The r.m.s. velocity of oxygen molecules at 273K is `460 m//s`. Find the r.m.s. velocity of argon at `40^(@)C`.

Explanation

To find the r.m.s. velocity of argon at 40°C, we can use the relationship between the r.m.s. velocities of two different gases at different temperatures. The formula for r.m.s. velocity (V_rms) is given by:

\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 1: Convert temperatures to Kelvin
- The temperature for argon is given as 40°C. To convert this to Kelvin:
\[ T_{argon} = 40 + 273 = 313 \, K \]

### Step 2: Identify the molar masses
- The molar mass of oxygen (O₂) is approximately 32 g/mol, which is equivalent to:
\[ M_{O_2} = 32 \times 10^{-3} \, kg/mol \]
- The molar mass of argon (Ar) is approximately 40 g/mol, which is equivalent to:
\[ M_{Ar} = 40 \times 10^{-3} \, kg/mol \]

### Step 3: Set up the ratio of r.m.s. velocities
Using the relationship between the r.m.s. velocities of argon and oxygen:
\[ \frac{V_{rms, Ar}}{V_{rms, O_2}} = \sqrt{\frac{T_{Ar}}{M_{Ar}}} \div \sqrt{\frac{T_{O_2}}{M_{O_2}}} \]
This simplifies to:
\[ V_{rms, Ar} = V_{rms, O_2} \cdot \sqrt{\frac{T_{Ar}}{M_{Ar}} \cdot \frac{M_{O_2}}{T_{O_2}}} \]

### Step 4: Substitute known values
- Given that \( V_{rms, O_2} = 460 \, m/s \) and \( T_{O_2} = 273 \, K \):
\[ V_{rms, Ar} = 460 \cdot \sqrt{\frac{313}{40 \times 10^{-3}} \cdot \frac{32 \times 10^{-3}}{273}} \]

### Step 5: Calculate the values
1. Calculate the ratio:
\[ \frac{T_{Ar}}{M_{Ar}} = \frac{313}{40 \times 10^{-3}} = \frac{313}{0.04} = 7825 \]
2. Calculate the ratio:
\[ \frac{M_{O_2}}{T_{O_2}} = \frac{32 \times 10^{-3}}{273} = \frac{0.032}{273} \approx 0.000117 \]
3. Now combine these:
\[ V_{rms, Ar} = 460 \cdot \sqrt{7825 \cdot 0.000117} \]
4. Calculate the square root:
\[ \sqrt{7825 \cdot 0.000117} \approx \sqrt{0.916} \approx 0.957 \]
5. Finally, calculate \( V_{rms, Ar} \):
\[ V_{rms, Ar} \approx 460 \cdot 0.957 \approx 440.5 \, m/s \]

### Final Answer
The r.m.s. velocity of argon at 40°C is approximately \( 440.5 \, m/s \).

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