Consider the reaction `""_(1)^(1)H+""_(62)^(150)Smto""_(61)^(147)Pm+""_(2)^(4)He` where the masses are `""_(1)^(1)H=1.007824u, ""_(2)^(4)He=4.002603u, ""_(62)^(150)Sm=149.917276u,""_(61)^(147)Pm=146.915108u`. How much energy is released in the reaction?

Options

A. 3.14MeV
B. 12.6MeV
C. 6.88MeV
D. 19.8MeV

Answer

To calculate the energy released in the nuclear reaction

\[ {}_{1}^{1}\text{H} + {}_{62}^{150}\text{Sm} \rightarrow {}_{61}^{147}\text{Pm} + {}_{2}^{4}\text{He}, \]

we will follow these steps:

### Step 1: Calculate the total mass of the reactants
The total mass of the reactants is the sum of the masses of hydrogen and samarium:

\[ \text{Mass of reactants} = m({}_{1}^{1}\text{H}) + m({}_{62}^{150}\text{Sm}) \]

Substituting the values:

\[ = 1.007824 \, \text{u} + 149.917276 \, \text{u} \]

Calculating this gives:

\[ = 150.925100 \, \text{u} \]

### Step 2: Calculate the total mass of the products
The total mass of the products is the sum of the masses of promethium and helium:

\[ \text{Mass of products} = m({}_{61}^{147}\text{Pm}) + m({}_{2}^{4}\text{He}) \]

Substituting the values:

\[ = 146.915108 \, \text{u} + 4.002603 \, \text{u} \]

Calculating this gives:

\[ = 150.917711 \, \text{u} \]

### Step 3: Calculate the mass defect
The mass defect (\(\Delta m\)) is the difference between the total mass of the reactants and the total mass of the products:

\[ \Delta m = \text{Mass of reactants} - \text{Mass of products} \]

Substituting the values:

\[ = 150.925100 \, \text{u} - 150.917711 \, \text{u} \]

Calculating this gives:

\[ = 0.007389 \, \text{u} \]

### Step 4: Convert mass defect to energy
To find the energy released, we use the equation \(E = \Delta m c^2\). The energy equivalent of 1 atomic mass unit (u) is approximately 931.5 MeV. Therefore, we can calculate the energy released:

\[ E = \Delta m \times 931.5 \, \text{MeV/u} \]

Substituting the mass defect:

\[ E = 0.007389 \, \text{u} \times 931.5 \, \text{MeV/u} \]

Calculating this gives:

\[ E \approx 6.88285 \, \text{MeV} \]

### Final Answer
The energy released in the reaction is approximately **6.88 MeV**.

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