Speed of sound in gas x is `1270 ms^(-1)?.` Calculate the speed of sound in the mixture of gas y and gas x in which they are mixed in 1: 8 ratio, given of ratio densities of the two gases is 6:1

Answer

To solve the problem of calculating the speed of sound in the mixture of gases X and Y, we will follow these steps:

### Step 1: Write down the given data
- Speed of sound in gas X, \( S_x = 1270 \, \text{m/s} \)
- Volume ratio of gas Y to gas X, \( \frac{V_y}{V_x} = \frac{1}{8} \)
- Density ratio of gas Y to gas X, \( \frac{\rho_y}{\rho_x} = \frac{6}{1} \)

### Step 2: Express the volumes in terms of a common variable
Let the volume of gas Y be \( V_y = V \) and the volume of gas X be \( V_x = 8V \).

### Step 3: Calculate the total mass of the mixture
Using the density and volume, the total mass of the mixture can be calculated as:
\[ \text{Total mass} = \text{mass of gas Y} + \text{mass of gas X} \]
\[ = \rho_y \cdot V_y + \rho_x \cdot V_x = \rho_y \cdot V + \rho_x \cdot 8V \]

### Step 4: Substitute the density ratio
From the density ratio \( \frac{\rho_y}{\rho_x} = 6 \), we can express \( \rho_y \) in terms of \( \rho_x \):
\[ \rho_y = 6 \rho_x \]
Substituting this into the total mass equation:
\[ \text{Total mass} = 6\rho_x \cdot V + \rho_x \cdot 8V = (6\rho_x + 8\rho_x)V = 14\rho_x V \]

### Step 5: Calculate the total volume
The total volume of the mixture is:
\[ V_{\text{total}} = V_y + V_x = V + 8V = 9V \]

### Step 6: Calculate the density of the mixture
The density of the mixture \( \rho_m \) is given by:
\[ \rho_m = \frac{\text{Total mass}}{\text{Total volume}} = \frac{14\rho_x V}{9V} = \frac{14\rho_x}{9} \]

### Step 7: Relate the speed of sound in the mixture to the speed of sound in gas X
The speed of sound in a medium is inversely proportional to the square root of its density:
\[ \frac{S_m}{S_x} = \sqrt{\frac{\rho_x}{\rho_m}} \]
Substituting for \( \rho_m \):
\[ \frac{S_m}{1270} = \sqrt{\frac{\rho_x}{\frac{14\rho_x}{9}}} = \sqrt{\frac{9}{14}} \]

### Step 8: Solve for \( S_m \)
Now, we can solve for \( S_m \):
\[ S_m = 1270 \cdot \sqrt{\frac{9}{14}} = 1270 \cdot \frac{3}{\sqrt{14}} = \frac{3810}{\sqrt{14}} \, \text{m/s} \]

### Step 9: Final answer
Thus, the speed of sound in the mixture of gases X and Y is:
\[ S_m = \frac{3810}{\sqrt{14}} \, \text{m/s} \]