Helium is a monatomic gas that has a density of `0.179 kg//m^(3)` at a pressure of `76 cm` of mercury and a temperature of `0^(@) C` . Find the speed of compressional waves (sound) in helium at this temperature and pressure.

Explanation

To find the speed of compressional waves (sound) in helium at the given temperature and pressure, we will use the formula derived from Newton's Laplace equation:

\[ V = \sqrt{\frac{\gamma P}{\rho}} \]

Where:
- \( V \) is the speed of sound,
- \( \gamma \) is the ratio of specific heats (for monatomic gases, \( \gamma = \frac{5}{3} \)),
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

### Step 1: Determine the value of \( \gamma \)
Since helium is a monatomic gas, we have:
\[ \gamma = \frac{5}{3} \]

### Step 2: Calculate the pressure \( P \)
The pressure is given in terms of a mercury column. We can calculate the pressure using the formula:
\[ P = \rho_{Hg} \cdot g \cdot h \]
Where:
- \( \rho_{Hg} \) is the density of mercury, approximately \( 13.6 \times 10^3 \, \text{kg/m}^3 \),
- \( g \) is the acceleration due to gravity, approximately \( 10 \, \text{m/s}^2 \),
- \( h \) is the height of the mercury column in meters, which is \( 76 \, \text{cm} = 0.76 \, \text{m} \).

Substituting the values:
\[ P = 13.6 \times 10^3 \cdot 10 \cdot 0.76 \]
\[ P = 13.6 \times 10^3 \cdot 7.6 \]
\[ P = 103.36 \times 10^3 \, \text{Pa} \]
\[ P = 103360 \, \text{Pa} \]

### Step 3: Use the density \( \rho \)
The density of helium is given as:
\[ \rho = 0.179 \, \text{kg/m}^3 \]

### Step 4: Substitute values into the speed of sound formula
Now we can substitute \( \gamma \), \( P \), and \( \rho \) into the speed of sound formula:
\[ V = \sqrt{\frac{\frac{5}{3} \cdot 103360}{0.179}} \]

### Step 5: Calculate the speed of sound
Calculating the numerator:
\[ \frac{5}{3} \cdot 103360 = \frac{516800}{3} \approx 172266.67 \]

Now, substituting this into the equation:
\[ V = \sqrt{\frac{172266.67}{0.179}} \]
\[ V \approx \sqrt{962145.64} \]
\[ V \approx 981.0 \, \text{m/s} \]

### Final Result
The speed of sound in helium at the given conditions is approximately:
\[ V \approx 981 \, \text{m/s} \]