Natural uranium is a mixture of three isotopes `_92^234U`, `_92^234U`, `_92^235U` and `_92^238U` with mass percentage 0.01%, 0.71% and 99.28% respectively. The half-life of three isotopes are `2.5xx10^5yr`, `7.1xx10^8yr` and `4.5xx10^9yr` respectively.
Determine the share of radioactivity of each isotope into the total activity of the natural uranium.

Explanation

To determine the share of radioactivity of each isotope of natural uranium, we will follow these steps:

### Step 1: Define the isotopes and their properties
We have three isotopes of uranium:
1. Uranium-234 (`_92^234U`): Mass percentage = 0.01%, Half-life = `2.5 x 10^5` years
2. Uranium-235 (`_92^235U`): Mass percentage = 0.71%, Half-life = `7.1 x 10^8` years
3. Uranium-238 (`_92^238U`): Mass percentage = 99.28%, Half-life = `4.5 x 10^9` years

### Step 2: Calculate the number of atoms for each isotope
Let the total mass of natural uranium be `M`. The number of atoms for each isotope can be calculated using the formula:
\[ N = \frac{m}{M} \times \frac{N_A}{A} \]
where:
- `m` is the mass of the isotope,
- `M` is the total mass of natural uranium,
- `N_A` is Avogadro's number (`6.022 x 10^{23} mol^{-1}`),
- `A` is the atomic mass of the isotope.

For each isotope:
- For `_92^234U`:
\[ N_A = \frac{0.01}{100} \times \frac{6.022 \times 10^{23}}{234} \]
- For `_92^235U`:
\[ N_B = \frac{0.71}{100} \times \frac{6.022 \times 10^{23}}{235} \]
- For `_92^238U`:
\[ N_C = \frac{99.28}{100} \times \frac{6.022 \times 10^{23}}{238} \]

### Step 3: Calculate the decay constants for each isotope
The decay constant `λ` is given by:
\[ λ = \frac{0.693}{T_{1/2}} \]
where `T_{1/2}` is the half-life of the isotope.

Calculating for each isotope:
- For `_92^234U`:
\[ λ_A = \frac{0.693}{2.5 \times 10^5} \]
- For `_92^235U`:
\[ λ_B = \frac{0.693}{7.1 \times 10^8} \]
- For `_92^238U`:
\[ λ_C = \frac{0.693}{4.5 \times 10^9} \]

### Step 4: Calculate the activity for each isotope
The activity `R` is given by:
\[ R = λ \times N \]
Calculating for each isotope:
- For `_92^234U`:
\[ R_A = λ_A \times N_A \]
- For `_92^235U`:
\[ R_B = λ_B \times N_B \]
- For `_92^238U`:
\[ R_C = λ_C \times N_C \]

### Step 5: Calculate the total activity
The total activity `E` is the sum of the activities of all isotopes:
\[ E = R_A + R_B + R_C \]

### Step 6: Calculate the share of radioactivity for each isotope
The share of radioactivity for each isotope can be calculated as:
- For `_92^234U`:
\[ \text{Share of } R_A = \frac{R_A}{E} \times 100 \]
- For `_92^235U`:
\[ \text{Share of } R_B = \frac{R_B}{E} \times 100 \]
- For `_92^238U`:
\[ \text{Share of } R_C = \frac{R_C}{E} \times 100 \]

### Final Calculation
Perform the calculations using the values obtained in the previous steps to find the percentage shares of radioactivity for each isotope.

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