A first order reaction: `A rarr` Products and a second order reaction: `2R rarr` Products both have half time of `20 min` when they are carried out taking `4 mol L^(-1)` of their respective reactants. The number of mole per litre of `A` and `R` remaining unreacted after`60 min` form the start of the reaction, respectively, will be

Options

A. `1` and `0.5 M`
B. `0.5 M` and negligible
C. `0.5` and `1 M`
D. `1` and `0.25 M`

Explanation

To solve the problem, we need to calculate the concentration of reactants A and R after 60 minutes for a first-order and a second-order reaction, respectively.

### Step 1: Calculate the concentration of A after 60 minutes (First-order reaction)

1. **Identify the half-life of the first-order reaction**: The half-life (t₁/₂) is given as 20 minutes.
2. **Determine how many half-lives fit into 60 minutes**:
- 60 minutes / 20 minutes = 3 half-lives.
3. **Calculate the concentration remaining after each half-life**:
- Initial concentration of A = 4 mol/L.
- After 1st half-life (20 min): 4 mol/L → 2 mol/L.
- After 2nd half-life (40 min): 2 mol/L → 1 mol/L.
- After 3rd half-life (60 min): 1 mol/L → 0.5 mol/L.

Thus, the concentration of A remaining after 60 minutes is **0.5 mol/L**.

### Step 2: Calculate the concentration of R after 60 minutes (Second-order reaction)

1. **Identify the half-life of the second-order reaction**: The half-life (t₁/₂) is also given as 20 minutes.
2. **Understand the relationship of half-life in second-order reactions**: The half-life for a second-order reaction is inversely proportional to the initial concentration.
- The formula for half-life of a second-order reaction is:
\[ t_{1/2} = \frac{1}{k[A]_0} \]
- Since t₁/₂ is constant at 20 minutes for an initial concentration of 4 mol/L, we can calculate the concentration after each half-life.
3. **Calculate the concentration remaining after each half-life**:
- After the first half-life (20 min):
- Initial concentration = 4 mol/L → After 20 min: 2 mol/L.
- After the second half-life (40 min):
- New t₁/₂ = 40 min (since concentration is now 2 mol/L).
- After 40 min: 2 mol/L → 1 mol/L.
- After the third half-life (60 min):
- New t₁/₂ = 60 min (since concentration is now 1 mol/L).
- After 60 min: 1 mol/L → 0.5 mol/L.

Thus, the concentration of R remaining after 60 minutes is **1 mol/L**.

### Final Answer:
- Concentration of A after 60 minutes: **0.5 mol/L**
- Concentration of R after 60 minutes: **1 mol/L**