Calculate the speed of longitudinal waves in the following gases at `0^(@) C` and `1 atm( = 10^(5) pa):`
(a) oxygen for which the bulk modulus is `1.41 xx 10^(5)` pa and density is `1.43 kg//m ^(3)` .
(b) helium for which the bulk modulus is `1.7xx 10^(5)` pa and density is `0.18 kg//m^(3)` .

Explanation

To calculate the speed of longitudinal waves in gases, we can use the formula:

\[ v = \sqrt{\frac{B}{\rho}} \]

where:
- \( v \) is the speed of sound in the gas,
- \( B \) is the bulk modulus of the gas,
- \( \rho \) is the density of the gas.

Let's solve the problem step by step for both gases.

### Part (a): Oxygen

1. **Identify the given values for oxygen:**
- Bulk modulus \( B = 1.41 \times 10^5 \, \text{Pa} \)
- Density \( \rho = 1.43 \, \text{kg/m}^3 \)

2. **Substitute the values into the formula:**
\[ v_{O_2} = \sqrt{\frac{1.41 \times 10^5}{1.43}} \]

3. **Calculate the value inside the square root:**
\[ \frac{1.41 \times 10^5}{1.43} \approx 98655.44 \]

4. **Take the square root:**
\[ v_{O_2} = \sqrt{98655.44} \approx 314 \, \text{m/s} \]

### Part (b): Helium

1. **Identify the given values for helium:**
- Bulk modulus \( B = 1.7 \times 10^5 \, \text{Pa} \)
- Density \( \rho = 0.18 \, \text{kg/m}^3 \)

2. **Substitute the values into the formula:**
\[ v_{He} = \sqrt{\frac{1.7 \times 10^5}{0.18}} \]

3. **Calculate the value inside the square root:**
\[ \frac{1.7 \times 10^5}{0.18} \approx 944444.44 \]

4. **Take the square root:**
\[ v_{He} = \sqrt{944444.44} \approx 972 \, \text{m/s} \]

### Final Answers:
- (a) The speed of longitudinal waves in oxygen is approximately \( 314 \, \text{m/s} \).
- (b) The speed of longitudinal waves in helium is approximately \( 972 \, \text{m/s} \).