For the following reaction:
Initial concentration: `overset(10"mol"//L)(2A)+overset(2"mol"//L)(B)to"product"`
`t_(1//2)` of the overall reaction is the time when

Options

A. half of A changes to product
B. half of B changes to product
C. half of each of A and B changes to product
D. 6 moles of A and B changes to product

Answer

To solve the question regarding the half-life of the overall reaction given the initial concentrations of reactants, we can follow these steps:

### Step 1: Write the balanced chemical equation
The reaction given is:
\[ 2A + B \rightarrow \text{Product} \]

### Step 2: Identify the initial concentrations
From the problem, we have:
- Initial concentration of \( A \) = \( 10 \, \text{mol/L} \)
- Initial concentration of \( B \) = \( 2 \, \text{mol/L} \)

### Step 3: Determine the stoichiometry of the reaction
According to the balanced equation, 2 moles of \( A \) react with 1 mole of \( B \). This means:
- For every 2 moles of \( A \), 1 mole of \( B \) is required.

### Step 4: Calculate the limiting reagent
To find the limiting reagent, we can compare the mole ratio of the reactants based on their initial concentrations:
- From the stoichiometry, we need \( 2 \) moles of \( A \) for every \( 1 \) mole of \( B \).
- Therefore, the required amount of \( A \) for \( 2 \, \text{mol/L} \) of \( B \) is:
\[ \text{Required } A = 2 \times 2 = 4 \, \text{mol/L} \]

Since we have \( 10 \, \text{mol/L} \) of \( A \), which is more than the required \( 4 \, \text{mol/L} \), \( B \) is the limiting reagent.

### Step 5: Determine the half-life of the reaction
The half-life of a reaction is defined as the time required for half of the limiting reagent to be consumed. Since \( B \) is the limiting reagent, we need to find out how much of \( B \) will be consumed when half of it reacts.

- Initial concentration of \( B \) = \( 2 \, \text{mol/L} \)
- Half of \( B \) = \( 1 \, \text{mol/L} \)

Thus, the half-life of the overall reaction is the time taken for \( 1 \, \text{mol/L} \) of \( B \) to be converted into product.

### Conclusion
The half-life of the overall reaction is the time when half of \( B \) changes to product.