The initial rate of reaction `A+5B+6Cto3L+3M` has been determined by measuring the rate of disappearance of A under the following conditions :
`{:("Expt. No.",,,[A]_(0)//M,,,[B]_(0)//M,,,[C]_(0)//M,,,"Initial rate"//"M min"^(-1)),(1,,,0.02,,,0.02,,,0.02,,,2.08xx10^(-3)),(2,,,0.01,,,0.02,,,0.02,,,1.04xx10^(-3)),(3,,,0.02,,,0.04,,,0.02,,,4.16xx10^(-3)),(4,,,0.02,,,0.02,,,0.04,,,8.32xx10^(-3)):}`
Determine the order of reaction with respect to each reactant and overall order of the reaction. What is the rate constant ? Calculate the initial rate of change in concentration of B and L.

Explanation

To solve the problem step by step, we need to determine the order of reaction with respect to each reactant (A, B, and C), the overall order of the reaction, the rate constant, and the initial rate of change in concentration of B and L.

### Step 1: Write the Rate Law Expression
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k [A]^{\alpha} [B]^{\beta} [C]^{\gamma} \]
where \( \alpha, \beta, \gamma \) are the orders of the reaction with respect to A, B, and C respectively.

### Step 2: Analyze the Experimental Data
We have the following experimental data:

| Expt. No. | [A]₀ (M) | [B]₀ (M) | [C]₀ (M) | Initial Rate (M/min) |
|-----------|----------|----------|----------|-----------------------|
| 1 | 0.02 | 0.02 | 0.02 | 2.08 x 10⁻³ |
| 2 | 0.01 | 0.02 | 0.02 | 1.04 x 10⁻³ |
| 3 | 0.02 | 0.04 | 0.02 | 4.16 x 10⁻³ |
| 4 | 0.02 | 0.02 | 0.04 | 8.32 x 10⁻³ |

### Step 3: Determine the Order with Respect to A
Using experiments 1 and 2:
- From Expt 1:
\[ \text{Rate}_1 = k [0.02]^{\alpha} [0.02]^{\beta} [0.02]^{\gamma} = 2.08 \times 10^{-3} \]
- From Expt 2:
\[ \text{Rate}_2 = k [0.01]^{\alpha} [0.02]^{\beta} [0.02]^{\gamma} = 1.04 \times 10^{-3} \]

Dividing the two equations:
\[ \frac{2.08 \times 10^{-3}}{1.04 \times 10^{-3}} = \frac{[0.02]^{\alpha} [0.02]^{\beta} [0.02]^{\gamma}}{[0.01]^{\alpha} [0.02]^{\beta} [0.02]^{\gamma}} \]
This simplifies to:
\[ 2 = \left(\frac{0.02}{0.01}\right)^{\alpha} \]
\[ 2 = 2^{\alpha} \]
Thus, \( \alpha = 1 \).

### Step 4: Determine the Order with Respect to B
Using experiments 1 and 3:
- From Expt 1:
\[ \text{Rate}_1 = 2.08 \times 10^{-3} \]
- From Expt 3:
\[ \text{Rate}_3 = k [0.02]^{\alpha} [0.04]^{\beta} [0.02]^{\gamma} = 4.16 \times 10^{-3} \]

Dividing the two equations:
\[ \frac{4.16 \times 10^{-3}}{2.08 \times 10^{-3}} = \frac{[0.02]^{\alpha} [0.04]^{\beta}}{[0.02]^{\alpha} [0.02]^{\gamma}} \]
This simplifies to:
\[ 2 = \left(\frac{0.04}{0.02}\right)^{\beta} \]
\[ 2 = 2^{\beta} \]
Thus, \( \beta = 1 \).

### Step 5: Determine the Order with Respect to C
Using experiments 1 and 4:
- From Expt 1:
\[ \text{Rate}_1 = 2.08 \times 10^{-3} \]
- From Expt 4:
\[ \text{Rate}_4 = 8.32 \times 10^{-3} \]

Dividing the two equations:
\[ \frac{8.32 \times 10^{-3}}{2.08 \times 10^{-3}} = \frac{[0.02]^{\alpha} [0.02]^{\beta} [0.04]^{\gamma}}{[0.02]^{\alpha} [0.02]^{\beta} [0.02]^{\gamma}} \]
This simplifies to:
\[ 4 = \left(\frac{0.04}{0.02}\right)^{\gamma} \]
\[ 4 = 2^{\gamma} \]
Thus, \( \gamma = 2 \).

### Step 6: Calculate Overall Order
The overall order of the reaction is:
\[ \text{Overall Order} = \alpha + \beta + \gamma = 1 + 1 + 2 = 4 \]

### Step 7: Calculate the Rate Constant (k)
Using any experiment, we can calculate k. Using Experiment 1:
\[ 2.08 \times 10^{-3} = k [0.02]^1 [0.02]^1 [0.02]^2 \]
\[ 2.08 \times 10^{-3} = k (0.02)(0.02)(0.02^2) \]
\[ 2.08 \times 10^{-3} = k (0.000008) \]
\[ k = \frac{2.08 \times 10^{-3}}{8 \times 10^{-6}} = 260 \, \text{M}^{-3} \text{min}^{-1} \]

### Step 8: Calculate Initial Rate of Change in Concentration of B and L
Using the rate of reaction:
- For B:
\[ -\frac{1}{5} \frac{d[B]}{dt} = \text{Rate} \]
\[ \frac{d[B]}{dt} = -5 \times 2.08 \times 10^{-3} = -1.04 \times 10^{-2} \, \text{M/min} \]

- For L:
\[ \frac{1}{3} \frac{d[L]}{dt} = \text{Rate} \]
\[ \frac{d[L]}{dt} = 3 \times 2.08 \times 10^{-3} = 6.24 \times 10^{-3} \, \text{M/min} \]

### Final Results
- Order with respect to A: 1
- Order with respect to B: 1
- Order with respect to C: 2
- Overall order: 4
- Rate constant (k): \( 260 \, \text{M}^{-3} \text{min}^{-1} \)
- Initial rate of change in concentration of B: \( -1.04 \times 10^{-2} \, \text{M/min} \)
- Initial rate of change in concentration of L: \( 6.24 \times 10^{-3} \, \text{M/min} \)