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:

$$ \mathrm{n} = 5$$