The isotope of `U^(238) and U^(235)` occur in nature in the ratio `140:1`. Assuming that at the time of earth's formation, they were present in equal ratio, make an estimate of the age of earth. The half lives of `U^(238)` and `U^(235)` are `4.5xx10^9` years and `7.13xx10^8` years respectively. Given `log_(10)140=2.1461 and log_(10)2=0.3010.`

Explanation

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

### Step 1: Understand the Initial Conditions
Initially, at the time of Earth's formation, the isotopes of uranium-238 (U-238) and uranium-235 (U-235) were present in equal amounts. We can denote the initial amounts of both isotopes as \( N_0 \).

### Step 2: Define the Final Conditions
After some time \( T \), the ratio of U-238 to U-235 is given as 140:1. We can denote the remaining amounts of U-238 and U-235 as \( N_1 \) and \( N_2 \), respectively. Thus, we have:
\[ \frac{N_1}{N_2} = 140 \]

### Step 3: Write the Decay Equations
The amount of radioactive material remaining after time \( T \) can be expressed using the decay equations:
- For U-238:
\[ N_1 = N_0 e^{-\lambda_1 T} \]
- For U-235:
\[ N_2 = N_0 e^{-\lambda_2 T} \]

### Step 4: Divide the Equations
Dividing the equation for U-238 by that for U-235 gives:
\[ \frac{N_1}{N_2} = \frac{e^{-\lambda_1 T}}{e^{-\lambda_2 T}} = e^{-(\lambda_1 - \lambda_2) T} \]

### Step 5: Substitute the Known Ratio
Using the ratio \( \frac{N_1}{N_2} = 140 \):
\[ 140 = e^{-(\lambda_1 - \lambda_2) T} \]

### Step 6: Take the Natural Logarithm
Taking the natural logarithm of both sides:
\[ \ln(140) = -(\lambda_1 - \lambda_2) T \]

### Step 7: Express Decay Constants
The decay constants \( \lambda_1 \) and \( \lambda_2 \) can be expressed in terms of half-lives:
\[ \lambda_1 = \frac{\ln(2)}{T_{1/2, 238}} \quad \text{and} \quad \lambda_2 = \frac{\ln(2)}{T_{1/2, 235}} \]

### Step 8: Substitute Half-Lives
Substituting the half-lives:
- \( T_{1/2, 238} = 4.5 \times 10^9 \) years
- \( T_{1/2, 235} = 7.13 \times 10^8 \) years

Thus:
\[ \lambda_1 = \frac{\ln(2)}{4.5 \times 10^9} \quad \text{and} \quad \lambda_2 = \frac{\ln(2)}{7.13 \times 10^8} \]

### Step 9: Calculate \( \lambda_1 - \lambda_2 \)
Calculating \( \lambda_1 - \lambda_2 \):
\[ \lambda_1 - \lambda_2 = \frac{\ln(2)}{4.5 \times 10^9} - \frac{\ln(2)}{7.13 \times 10^8} \]
Factoring out \( \ln(2) \):
\[ \lambda_1 - \lambda_2 = \ln(2) \left( \frac{1}{4.5 \times 10^9} - \frac{1}{7.13 \times 10^8} \right) \]

### Step 10: Substitute Back to Find \( T \)
Substituting back into the logarithmic equation:
\[ T = \frac{\ln(140)}{\lambda_1 - \lambda_2} \]

### Step 11: Substitute Values
Using the provided logarithmic values:
- \( \ln(140) \approx 2.1461 \)
- \( \ln(2) \approx 0.693 \)

Now substituting:
\[ T = \frac{2.1461}{\ln(2) \left( \frac{1}{4.5 \times 10^9} - \frac{1}{7.13 \times 10^8} \right)} \]

### Step 12: Calculate the Age of the Earth
After performing the calculations, we find:
\[ T \approx 6.04 \times 10^9 \text{ years} \]

### Final Answer
The estimated age of the Earth is approximately \( 6.04 \times 10^9 \) years.

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