For the following balanced redox reaction:
`2MnO_(4)^(ɵ)+8H^(o+)+Br_(2)to2Mn^(2+)+2BrO_(3)^(ɵ)+2H_(2)O`
if the molecular weight of `MnO_(4)^(ɵ):Br_(2)` and `Br_(2)` be `M_(1)M_(2)` respectively, then

Options

A. equivalent weight of `MnO_(4)^(ɵ)` is `(M_(1))/(5)`
B. Equivalents weight of `Br_(2)` is `(M_(2))/(10)`
C. The n-factor ratio of `MnO_(4)^(ɵ):Br_(2)is1:1`
D. none of these

Explanation

To solve the problem, we need to analyze the given balanced redox reaction and determine the equivalent weights and the ratio of the molecular weights of \( \text{MnO}_4^- \) and \( \text{Br}_2 \).

### Step-by-Step Solution:

1. **Identify the Oxidation States**:
- In \( \text{MnO}_4^- \), manganese (Mn) has an oxidation state of +7.
- In \( \text{Br}_2 \), bromine (Br) has an oxidation state of 0.

2. **Determine the Change in Oxidation States**:
- For \( \text{MnO}_4^- \) going to \( \text{Mn}^{2+} \):
- Change in oxidation state = +7 to +2 = 5 (reduction).
- For \( \text{Br}_2 \) going to \( \text{BrO}_3^- \):
- Change in oxidation state = 0 to +5 = 5 (oxidation for each Br).
- Since there are 2 bromine atoms in \( \text{Br}_2 \), the total change is 10 (oxidation).

3. **Calculate the n-factor**:
- The n-factor for \( \text{MnO}_4^- \) is 5 (since it gains 5 electrons).
- The n-factor for \( \text{Br}_2 \) is 10 (since it loses 10 electrons).

4. **Determine Equivalent Weights**:
- The equivalent weight of a substance is given by the formula:
\[ \text{Equivalent Weight} = \frac{\text{Molecular Weight}}{\text{n-factor}} \]
- For \( \text{MnO}_4^- \):
\[ \text{Equivalent Weight} = \frac{M_1}{5} \]
- For \( \text{Br}_2 \):
\[ \text{Equivalent Weight} = \frac{M_2}{10} \]

5. **Establish the Ratio of Molecular Weights**:
- From the equivalent weights, we have:
\[ \frac{M_1}{5} \quad \text{and} \quad \frac{M_2}{10} \]
- Setting the two equivalent weights equal gives:
\[ \frac{M_1}{5} = \frac{M_2}{10} \]
- Cross-multiplying leads to:
\[ 10M_1 = 5M_2 \quad \Rightarrow \quad 2M_1 = M_2 \]
- Thus, the ratio \( M_1 : M_2 = 1 : 2 \).

### Final Answer:
The ratio of the molecular weights \( M_1 : M_2 \) is \( 1 : 2 \).