A nuclear reaction that uses one nucleus of `""_(92)^(236)U` generates 170 Mev. How much energy is released when 5.0 kg of this isotope is udeD?

Options

A. `1.4x10^(10)J`
B. `3.5xx10^(14)J`
C. `6.9xx10^(13)J`
D. `8.9xx10^(14)J`

Answer

To solve the problem, we need to calculate the total energy released when 5.0 kg of the isotope \( _{92}^{236}U \) is used. The energy released from one nucleus of \( _{92}^{236}U \) is given as 170 MeV. Here’s a step-by-step solution:

### Step 1: Convert the mass of uranium to grams
Given mass of uranium = 5.0 kg

To convert kilograms to grams:
\[ 5.0 \, \text{kg} = 5.0 \times 1000 \, \text{g} = 5000 \, \text{g} \]

### Step 2: Calculate the number of moles of \( _{92}^{236}U \)
The molar mass of \( _{92}^{236}U \) is approximately 236 g/mol.

Using the formula for the number of moles:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
\[ \text{Number of moles} = \frac{5000 \, \text{g}}{236 \, \text{g/mol}} \approx 21.18 \, \text{moles} \]

### Step 3: Calculate the total energy released from one mole
The energy released from one nucleus is 170 MeV.

To find the energy released from one mole, we use Avogadro's number (\( N_A \approx 6.022 \times 10^{23} \)):
\[ \text{Energy from one mole} = 170 \, \text{MeV} \times N_A \]
\[ \text{Energy from one mole} = 170 \, \text{MeV} \times 6.022 \times 10^{23} \, \text{nuclei/mol} \]

### Step 4: Convert MeV to Joules
1 MeV = \( 1.6 \times 10^{-13} \) Joules. Thus,
\[ \text{Energy from one mole in Joules} = 170 \times 6.022 \times 10^{23} \times 1.6 \times 10^{-13} \, \text{J} \]

### Step 5: Calculate total energy for 21.18 moles
Now we can find the total energy released when using 21.18 moles:
\[ \text{Total energy} = \text{Energy from one mole} \times \text{Number of moles} \]
\[ \text{Total energy} = \left(170 \times 6.022 \times 10^{23} \times 1.6 \times 10^{-13}\right) \times 21.18 \]

### Step 6: Calculate the final energy value
Calculating the above expression gives:
\[ \text{Total energy} \approx 3.4698 \times 10^{14} \, \text{J} \]

### Conclusion
The total energy released when 5.0 kg of \( _{92}^{236}U \) is used is approximately \( 3.47 \times 10^{14} \, \text{J} \).

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