QJEEM24Ph1f26: JEE Main 2024 Physics - Class 12 JEEM24Ph1f26 Solution: A star has helium composition. It starts to convert three into…

To determine the rate of converting ${ }^4 \mathrm{He}$ to ${ }^{12} \mathrm{C}$ , we need to calculate the energy released per reaction and use the given power production of the star to find the rate of reactions. The relevant nuclear reaction is:

${ }^4 \mathrm{He} + { }^4 \mathrm{He} + { }^4 \mathrm{He} \rightarrow { }^{12} \mathrm{C} + \mathrm{Q}$

The masses involved in the reaction are given:

• Mass of ${ }^4 \mathrm{He} = 4.0026 \, \mathrm{u}$
• Mass of ${ }^{12} \mathrm{C} = 12 \, \mathrm{u}$

First, we calculate the mass defect (difference between the mass of reactants and products) which will give us the energy released in each reaction:

Mass of reactants: $3 \times 4.0026 \, \mathrm{u} = 12.0078 \, \mathrm{u}$

Mass of product: $12 \, \mathrm{u}$

Mass defect: $12.0078 \, \mathrm{u} - 12 \, \mathrm{u} = 0.0078 \, \mathrm{u}$

We use Einstein's mass-energy equivalence principle, $E = mc^2$ , to find the energy released per reaction. The conversion factor between atomic mass units and energy is $1 \, \mathrm{u} = 931.5 \, \mathrm{MeV}$ .

Energy released per reaction: $0.0078 \, \mathrm{u} \times 931.5 \, \mathrm{MeV/u} = 7.2627 \, \mathrm{MeV}$

We convert this energy into joules. $1 \, \mathrm{MeV} = 1.60218 \times 10^{-13} \, \mathrm{J}$ :

Energy per reaction: $7.2627 \, \mathrm{MeV} \times 1.60218 \times 10^{-13} \, \mathrm{J/MeV} = 1.163 \times 10^{-12} \, \mathrm{J}$

The power generated by the star is given as $5.808 \times 10^{30} \, \mathrm{W}$ . The rate of the reaction is the power divided by the energy per reaction:

$\text{Rate of reactions} = \frac{\text{Power}}{\text{Energy per reaction}}$

$\text{Rate} = \frac{5.808 \times 10^{30} \, \mathrm{W}}{1.163 \times 10^{-12} \, \mathrm{J}}$

$\text{Rate} = 4.99 \times 10^{42} \, \mathrm{s^{-1}} \simeq 5 \times 10^{42} \mathrm{~s}^{-1}$

Thus, the rate of converting ${ }^4 \mathrm{He}$ to ${ }^{12} \mathrm{C}$ is:

$\mathrm{n} = 5$