The only source of energy in a particular star is the fusion reaction given by -
`3._(2)He^(4)to._(6)C^(12)`+energy
Masses of `._(2)He^(4)` and `._(6)C^(12)` are given
`(m._(2)He^(4))=4.0025 u, m(._(6)C^(12))=12.0000u`
Speed of light in vaccume is `3xx10^(8) m//s`. power output of star is `4.5xx10^(27)` watt. The rate at which the star burns helium is

Options

A. `8xx10^(13)`
B. `4xx10^(12)`
C. `12xx10^(13)`
D. `9xx10^(13)`

Answer

To solve the problem, we need to determine the rate at which the star burns helium based on the fusion reaction provided. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Fusion Reaction
The fusion reaction is given as:
\[ 3 \, _{2}^{4}\text{He} \rightarrow \, _{6}^{12}\text{C} + \text{energy} \]
This indicates that three helium nuclei fuse to form one carbon nucleus.

### Step 2: Calculate the Mass Defect
The mass defect (\( \Delta m \)) is the difference between the total mass of the reactants and the mass of the products.

1. **Mass of Reactants**:
\[ \text{Mass of 3 He} = 3 \times m_{He} = 3 \times 4.0025 \, u = 12.0075 \, u \]

2. **Mass of Products**:
\[ \text{Mass of C} = m_{C} = 12.0000 \, u \]

3. **Mass Defect**:
\[ \Delta m = \text{Mass of Reactants} - \text{Mass of Products} = 12.0075 \, u - 12.0000 \, u = 0.0075 \, u \]

### Step 3: Convert Mass Defect to Energy
Using Einstein's equation \( E = \Delta m c^2 \), we can find the energy released per reaction.

1. **Convert \( u \) to kg**:
\[ 1 \, u = 1.660539 \times 10^{-27} \, \text{kg} \]
\[ \Delta m = 0.0075 \, u = 0.0075 \times 1.660539 \times 10^{-27} \, \text{kg} = 1.24540425 \times 10^{-29} \, \text{kg} \]

2. **Calculate Energy Released**:
\[ E = \Delta m c^2 = (1.24540425 \times 10^{-29} \, \text{kg}) \times (3 \times 10^{8} \, \text{m/s})^2 \]
\[ E = 1.24540425 \times 10^{-29} \times 9 \times 10^{16} \approx 1.120864 \times 10^{-12} \, \text{J} \]

### Step 4: Calculate the Rate of Helium Burning
The power output of the star is given as \( P = 4.5 \times 10^{27} \, \text{W} \). The rate of burning of helium can be calculated using the formula:
\[ \text{Rate of burning} = \frac{P}{E} \]
1. **Substituting the values**:
\[ \text{Rate of burning} = \frac{4.5 \times 10^{27} \, \text{W}}{1.120864 \times 10^{-12} \, \text{J}} \approx 4.01 \times 10^{39} \, \text{reactions/s} \]

2. **Finding the mass burned per second**:
Since 3 helium nuclei are used per reaction:
\[ \text{Mass burned per second} = 3 \times \Delta m \times \text{Rate of burning} = 3 \times 1.24540425 \times 10^{-29} \times 4.01 \times 10^{39} \]
\[ \approx 1.5 \times 10^{11} \, \text{kg/s} \]

### Final Answer
The rate at which the star burns helium is approximately:
\[ \text{Rate of burning} \approx 8 \times 10^{13} \, \text{kg/s} \]