What is the minimum energy that is required to break a nucleus of `""^(12)C` (of mass 11.996 71 u) into three nuclei of `""^(4)He` (of mass 4.001 51 u each)?

Answer

To calculate the minimum energy required to break a nucleus of \( ^{12}C \) into three nuclei of \( ^{4}He \), we will follow these steps:

### Step 1: Identify the masses involved
- The mass of the \( ^{12}C \) nucleus is given as \( 11.99671 \, u \).
- The mass of a \( ^{4}He \) nucleus is given as \( 4.00151 \, u \).

### Step 2: Calculate the total mass of the final products
Since we are breaking \( ^{12}C \) into three \( ^{4}He \) nuclei, we need to calculate the total mass of the three \( ^{4}He \) nuclei:
\[ \text{Total mass of final products} = 3 \times \text{mass of } ^{4}He = 3 \times 4.00151 \, u = 12.00453 \, u \]

### Step 3: Calculate the mass defect (\( \Delta m \))
The mass defect is the difference between the initial mass and the total final mass:
\[ \Delta m = \text{mass of } ^{12}C - \text{total mass of final products} = 11.99671 \, u - 12.00453 \, u = -0.00782 \, u \]

### Step 4: Convert mass defect to energy
To find the energy equivalent of the mass defect, we use Einstein's equation \( E = \Delta m c^2 \). In nuclear physics, we often convert mass in atomic mass units (u) to energy in MeV using the conversion factor \( 1 \, u = 931.5 \, \text{MeV}/c^2 \):
\[ E = \Delta m \times 931.5 \, \text{MeV}/u = -0.00782 \, u \times 931.5 \, \text{MeV}/u \]
Calculating this gives:
\[ E = -7.28 \, \text{MeV} \]

### Step 5: Interpret the result
The negative sign indicates that energy must be supplied to break the nucleus apart. Thus, the minimum energy required to break the \( ^{12}C \) nucleus into three \( ^{4}He \) nuclei is:
\[ E = 7.28 \, \text{MeV} \]

### Final Answer
The minimum energy required to break a nucleus of \( ^{12}C \) into three nuclei of \( ^{4}He \) is \( 7.28 \, \text{MeV} \).

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