The speed of longitudinal waves in a mixture of He and Ne was found to be `760 ms^(-1)` at `30^@C.` Find the composition of the mixture. Both He and Ne ate monoatomic. Assume their molecular masses, Molar-mass of He=0.004 kg, Molar mass of ne=0.02 kg

Answer

To solve the problem of finding the composition of the mixture of helium (He) and neon (Ne) based on the speed of longitudinal waves, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- Speed of longitudinal waves (v) = 760 m/s
- Temperature (T) = 30°C = 303 K
- Molar mass of He (M_He) = 0.004 kg/mol
- Molar mass of Ne (M_Ne) = 0.02 kg/mol
- Both gases are monoatomic, so γ (gamma) = 1.67.

### Step 2: Use the Formula for the Speed of Sound in a Gas
The speed of sound in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( R \) = universal gas constant = 8.31 J/(mol·K)
- \( M \) = molar mass of the gas mixture.

### Step 3: Calculate the Molar Mass of the Mixture
Let the composition of neon in the mixture be \( x \) (in kg), then the composition of helium will be \( 1 - x \) (in kg). The average molar mass \( M \) of the mixture can be expressed as:
\[ M = x \cdot M_{Ne} + (1 - x) \cdot M_{He} \]
Substituting the values:
\[ M = x \cdot 0.02 + (1 - x) \cdot 0.004 \]

### Step 4: Substitute Values into the Speed of Sound Formula
Substituting \( M \) into the speed of sound formula:
\[ 760 = \sqrt{\frac{1.67 \cdot 8.31 \cdot 303}{M}} \]
Squaring both sides gives:
\[ 760^2 = \frac{1.67 \cdot 8.31 \cdot 303}{M} \]
Calculating \( 760^2 \):
\[ 577600 = \frac{1.67 \cdot 8.31 \cdot 303}{M} \]

### Step 5: Calculate M
Rearranging the equation to find \( M \):
\[ M = \frac{1.67 \cdot 8.31 \cdot 303}{577600} \]
Calculating the numerator:
\[ 1.67 \cdot 8.31 \cdot 303 \approx 4241.83 \]
Thus:
\[ M \approx \frac{4241.83}{577600} \approx 0.00735 \text{ kg/mol} \]

### Step 6: Set Up the Equation for Molar Mass
Now, we have:
\[ 0.00735 = 0.02x + 0.004(1 - x) \]
Expanding this gives:
\[ 0.00735 = 0.02x + 0.004 - 0.004x \]
Combining like terms:
\[ 0.00735 = 0.016x + 0.004 \]
Subtracting 0.004 from both sides:
\[ 0.00335 = 0.016x \]
Solving for \( x \):
\[ x = \frac{0.00335}{0.016} \approx 0.209375 \]

### Step 7: Find the Composition
Now, we can find the percentage of each gas in the mixture:
- Percentage of Ne = \( x \times 100 \approx 20.94\% \)
- Percentage of He = \( (1 - x) \times 100 \approx 79.06\% \)

### Final Answer
The composition of the mixture is approximately:
- Neon (Ne): 20.94%
- Helium (He): 79.06%