A star has $100 \%$ helium composition. It starts to convert three ${ }^4 \mathrm{He}$ into one ${ }^{12} \mathrm{C}$ via triple alpha process as $${ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}$$. The mass of the star is $2.0 \times 10^{32} \mathrm{~kg}$ and it generates energy at the rate of $5.808 \times 10^{30} \mathrm{~W}$. The rate of converting these ${ }^4 \mathrm{He}$ to ${ }^{12} \mathrm{C}$ is $\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}$, where $\mathrm{n}$ is _________. [ Take, mass of ${ }^4 \mathrm{He}=4.0026 \mathrm{u}$, mass of ${ }^{12} \mathrm{C}=12 \mathrm{u}$]

<p>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:</p> <p> <p>$${ }^4 \mathrm{He} + { }^4 \mathrm{He} + { }^4 \mathrm{He} \rightarrow { }^{12} \mathrm{C} + \mathrm{Q}$$</p> </p> <p>The masses involved in the reaction are given: <ul> <li>Mass of ${ }^4 \mathrm{He} = 4.0026 \, \mathrm{u}$</li> <li>Mass of ${ }^{12} \mathrm{C} = 12 \, \mathrm{u}$</li> </ul> </p> <p>First, we calculate the mass defect (difference between the mass of reactants and products) which will give us the energy released in each reaction:</p> <p> <p>Mass of reactants: $3 \times 4.0026 \, \mathrm{u} = 12.0078 \, \mathrm{u}$</p> </p> <p> <p>Mass of product: $12 \, \mathrm{u}$</p> </p> <p> <p>Mass defect: $12.0078 \, \mathrm{u} - 12 \, \mathrm{u} = 0.0078 \, \mathrm{u}$</p> </p> <p>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 $1 \, \mathrm{u} = 931.5 \, \mathrm{MeV}$.</p> <p> <p>Energy released per reaction: $0.0078 \, \mathrm{u} \times 931.5 \, \mathrm{MeV/u} = 7.2627 \, \mathrm{MeV}$</p> </p> <p>We convert this energy into joules. $1 \, \mathrm{MeV} = 1.60218 \times 10^{-13} \, \mathrm{J}$:</p> <p> <p>Energy per reaction: $$7.2627 \, \mathrm{MeV} \times 1.60218 \times 10^{-13} \, \mathrm{J/MeV} = 1.163 \times 10^{-12} \, \mathrm{J}$$</p> </p> <p>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:</p> <p> <p>$\text{Rate of reactions} = \frac{\text{Power}}{\text{Energy per reaction}}$</p> </p> <p> <p>$$ \text{Rate} = \frac{5.808 \times 10^{30} \, \mathrm{W}}{1.163 \times 10^{-12} \, \mathrm{J}}$$</p> </p> <p> <p>$$ \text{Rate} = 4.99 \times 10^{42} \, \mathrm{s^{-1}} \simeq 5 \times 10^{42} \mathrm{~s}^{-1}$$</p> </p> <p>Thus, the rate of converting ${ }^4 \mathrm{He}$ to ${ }^{12} \mathrm{C}$ is: <p>$\mathrm{n} = 5$</p></p>