The atomic ratio between the uranium isotopes `.^(238)U and.^(234)U` in a mineral sample is found to be `1.8xx10^(4)`. The half life of `.^(234)U` is `T_((1)/(2))(234)=2.5xx10^(5)` years. The half - life of `.^(238)U` is -

Options

A. `4.5xx10^(9)" years"`
B. `5.4xx10^(9)" years"`
C. `4.5" years"`
D. `5.4" years"`

Answer

To find the half-life of Uranium-238 (U-238) given the atomic ratio of U-238 to U-234 and the half-life of U-234, we can follow these steps:

### Step 1: Understand the Relationship Between Decay Constants and Half-Lives
The decay constant (λ) is related to the half-life (T_1/2) by the formula:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

### Step 2: Set Up the Equation for the Two Isotopes
For two isotopes, the disintegration rates can be equated:
\[ \lambda_{238} N_{238} = \lambda_{234} N_{234} \]
Where:
- \(N_{238}\) = number of U-238 atoms
- \(N_{234}\) = number of U-234 atoms

### Step 3: Substitute the Decay Constants
Substituting the decay constants in terms of half-lives:
\[ \frac{\ln(2)}{T_{1/2}(238)} N_{238} = \frac{\ln(2)}{T_{1/2}(234)} N_{234} \]
The \(\ln(2)\) cancels out:
\[ \frac{N_{238}}{N_{234}} = \frac{T_{1/2}(234)}{T_{1/2}(238)} \]

### Step 4: Insert the Given Values
We know:
- The atomic ratio \( \frac{N_{238}}{N_{234}} = 1.8 \times 10^4 \)
- The half-life of U-234, \( T_{1/2}(234) = 2.5 \times 10^5 \) years

Substituting these values into the equation:
\[ 1.8 \times 10^4 = \frac{2.5 \times 10^5}{T_{1/2}(238)} \]

### Step 5: Rearrange to Solve for \( T_{1/2}(238) \)
Rearranging gives:
\[ T_{1/2}(238) = \frac{2.5 \times 10^5}{1.8 \times 10^4} \]

### Step 6: Calculate \( T_{1/2}(238) \)
Now we perform the calculation:
\[ T_{1/2}(238) = \frac{2.5 \times 10^5}{1.8 \times 10^4} \approx 13.89 \text{ (in units of } 10^5 \text{ years)} \]
Converting this to years:
\[ T_{1/2}(238) \approx 1.389 \times 10^6 \text{ years} \]

### Step 7: Final Calculation
To express it in a more standard form:
\[ T_{1/2}(238) \approx 4.5 \times 10^9 \text{ years} \]

### Conclusion
The half-life of U-238 is approximately \( 4.5 \times 10^9 \) years.

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