The energy released by fission of one `U^(235)` atom is 200 MeV. Calculate the energy released in kWh, when one gram of uranium undergoes fission.

Options

A. `2.270 xx 10^(4) kWh`
B. `2.278 xx 10^4 kWh`
C. `2.268 xx 10^2 kWh`
D. `2.288 xx 10^4 kWh`

Answer

To solve the problem of calculating the energy released in kilowatt-hours when one gram of uranium-235 undergoes fission, we can follow these steps:

### Step 1: Calculate the number of atoms in 1 gram of uranium-235.
The number of atoms (n) can be calculated using Avogadro's number:

\[ n = \frac{N_A}{A} \]

Where:
- \( N_A = 6.022 \times 10^{23} \) atoms/mol (Avogadro's number)
- \( A = 235 \) g/mol (atomic mass of uranium-235)

Substituting the values:

\[ n = \frac{6.022 \times 10^{23}}{235} \approx 2.56 \times 10^{21} \text{ atoms} \]

### Step 2: Calculate the total energy released from the fission of these atoms.
Each uranium-235 atom releases 200 MeV during fission. We need to convert this energy into Joules:

\[ \text{Energy per atom} = 200 \text{ MeV} = 200 \times 10^6 \text{ eV} \]

To convert electron volts to Joules, we use the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \):

\[ \text{Energy per atom in Joules} = 200 \times 10^6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} \approx 3.2 \times 10^{-11} \text{ J} \]

Now, we can calculate the total energy released by all the atoms in 1 gram of uranium:

\[ \text{Total energy} = n \times \text{Energy per atom} \]

Substituting the values:

\[ \text{Total energy} = 2.56 \times 10^{21} \text{ atoms} \times 3.2 \times 10^{-11} \text{ J/atom} \approx 8.19 \times 10^{10} \text{ J} \]

### Step 3: Convert the total energy from Joules to kilowatt-hours.
To convert Joules to kilowatt-hours, we use the conversion factor \( 1 \text{ kWh} = 3.6 \times 10^6 \text{ J} \):

\[ \text{Energy in kWh} = \frac{\text{Total energy in J}}{3.6 \times 10^6 \text{ J/kWh}} \]

Substituting the total energy:

\[ \text{Energy in kWh} = \frac{8.19 \times 10^{10} \text{ J}}{3.6 \times 10^6 \text{ J/kWh}} \approx 2.28 \times 10^4 \text{ kWh} \]

### Final Answer:
The energy released when one gram of uranium-235 undergoes fission is approximately \( 2.28 \times 10^4 \text{ kWh} \).

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