Half life of 90Sr is 6.93 years. In a child body 1g of 90Sr dopped in place of calcium, how many years will it take to reduce its concentration by 90% (Assume no involvement of Sr in metabolism)

Options

A. (a) 23.03 yr
B. (b) 15.42 yr
C. (c) 35.10 yr
D. (d) 30.02 yr

Answer

To solve the problem of how long it will take for the concentration of 90Sr to reduce by 90%, we can follow these steps:

### Step 1: Understand the Half-Life
The half-life of 90Sr is given as 6.93 years. This means that every 6.93 years, half of the remaining quantity of 90Sr will decay.

### Step 2: Determine the Initial and Remaining Concentrations
We start with an initial concentration of 1g of 90Sr. To find out how long it takes to reduce this concentration by 90%, we need to find the remaining concentration after 90% decay:
- Initial concentration (A) = 1g
- Remaining concentration after 90% decay = 10% of 1g = 0.1g

### Step 3: Use the First-Order Kinetics Formula
Radioactive decay follows first-order kinetics. The formula for the relationship between half-life (t₁/₂), rate constant (k), and the concentration is:
\[ k = \frac{0.693}{t_{1/2}} \]
Where \( t_{1/2} \) is the half-life.

### Step 4: Calculate the Rate Constant (k)
Using the half-life of 90Sr:
\[ k = \frac{0.693}{6.93 \text{ years}} \]
Calculating this gives:
\[ k \approx 0.100 \text{ year}^{-1} \]

### Step 5: Set Up the Equation for 90% Decay
We need to find the time (t) it takes for the concentration to drop from 1g to 0.1g. We can use the integrated first-order rate equation:
\[ \ln\left(\frac{A}{A - x}\right) = kt \]
Where:
- A = initial concentration = 1g
- x = amount decayed = 1g - 0.1g = 0.9g

Substituting the values:
\[ \ln\left(\frac{1}{1 - 0.9}\right) = kt \]
\[ \ln\left(\frac{1}{0.1}\right) = kt \]
\[ \ln(10) = kt \]

### Step 6: Solve for Time (t)
Now substitute k into the equation:
\[ \ln(10) = 0.100 \cdot t \]
Calculating \( \ln(10) \) gives approximately 2.303. Therefore:
\[ 2.303 = 0.100 \cdot t \]
\[ t = \frac{2.303}{0.100} \]
\[ t \approx 23.03 \text{ years} \]

### Conclusion
It will take approximately **23.03 years** for the concentration of 90Sr to reduce by 90%.

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