At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the r.m.s. speed of a helium gas atom at `-20^(@) C` ? (Atomic mass of Ar = 39.9 u, of He = 4.0 u).

Options

A. `2.52xx10^(3)K`
B. `2.52xx10^(2)K`
C. `4.03xx10^(3)K`
D. `4.03xx10^(2)K`

Explanation

To solve the problem, we need to find the temperature at which the root mean square (r.m.s.) speed of an argon atom equals the r.m.s. speed of a helium atom at -20°C.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Atomic mass of Argon (Ar) = 39.9 u
- Atomic mass of Helium (He) = 4.0 u
- Temperature of Helium (T') = -20°C = 253 K (after converting to Kelvin)

2. **Understand the Formula for r.m.s. Speed:**
The r.m.s. speed (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- R = universal gas constant (8.314 J/(mol·K))
- T = absolute temperature in Kelvin
- M = molar mass of the gas in kg/mol

3. **Set Up the Equation:**
Since we need the r.m.s. speed of Argon to equal the r.m.s. speed of Helium, we can set up the equation:
\[ \sqrt{\frac{3RT_a}{M_a}} = \sqrt{\frac{3RT'}{M'}} \]
where:
- T_a = temperature of Argon (what we need to find)
- M_a = molar mass of Argon = 39.9 u = 0.0399 kg/mol (conversion from atomic mass unit to kg)
- T' = 253 K (temperature of Helium)
- M' = molar mass of Helium = 4.0 u = 0.004 kg/mol

4. **Square Both Sides:**
To eliminate the square root, we square both sides of the equation:
\[ \frac{3RT_a}{M_a} = \frac{3RT'}{M'} \]

5. **Cancel Out Common Terms:**
The factor of 3R can be canceled from both sides:
\[ \frac{T_a}{M_a} = \frac{T'}{M'} \]

6. **Substitute the Known Values:**
Now substitute the known values into the equation:
\[ \frac{T_a}{39.9} = \frac{253}{4} \]

7. **Cross Multiply to Solve for T_a:**
Cross multiplying gives us:
\[ T_a = \frac{253 \times 39.9}{4} \]

8. **Calculate T_a:**
Now perform the calculation:
\[ T_a = \frac{253 \times 39.9}{4} = \frac{10069.7}{4} = 2517.425 \text{ K} \]

9. **Final Result:**
Therefore, the temperature at which the r.m.s. speed of an Argon atom equals that of a Helium atom at -20°C is approximately:
\[ T_a \approx 2517.4 \text{ K} \]