An explosion of atomic bomb release an energy of `7.6 xx 10^13`J. If 200 Mev energy is released on fission of one `.^235U` atom calculate (i) the number of uranium atoms undergoing fission, (ii) the mass of uranium used in the bomb.

Options

A. `4.375xx10^24`, 926.66 g
B. `2.375xx10^24` , 826.66 g
C. `3.375xx10^24` , 926.66 g
D. `2.375xx10^24` , 926.66 g

Answer

To solve the problem step by step, we will follow these calculations:

### Given Data:
- Energy released by the atomic bomb, \( E = 7.6 \times 10^{13} \) J
- Energy released per fission of one \( ^{235}U \) atom, \( E_{fission} = 200 \) MeV

### Step 1: Convert Energy from MeV to Joules
1. **Convert 200 MeV to Joules**:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
\[ 200 \text{ MeV} = 200 \times 10^6 \text{ eV} = 200 \times 10^6 \times 1.6 \times 10^{-19} \text{ J} \]
\[ = 3.2 \times 10^{-11} \text{ J} \]

### Step 2: Calculate the Number of Uranium Atoms Undergoing Fission
2. **Calculate the number of uranium atoms**:
Using the formula:
\[ N = \frac{E}{E_{fission}} \]
where \( N \) is the number of uranium atoms.
\[ N = \frac{7.6 \times 10^{13} \text{ J}}{3.2 \times 10^{-11} \text{ J}} \]
\[ N = 2.375 \times 10^{24} \text{ atoms} \]

### Step 3: Calculate the Mass of Uranium Used
3. **Calculate the mass of uranium**:
- The mass of one \( ^{235}U \) atom is approximately \( 3.95 \times 10^{-22} \) grams.
- The total mass of uranium used can be calculated as:
\[ \text{Mass} = N \times \text{mass of one atom} \]
\[ \text{Mass} = 2.375 \times 10^{24} \text{ atoms} \times 3.95 \times 10^{-22} \text{ g} \]
\[ \text{Mass} \approx 927.625 \text{ g} \approx 927 \text{ g} \]

### Final Answers:
(i) The number of uranium atoms undergoing fission is approximately \( 2.375 \times 10^{24} \) atoms.

(ii) The mass of uranium used in the bomb is approximately \( 927 \) grams.