Find speed of sound in hydrogen gas at `27^(@)` . Ratio `C_(p)//C_(V)` for `H_(2)` is `1.4` . Gas constant `R =8.31 J//mol -K` .

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:

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

Where:
- \( V \) = speed of sound
- \( \gamma \) = ratio of specific heats \( \frac{C_p}{C_v} \)
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 1: Convert the given temperature to Kelvin
The temperature in Celsius is given as 27°C. To convert this to Kelvin:

\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the values for \( \gamma \), \( R \), and \( M \)
From the problem statement:
- \( \gamma = 1.4 \) (for hydrogen)
- \( R = 8.31 \, \text{J/(mol·K)} \)
- The molar mass of hydrogen \( H_2 \) is \( 2 \, \text{g/mol} = 2 \times 10^{-3} \, \text{kg/mol} \)

### Step 3: Substitute the values into the speed of sound formula
Now we can substitute the values into the formula:

\[ V = \sqrt{\frac{1.4 \times 8.31 \times 300}{2 \times 10^{-3}}} \]

### Step 4: Calculate the value inside the square root
First, calculate the numerator:

\[ 1.4 \times 8.31 \times 300 = 3492.6 \]

Now, calculate the denominator:

\[ 2 \times 10^{-3} = 0.002 \]

Now, divide the numerator by the denominator:

\[ \frac{3492.6}{0.002} = 1746300 \]

### Step 5: Take the square root
Now, take the square root of the result:

\[ V = \sqrt{1746300} \approx 1321 \, \text{m/s} \]

### Final Answer
The speed of sound in hydrogen gas at 27°C is approximately:

\[ V \approx 1321 \, \text{m/s} \]

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