For a reaction A +B → C, the experimental rate law is found to be `R=k[A]^1[B]^(1/2)`. Find the rate of the reaction when [A] = 0.5 M, [B] = 0.1 M and k=0.03.

Options

A. 4.74 ×` 10^-2 (Lmol^-1)^(1/2) s^-1`
B. 5.38 ×` 10^-2 (Lmol^-1)^(1/2) s^-1`
C. 5.748 ×` 10^-2 (Lmol^-1)^(1/2) s^-1`
D. 4.86 ×` 10^-2 (Lmol^-1)^(1/2) s^-1`

Answer

To find the rate of the reaction given the rate law and the concentrations of reactants, we can follow these steps:

### Step 1: Write down the rate law
The rate law for the reaction A + B → C is given as:
\[ R = k[A]^1[B]^{1/2} \]

### Step 2: Substitute the values into the rate law
We need to substitute the values of the concentrations of A and B, and the rate constant k into the rate law. Given:
- \([A] = 0.5 \, M\)
- \([B] = 0.1 \, M\)
- \(k = 0.03 \, L^{1.5} \, mol^{-0.5} \, s^{-1}\)

So, substituting these values into the rate law:
\[ R = 0.03 \times (0.5)^1 \times (0.1)^{1/2} \]

### Step 3: Calculate \((0.1)^{1/2}\)
Calculate the square root of 0.1:
\[ (0.1)^{1/2} = 0.3162 \]

### Step 4: Substitute the calculated value back into the equation
Now substitute this value back into the rate equation:
\[ R = 0.03 \times 0.5 \times 0.3162 \]

### Step 5: Perform the multiplication
Now, calculate the rate:
1. First, calculate \(0.03 \times 0.5 = 0.015\)
2. Then, multiply \(0.015 \times 0.3162\):
\[ R = 0.015 \times 0.3162 = 0.004743 \]

### Step 6: Express the final answer in scientific notation
Expressing \(0.004743\) in scientific notation gives:
\[ R \approx 4.74 \times 10^{-3} \, L \, mol^{-1} \, s^{-1} \]

### Final Answer:
The rate of the reaction is approximately \(4.74 \times 10^{-3} \, L \, mol^{-1} \, s^{-1}\).

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