Question

A STAR CONVERTS ALL ITS HYDROGEN INTO HELIUM. IT THEN CONVERTS HELIUM TO CARBON VIA THE REACTION:

3 ⁴He₂ TO¹²C₆ + 7.27 MeV

THE MASS OF THE STAR IS 5*10³²kg AND IT GENERATES ENERGY AT THE RATE OF 5*10³⁰ W. HOW LONG WILL IT TAKE TO CONVERT ALL THE HELIUM INTO CARBON AT THIS RATE?

Answer 1

$$\begin{gathered} 4gmofHe=6.023\times {10}^{23}Heatoms \\ \Rightarrow 5\times {10}^{32}kgofHe=\frac{6.023\times {10}^{23}}{4\times {10}^{-3}}\times 5\times {10}^{32}=7.52875\times {10}^{58}Heatoms \end{gathered}$$
Now total energy can be produced by the star,
$$\begin{gathered} =\frac{1}{3}\times 7.52875\times {10}^{58}\times 7.27MeV \\ =\frac{1}{3}\times 7.52875\times {10}^{58}\times 7.27\times 1.6\times {10}^{-13}J \\ =29.2\times {10}^{45}J \end{gathered}$$
So the star will live for,
$$\begin{gathered} =\frac{29.2\times {10}^{45}}{5\times {10}^{30}}sec \\ =5.84\times {10}^{15}sec \\ =\frac{5.84\times {10}^{15}}{60\times 60\times 24\times 365}years=185\times {10}^{6}years \end{gathered}$$