Find the speed of sound in a mixture of 1 mole of helium and 2 moles of oxygen at `27^(@)C`. If the temperature is raised by 1K from 300K, find the percentage change in the speed of sound in the gaseous mixture.
Take `R=8.31Jmo l e^(-1)K^(-1)`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the Molar Mass of the Mixture
The molar mass \( M \) of the gas mixture can be calculated using the formula:
\[ M = \frac{n_1 \cdot m_1 + n_2 \cdot m_2}{n_1 + n_2} \]
Where:
- \( n_1 = 1 \) mole of Helium
- \( n_2 = 2 \) moles of Oxygen
- \( m_1 = 4 \) g/mol (molar mass of Helium)
- \( m_2 = 32 \) g/mol (molar mass of Oxygen)

Substituting the values:
\[ M = \frac{1 \cdot 4 + 2 \cdot 32}{1 + 2} = \frac{4 + 64}{3} = \frac{68}{3} \text{ g/mol} \]
Converting to kg/mol:
\[ M = \frac{68}{3} \times 10^{-3} \text{ kg/mol} \]

### Step 2: Calculate the Specific Heat Capacities
For the specific heat capacities, we need to find \( C_v \) and \( C_p \).

- For Helium (monatomic gas):
\[ C_{v1} = \frac{3}{2} R \]
- For Oxygen (diatomic gas):
\[ C_{v2} = \frac{5}{2} R \]

The average \( C_v \) for the mixture is:
\[ C_v = \frac{n_1 \cdot C_{v1} + n_2 \cdot C_{v2}}{n_1 + n_2} = \frac{1 \cdot \frac{3}{2} R + 2 \cdot \frac{5}{2} R}{1 + 2} = \frac{\frac{3}{2} R + 5 R}{3} = \frac{13}{6} R \]

Now, we can find \( C_p \):
\[ C_p = C_v + R = \frac{13}{6} R + R = \frac{19}{6} R \]

### Step 3: Calculate \( \gamma \)
\[ \gamma = \frac{C_p}{C_v} = \frac{\frac{19}{6} R}{\frac{13}{6} R} = \frac{19}{13} \]

### Step 4: Calculate the Speed of Sound
The speed of sound \( V \) in the gas mixture is given by:
\[ V = \sqrt{\frac{\gamma R T}{M}} \]
Where:
- \( T = 300 \) K (27°C)
- \( R = 8.31 \, \text{J/mol K} \)

Substituting the values:
\[ V = \sqrt{\frac{\frac{19}{13} \cdot 8.31 \cdot 300}{\frac{68}{3} \times 10^{-3}}} \]

Calculating the above expression:
1. Calculate \( \frac{19}{13} \cdot 8.31 \cdot 300 \)
2. Divide by \( \frac{68}{3} \times 10^{-3} \)
3. Take the square root to find \( V \).

After performing the calculations, we find:
\[ V \approx 400.9 \, \text{m/s} \]

### Step 5: Calculate the Percentage Change in Speed of Sound
Using the formula for percentage change:
\[ \frac{\Delta V}{V} = \frac{1}{2} \cdot \frac{\Delta T}{T} \]
Where \( \Delta T = 1 \, \text{K} \) and \( T = 300 \, \text{K} \).

Calculating:
\[ \frac{\Delta V}{V} = \frac{1}{2} \cdot \frac{1}{300} \]
\[ \Delta V \text{ (percentage)} = \frac{1}{2} \cdot \frac{1}{300} \cdot 100 \approx 0.167\% \]

### Final Answers
1. The speed of sound in the mixture at \( 27^\circ C \) is approximately \( 400.9 \, \text{m/s} \).
2. The percentage change in the speed of sound when the temperature is raised by \( 1 \, \text{K} \) is approximately \( 0.167\% \).