Uranium ores on the earth at the present time typically have a composition consisting of 99.3% of the isotope `_92U^238` and 0.7% of the isotope `_92U^235`. The half-lives of these isotopes are `4.47xx10^9yr` and `7.04xx10^8yr`, respectively. If these isotopes were equally abundant when the earth was formed, estimate the age of the earth.

Explanation

To estimate the age of the Earth using the isotopes of uranium, we can follow these steps:

### Step 1: Define Initial Conditions
Let \( N_0 \) be the initial number of atoms of each isotope of uranium when the Earth was formed.

### Step 2: Define Current Conditions
At present, the number of atoms of uranium-238 (\( _{92}^{238}U \)) is \( N_1 \) and the number of atoms of uranium-235 (\( _{92}^{235}U \)) is \( N_2 \). Given the current composition:
- \( N_1 = 99.3\% \) of total uranium
- \( N_2 = 0.7\% \) of total uranium

### Step 3: Express the Number of Atoms in Terms of Time
The number of atoms of each isotope at time \( t \) can be expressed using the decay formula:
- For uranium-238:
\[ N_1 = N_0 e^{-\lambda_1 t} \]
- For uranium-235:
\[ N_2 = N_0 e^{-\lambda_2 t} \]

### Step 4: Set Up the Ratio of Current Atoms
The ratio of the current number of atoms of uranium-238 to uranium-235 is:
\[ \frac{N_1}{N_2} = \frac{99.3}{0.7} \]
Substituting the expressions from Step 3, we have:
\[ \frac{N_0 e^{-\lambda_1 t}}{N_0 e^{-\lambda_2 t}} = \frac{99.3}{0.7} \]
This simplifies to:
\[ \frac{e^{-\lambda_1 t}}{e^{-\lambda_2 t}} = \frac{99.3}{0.7} \]
Which can be rewritten as:
\[ e^{(\lambda_2 - \lambda_1) t} = \frac{99.3}{0.7} \]

### Step 5: Calculate Decay Constants
The decay constants \( \lambda_1 \) and \( \lambda_2 \) can be calculated using the half-lives:
\[ \lambda_1 = \frac{0.693}{4.47 \times 10^9 \text{ years}} \]
\[ \lambda_2 = \frac{0.693}{7.04 \times 10^8 \text{ years}} \]

### Step 6: Substitute and Solve for \( t \)
Now, we can substitute \( \lambda_1 \) and \( \lambda_2 \) into the equation from Step 4:
\[ (\lambda_2 - \lambda_1) t = \ln\left(\frac{99.3}{0.7}\right) \]
Thus,
\[ t = \frac{\ln\left(\frac{99.3}{0.7}\right)}{\lambda_2 - \lambda_1} \]

### Step 7: Calculate \( t \)
Now substituting the values of \( \lambda_1 \) and \( \lambda_2 \):
1. Calculate \( \lambda_1 \):
\[ \lambda_1 = \frac{0.693}{4.47 \times 10^9} \approx 1.55 \times 10^{-10} \text{ year}^{-1} \]
2. Calculate \( \lambda_2 \):
\[ \lambda_2 = \frac{0.693}{7.04 \times 10^8} \approx 9.83 \times 10^{-9} \text{ year}^{-1} \]
3. Calculate \( \lambda_2 - \lambda_1 \):
\[ \lambda_2 - \lambda_1 \approx 9.83 \times 10^{-9} - 1.55 \times 10^{-10} \approx 9.67 \times 10^{-9} \text{ year}^{-1} \]
4. Calculate \( \ln\left(\frac{99.3}{0.7}\right) \):
\[ \ln\left(\frac{99.3}{0.7}\right) \approx \ln(142.857) \approx 4.966 \]
5. Finally, calculate \( t \):
\[ t \approx \frac{4.966}{9.67 \times 10^{-9}} \approx 5.14 \times 10^8 \text{ years} \]

### Final Result
The estimated age of the Earth is approximately \( 5.14 \times 10^9 \) years.