A solution containing active cobalt `60/27` Co having activity of `0.8 muCi` and decay constant `lambda` is injected in an animal's body. If `1cm^(3)` of blood is drawn from the animal's body after 10hrs of injection , the activity found was 300 decays per minute. What is the volume of blood that is flowing in the body ? (`1 Ci = 3.7 xx 10^(10)` decays per second and at `t = 10` hrs `e^(-lambdat) = 0.84`)

Options

A. 4 L
B. 6 L
C. 5 L
D. 7 L

Answer

To solve the problem, we will follow these steps:

### Step 1: Convert the initial activity from microcuries to decays per second
Given:
- Initial activity \( R_0 = 0.8 \, \mu Ci \)
- \( 1 \, Ci = 3.7 \times 10^{10} \, \text{decays/second} \)

Calculating \( R_0 \):
\[ R_0 = 0.8 \times 10^{-6} \, Ci \times 3.7 \times 10^{10} \, \text{decays/second} \]
\[ R_0 = 0.8 \times 3.7 \times 10^{4} \, \text{decays/second} \]
\[ R_0 = 2.96 \times 10^{4} \, \text{decays/second} \]

### Step 2: Use the decay formula to find the activity after 10 hours
The decay formula is given by:
\[ R = R_0 e^{-\lambda t} \]
Given \( e^{-\lambda t} = 0.84 \) and \( t = 10 \, \text{hours} = 36000 \, \text{seconds} \):
\[ R = R_0 \times 0.84 \]
Substituting \( R_0 \):
\[ R = 2.96 \times 10^{4} \times 0.84 \, \text{decays/second} \]
\[ R \approx 2.4864 \times 10^{4} \, \text{decays/second} \]

### Step 3: Convert the activity found in blood to decays per second
The activity found in the blood is given as:
\[ 300 \, \text{decays/minute} \]
Converting this to decays per second:
\[ R = \frac{300}{60} = 5 \, \text{decays/second} \]

### Step 4: Set up the relationship for the volume of blood
Let \( V \) be the volume of blood in liters. The relationship between the activity in the blood and the total activity in the body is:
\[ \frac{R}{R_0 e^{-\lambda t}} = \frac{1 \, \text{cm}^3}{V} \]
Substituting the values:
\[ \frac{5}{2.4864 \times 10^{4}} = \frac{1 \times 10^{-3}}{V} \]
Rearranging gives:
\[ V = \frac{1 \times 10^{-3} \times 2.4864 \times 10^{4}}{5} \]
Calculating \( V \):
\[ V = \frac{2.4864 \times 10^{1}}{5} = 4.9728 \, \text{liters} \]
Rounding gives:
\[ V \approx 5 \, \text{liters} \]

### Final Answer
The volume of blood flowing in the body is approximately **5 liters**.

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