A star converts all its hydrogen to helium, achieving 100% helium composition. It then converts the helium to carbon via the reaction `._2He^4 + ._2He^4+ ._2He^4 to._6C^(12)+7.27MeV`
The mass of the star is `5.0xx10^(32)kg` and it generates energy at the rate of `5xx10^(30)` watt. How long will it take to convert all the helium to carbon at this rate?

Explanation

Number of nuclei in 4g of He = Avogadro number = `6 xx 10^(23)`
Number of nuclei in `3 xx 10^(32)` kg or `3 xx 10^(36)` g of helium
`=(6 xx 10^(23) xx 3 xx 10^(35))/4 = 4.5 xx 10^(58)`
The given fission reaction is
`""_(2)He^(4) + ""_(2)He^(4) + ""_(2)He^(4) to ""_(6)C^(12) + 7.27` MeV
Energy released in the fusion of `4.5 xx 10^(58)` nuclei
`=(7.27 xx 4.5 xx 10^(58) xx 1.6 xx 10^(-13))/3` J
`=1.75 xx 10^(46)` J
Also, given energy generated per second = 4 x 103° J Time taken to convert all helium nuclei into carbon
`=(1.75 xx 10^(46))/(4 xx 10^(30))s = 0.438 xx 10^(16) s`
`=(0.438 xx 10^(16))` S
`=1.4 xx 10^(8)` years.