Atoms & Nuclei

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190 of 239

Question 190 of 239

A star has 100%100 \% helium composition. It starts to convert three 4He{ }^4 \mathrm{He} into one 12C{ }^{12} \mathrm{C} via triple alpha process as 4He+4He+4He→12C+Q{ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}. The mass of the star is 2.0×1032 kg2.0 \times 10^{32} \mathrm{~kg} and it generates energy at the rate of 5.808×1030 W5.808 \times 10^{30} \mathrm{~W}. The rate of converting these 4He{ }^4 \mathrm{He} to 12C{ }^{12} \mathrm{C} is n×1042 s−1\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}, where n\mathrm{n} is _________
[ Take, mass of
4He=4.0026u{ }^4 \mathrm{He}=4.0026 \mathrm{u}, mass of 12C=12u{ }^{12} \mathrm{C}=12 \mathrm{u} ]

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Answer: 15

4He+4He+4He→12C+Q{ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}
power generated
=NtQ=\frac{\mathrm{N}}{\mathrm{t}} \mathrm{Q}
where,
N→\mathrm{N} \rightarrow No. of reaction /sec/ \mathrm{sec}.
Q=(3 mHe−mC)C2Q=(3×4.0026−12)(3×108)2Q=7.266MeV\begin{aligned} & \mathrm{Q}=\left(3 \mathrm{~m}_{\mathrm{He}}-\mathrm{m}_{\mathrm{C}}\right) \mathrm{C}^2 \\ & \mathrm{Q}=(3 \times 4.0026-12)\left(3 \times 10^8\right)^2 \\ & \mathrm{Q}=7.266 \mathrm{MeV}\end{aligned}
Nt= power Q=5.808×10307.266×106×1.6×10−19Nt=5×1042\begin{aligned} & \frac{\mathrm{N}}{\mathrm{t}}=\frac{\text { power }}{\mathrm{Q}}=\frac{5.808 \times 10^{30}}{7.266 \times 10^6 \times 1.6 \times 10^{-19}} \\ & \frac{\mathrm{N}}{\mathrm{t}}=5 \times 10^{42}\end{aligned}
rate of conversion of
4He{ }^4 \mathrm{He} into 12C=15×1042{ }^{12} \mathrm{C}=15 \times 10^{42} Hence, n=15\mathrm{n}=15

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