What will be the relation between the `T_(1)` of gas 1 with `M_(1)=56` and `T_(2)` of gas 2 with `M_(2)=44` if the average speed of gas 1 is equal to most probable speed of gas 2?

Options

A. `T_(1)=T_(2)^(2)`
B. `T_(1)=T_(2)`
C. `T_(1)=(T_(2))^((1)/(2))`
D. `T_(1)=(1)/(T_(2))`

Answer

To solve the problem, we need to establish the relationship between the temperatures \( T_1 \) and \( T_2 \) of two gases based on their average and most probable speeds.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Molar mass of gas 1, \( M_1 = 56 \) g/mol
- Molar mass of gas 2, \( M_2 = 44 \) g/mol
- Average speed of gas 1 is equal to the most probable speed of gas 2.

2. **Write the Formulas for Speeds:**
- The average speed \( V_{avg} \) of a gas is given by:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
- The most probable speed \( V_{mp} \) of a gas is given by:
\[ V_{mp} = \sqrt{\frac{2RT}{M}} \]

3. **Set Up the Equation:**
- According to the problem, we have:
\[ V_{avg} (gas 1) = V_{mp} (gas 2) \]
- Substituting the formulas:
\[ \sqrt{\frac{8RT_1}{\pi M_1}} = \sqrt{\frac{2RT_2}{M_2}} \]

4. **Square Both Sides:**
- To eliminate the square roots, we square both sides:
\[ \frac{8RT_1}{\pi M_1} = \frac{2RT_2}{M_2} \]

5. **Cancel Out the Universal Gas Constant \( R \):**
- Since \( R \) is present on both sides, we can cancel it out:
\[ \frac{8T_1}{\pi M_1} = \frac{2T_2}{M_2} \]

6. **Substitute the Molar Mass Values:**
- Plugging in \( M_1 = 56 \) and \( M_2 = 44 \):
\[ \frac{8T_1}{\pi \cdot 56} = \frac{2T_2}{44} \]

7. **Cross-Multiply to Solve for \( T_1 \) and \( T_2 \):**
- Cross-multiplying gives:
\[ 8T_1 \cdot 44 = 2T_2 \cdot \pi \cdot 56 \]
- Simplifying this:
\[ 352T_1 = 112\pi T_2 \]

8. **Isolate \( T_1 \) in Terms of \( T_2 \):**
- Dividing both sides by 352:
\[ T_1 = \frac{112\pi T_2}{352} \]
- Simplifying further:
\[ T_1 = \frac{112\pi}{352} T_2 = \frac{\pi}{3.142857} T_2 \approx 0.3183 T_2 \]

### Final Relation:
Thus, the relation between \( T_1 \) and \( T_2 \) is:
\[ T_1 \approx 0.3183 T_2 \]