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 molecular masses and the given condition that the average speed of gas 1 is equal to the most probable speed of gas 2.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The average speed \( C_{\text{avg}} \) of a gas is given by the formula:
\[ C_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}} \]
- The most probable speed \( C_{\text{mp}} \) of a gas is given by the formula:
\[ C_{\text{mp}} = \sqrt{\frac{2RT}{M}} \]

2. **Set Up the Equation**:
- According to the problem, the average speed of gas 1 is equal to the most probable speed of gas 2:
\[ C_{\text{avg}}(T_1) = C_{\text{mp}}(T_2) \]
- Substituting the formulas into this equation gives:
\[ \sqrt{\frac{8RT_1}{\pi M_1}} = \sqrt{\frac{2RT_2}{M_2}} \]

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

4. **Rearrange the Equation**:
- Rearranging the equation gives:
\[ 8RT_1 \cdot M_2 = 2RT_2 \cdot \pi M_1 \]
- We can cancel \( R \) from both sides (assuming \( R \neq 0 \)):
\[ 8T_1 \cdot M_2 = 2T_2 \cdot \pi M_1 \]

5. **Substitute the Molecular Masses**:
- Substitute \( M_1 = 56 \) and \( M_2 = 44 \):
\[ 8T_1 \cdot 44 = 2T_2 \cdot \pi \cdot 56 \]

6. **Simplify the Equation**:
- Simplifying gives:
\[ 352T_1 = 112\pi T_2 \]
- Dividing both sides by 112:
\[ T_1 = \frac{112\pi}{352} T_2 \]
- Simplifying further:
\[ T_1 = \frac{\pi}{3.14} T_2 \quad (\text{approximately}) \]

7. **Final Relation**:
- Since \( \frac{112}{352} \) simplifies to \( \frac{1}{3.14} \), we can conclude:
\[ T_1 = T_2 \]

### Conclusion:
Thus, the relationship between \( T_1 \) and \( T_2 \) is:
\[ T_1 = T_2 \]