \( v \) is the speed of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 1: Convert the temperature from Celsius to Kelvin
Given:
- Temperature \( T = 27^\circ C \)

To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the values needed for the calculation
From the problem statement, we have:
- Molar mass of hydrogen \( M = 2 \, kg/(K \cdot mol) \)
- Universal gas constant \( R = 8.31 \times 10^3 \, J/(kg \cdot K \cdot mol) \)
- Adiabatic index for diatomic molecules \( \gamma = 1.4 \)

### Step 3: Substitute the values into the speed of sound formula
Now we can substitute the values into the formula:
\[ v = \sqrt{\frac{\gamma R T}{M}} = \sqrt{\frac{1.4 \times 8.31 \times 10^3 \times 300}{2}} \]

### Step 4: Calculate the numerator
First, calculate the numerator:
\[ 1.4 \times 8.31 \times 10^3 \times 300 = 1.4 \times 2493000 = 3480200 \]

### Step 5: Divide by the molar mass
Now divide this result by the molar mass:
\[ \frac{3480200}{2} = 1740100 \]

### Step 6: Take the square root
Finally, take the square root to find the speed of sound:
\[ v = \sqrt{1740100} \approx 1321 \, m/s \]

### Final Answer:
The speed of sound in hydrogen at 27°C is approximately \( 1321 \, m/s \).

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