Compare the velocities of sound in Argon and Oxygen. `gamma" of "O_2` and Argon are 1.4 and 1.67 respectively.

Answer

To compare the velocities of sound in Argon and Oxygen, we can use the formula for the velocity of sound in a gas, which is given by:

\[ V = \sqrt{\frac{\gamma RT}{M}} \]

Where:
- \( V \) is the velocity of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Write the formula for the ratio of velocities
We can express the ratio of the velocities of sound in Argon (\( V_{Ar} \)) and Oxygen (\( V_{O_2} \)) as follows:

\[ \frac{V_{Ar}}{V_{O_2}} = \sqrt{\frac{\gamma_{Ar} R T / M_{Ar}}{\gamma_{O_2} R T / M_{O_2}}} \]

Since \( R \) and \( T \) are constants for both gases, they cancel out:

\[ \frac{V_{Ar}}{V_{O_2}} = \sqrt{\frac{\gamma_{Ar}}{\gamma_{O_2}} \cdot \frac{M_{O_2}}{M_{Ar}}} \]

### Step 2: Substitute the known values
Given:
- \( \gamma_{Ar} = 1.67 \)
- \( \gamma_{O_2} = 1.4 \)
- Molar mass of Oxygen (\( M_{O_2} \)) = 32 g/mol
- Molar mass of Argon (\( M_{Ar} \)) = 40 g/mol

Substituting these values into the equation:

\[ \frac{V_{Ar}}{V_{O_2}} = \sqrt{\frac{1.67}{1.4} \cdot \frac{32}{40}} \]

### Step 3: Calculate the ratio
Now, we can calculate the values step by step:

1. Calculate \( \frac{1.67}{1.4} \):
\[ \frac{1.67}{1.4} \approx 1.192857 \]

2. Calculate \( \frac{32}{40} \):
\[ \frac{32}{40} = 0.8 \]

3. Now, multiply these two results:
\[ 1.192857 \cdot 0.8 \approx 0.954286 \]

4. Finally, take the square root:
\[ \sqrt{0.954286} \approx 0.977 \]

### Step 4: Write the final ratio
Thus, we have:

\[ \frac{V_{Ar}}{V_{O_2}} \approx 0.977 \]

This means:

\[ V_{Ar} \approx 0.977 \cdot V_{O_2} \]

### Conclusion
The velocity of sound in Argon is approximately 0.977 times that in Oxygen.

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