A radioactive elements has a half life of 2500 yrs. In how many years will its mass decay by 90% of its initial mass?

Answer

To solve the problem of how many years it will take for a radioactive element with a half-life of 2500 years to decay by 90% of its initial mass, we can follow these steps:

### Step 1: Understand the Problem
We know that the half-life (T) of the radioactive element is 2500 years. If the mass decays by 90%, this means that only 10% of the initial mass remains.

### Step 2: Define the Initial Mass
Let the initial mass of the radioactive element be \( N_0 \). After decaying by 90%, the remaining mass \( N \) will be:
\[ N = 0.1 N_0 \]

### Step 3: Use the Exponential Decay Formula
The relationship between the remaining mass and time can be expressed using the formula:
\[ N = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant and \( t \) is the time elapsed.

### Step 4: Substitute the Values
Substituting \( N \) and \( N_0 \) into the equation gives:
\[ 0.1 N_0 = N_0 e^{-\lambda t} \]
We can cancel \( N_0 \) from both sides (assuming \( N_0 \neq 0 \)):
\[ 0.1 = e^{-\lambda t} \]

### Step 5: Take the Natural Logarithm
Taking the natural logarithm of both sides:
\[ \ln(0.1) = -\lambda t \]

### Step 6: Relate Decay Constant to Half-Life
The decay constant \( \lambda \) can be related to the half-life \( T \) using the formula:
\[ \lambda = \frac{0.693}{T} \]
Substituting \( T = 2500 \) years:
\[ \lambda = \frac{0.693}{2500} \]

### Step 7: Substitute \( \lambda \) Back into the Equation
Now substituting \( \lambda \) back into the logarithmic equation:
\[ \ln(0.1) = -\left(\frac{0.693}{2500}\right) t \]

### Step 8: Solve for \( t \)
Rearranging gives:
\[ t = -\frac{2500 \ln(0.1)}{0.693} \]

### Step 9: Calculate \( \ln(0.1) \)
We know that:
\[ \ln(0.1) = \ln\left(\frac{1}{10}\right) = -\ln(10) \]
Using \( \ln(10) \approx 2.303 \):
\[ \ln(0.1) \approx -2.303 \]

### Step 10: Substitute and Calculate \( t \)
Substituting \( \ln(0.1) \) into the equation:
\[ t = -\frac{2500 \times (-2.303)}{0.693} \]
Calculating this gives:
\[ t \approx \frac{2500 \times 2.303}{0.693} \approx 8305.6 \text{ years} \]

Thus, it will take approximately **8305.6 years** for the mass of the radioactive element to decay by 90%.

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