Suppose `50` bacterial are placed in a flask containing nutrients for the bacteria so that they can multiply. A study at `35^(@)C` gave the following results:
Time: (minutes) `0, 15, 30, 45, 60`
Number of bacteria `100,, 200, 400, 800, 1600`
The rate of reaction initially is:

Options

A. `4.62` bacterias per minute
B. `23.1` bacterias per minute
C. `23.1` bacterias per sec
D. `0.231` bacterias per sec

Explanation

To find the initial rate of reaction of bacterial growth, we will follow these steps:

### Step 1: Identify the initial number of bacteria
At time \( t = 0 \) minutes, the number of bacteria is given as \( 100 \).

### Step 2: Identify the number of bacteria after a specific time
At \( t = 15 \) minutes, the number of bacteria doubles to \( 200 \).

### Step 3: Determine the time taken for the bacteria to double
From the data provided, we can see that the bacteria double every \( 15 \) minutes (from \( 100 \) to \( 200 \), and again from \( 200 \) to \( 400 \), etc.).

### Step 4: Calculate the rate constant (k)
In first-order kinetics, the relationship between the half-life (\( t_{1/2} \)) and the rate constant (\( k \)) is given by the formula:
\[ t_{1/2} = \frac{\ln(2)}{k} \]
Since we know that the doubling time is \( 15 \) minutes, we can rearrange the formula to find \( k \):
\[ k = \frac{\ln(2)}{15} \]

### Step 5: Calculate the initial rate of reaction
The rate of reaction can be expressed as:
\[ r = k \cdot [A_0] \]
Where \( [A_0] \) is the initial concentration of bacteria at \( t = 0 \) minutes, which is \( 100 \). Thus:
\[ r = \left(\frac{\ln(2)}{15}\right) \cdot 100 \]

### Step 6: Compute the value of r
Using the value of \( \ln(2) \approx 0.693 \):
\[ r = \left(\frac{0.693}{15}\right) \cdot 100 \approx 4.62 \text{ bacteria per minute} \]

### Conclusion
The initial rate of reaction is approximately \( 4.62 \) bacteria per minute.

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