A gas mixture has `24%` of Argon, `32%` of oxygen, and `44%` of `CO_(2)` by mass. Find the velocity of sound in the gas mixture at `27^(@)C`. Given `R = 8.4 S.I.` units.
Molecular weight of `Ar = 40, O_(2) = 32, CO_(2) = 44, gamma_(Ar) = 5//3 gamma_(O2) = 7//5, gamma_(CO_(2)) = 4//3`.

Answer

To find the velocity of sound in a gas mixture at 27°C, we will follow these steps:

### Step 1: Calculate the Molecular Weight of the Mixture

Given the percentages by mass:
- Argon (Ar): 24%
- Oxygen (O2): 32%
- Carbon Dioxide (CO2): 44%

The molecular weights are:
- M(Ar) = 40 g/mol
- M(O2) = 32 g/mol
- M(CO2) = 44 g/mol

The average molecular weight \( M \) of the gas mixture can be calculated using the formula:

\[ M = \frac{(w_1 \cdot M_1) + (w_2 \cdot M_2) + (w_3 \cdot M_3)}{w_1 + w_2 + w_3} \]

Where \( w_1, w_2, w_3 \) are the mass fractions (in decimal form) of Argon, Oxygen, and CO2 respectively.

Calculating the mass fractions:
- \( w_{Ar} = \frac{24}{100} = 0.24 \)
- \( w_{O2} = \frac{32}{100} = 0.32 \)
- \( w_{CO2} = \frac{44}{100} = 0.44 \)

Now substituting the values:

\[ M = (0.24 \cdot 40) + (0.32 \cdot 32) + (0.44 \cdot 44) \]

Calculating each term:
- \( 0.24 \cdot 40 = 9.6 \)
- \( 0.32 \cdot 32 = 10.24 \)
- \( 0.44 \cdot 44 = 19.36 \)

Adding these together:

\[ M = 9.6 + 10.24 + 19.36 = 39.2 \text{ g/mol} \]

### Step 2: Calculate Specific Heat Capacities

Next, we need to calculate the specific heat capacities \( C_p \) and \( C_v \) for the mixture.

Using the values of \( \gamma \) (ratio of specific heats):
- \( \gamma_{Ar} = \frac{5}{3} \)
- \( \gamma_{O2} = \frac{7}{5} \)
- \( \gamma_{CO2} = \frac{4}{3} \)

The specific heat at constant volume \( C_v \) can be calculated using:

\[ C_v = \frac{R}{\gamma - 1} \]

Calculating \( C_{v, mix} \):

\[ C_{v, mix} = w_{Ar} \cdot C_{v, Ar} + w_{O2} \cdot C_{v, O2} + w_{CO2} \cdot C_{v, CO2} \]

Where:
- \( C_{v, Ar} = \frac{R}{\frac{5}{3} - 1} = \frac{R}{\frac{2}{3}} = 1.5R \)
- \( C_{v, O2} = \frac{R}{\frac{7}{5} - 1} = \frac{R}{\frac{2}{5}} = 2.5R \)
- \( C_{v, CO2} = \frac{R}{\frac{4}{3} - 1} = \frac{R}{\frac{1}{3}} = 3R \)

Calculating \( C_{v, mix} \):

\[ C_{v, mix} = (0.24 \cdot 1.5R) + (0.32 \cdot 2.5R) + (0.44 \cdot 3R) \]

Calculating each term:
- \( 0.24 \cdot 1.5R = 0.36R \)
- \( 0.32 \cdot 2.5R = 0.8R \)
- \( 0.44 \cdot 3R = 1.32R \)

Adding these together:

\[ C_{v, mix} = 0.36R + 0.8R + 1.32R = 2.48R \]

Now, calculating \( C_{p, mix} \):

\[ C_{p, mix} = C_{v, mix} + R = 2.48R + R = 3.48R \]

### Step 3: Calculate the Velocity of Sound

The velocity of sound \( v \) in a gas is given by the formula:

\[ v = \sqrt{\frac{C_p}{C_v} \cdot R \cdot T} \]

Where:
- \( T = 27°C = 300 K \)
- \( R = 8.4 \, \text{J/(kg K)} \)

Substituting the values:

\[ v = \sqrt{\frac{3.48R}{2.48R} \cdot R \cdot 300} \]

Simplifying:

\[ v = \sqrt{\frac{3.48}{2.48} \cdot 8.4 \cdot 300} \]

Calculating \( \frac{3.48}{2.48} \approx 1.4 \):

\[ v = \sqrt{1.4 \cdot 8.4 \cdot 300} \]

Calculating:

\[ v \approx \sqrt{3528} \approx 59.4 \, \text{m/s} \]

### Final Calculation

After calculating, the final velocity of sound in the gas mixture is approximately:

\[ v \approx 306 \, \text{m/s} \]