Question

In certain stars the carbon cycle is more effective than the proton–proton cycle in generating energy.This carbon cycle is

$$\begin{gathered} {}^{\mathbf{12}}\mathbf{C}\mathbf{+}{}^{\mathbf{1}}\mathbf{H}{\mathbf{\to }}^{\mathbf{13}}\mathbf{N}\mathbf{+}\mathbf{\gamma }\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{95}\mathbf{}\mathbf{MeV}\mathbf{,} \\ {}^{\mathbf{13}}\mathbf{N}{\mathbf{\to }}^{\mathbf{13}}\mathbf{C}\mathbf{+}{\mathbf{e}}^{\mathbf{+}}\mathbf{+}\mathbf{v}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{19}\mathbf{,} \\ {}^{\mathbf{13}}\mathbf{C}\mathbf{+}{}^{\mathbf{1}}\mathbf{H}{\mathbf{\to }}^{\mathbf{14}}\mathbf{N}\mathbf{+}\mathbf{\gamma }\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{3}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{55}\mathbf{,} \\ {}^{\mathbf{14}}\mathbf{C}\mathbf{+}{}^{\mathbf{1}}\mathbf{H}{\mathbf{\to }}^{\mathbf{15}}\mathbf{O}\mathbf{+}\mathbf{\gamma }\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{4}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{30}\mathbf{,}{ \\ }^{\mathbf{15}}\mathbf{O}\mathbf{}{\mathbf{\to }}^{\mathbf{15}}\mathbf{N}\mathbf{+}{\mathbf{e}}^{\mathbf{+}}\mathbf{+}\mathbf{v}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{5}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{73}\mathbf{,} \\ {}^{\mathbf{15}}\mathbf{C}\mathbf{+}{}^{\mathbf{1}}\mathbf{H}{\mathbf{\to }}^{\mathbf{12}}\mathbf{C}{\mathbf{+}}^{\mathbf{4}}\mathbf{He}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{Q}}_{\mathbf{6}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{97} \end{gathered}$$

(a) Show that this cycle is exactly equivalent in its overall effects to the proton–proton cycle of Fig. 43-11. (b) Verify that the two cycles, as expected, have the same Q value.

Short answer

(a) The carbon cycle is exactly equivalent to proton-proton cycle.

(b) It is verified that the two cycles have the same Q value.

Step-by-step solution

Describe the expression for energy

The expression for energy is given by,

$\mathit{Q}\mathbf{=}\mathbf{-}\mathbf{∆}\mathit{m}{\mathit{c}}^{\mathbf{2}}$

Here, is the energy release in a reaction,$\mathbf{∆}\mathit{m}$ is the mass difference between the parent nuclei and the daughter nuclei, and c is the velocity of light.

Show that carbon-carbon cycle is exactly equivalent to the proton–proton cycle(a)

If can be carefully observed thatin the carbon-carbon cycle, the products are two positrons$\left(2e^{+}\right)$ , two neutrinos$\left(2v\right)$ , and one helium nucleus$\left({}^{4}He\right)$ .

The same products are also obtained in the proton-proton cycle.

Therefore, the carbon cycle is exactly equivalent to proton-proton cycle.

Verify that the two cycles have the same Q value(b)

The value of the energy is the sum of the Q values of the carbon cycle.

$$\begin{gathered} Q_c=\sum _{n=1}^6{Q}_n \\ =Q_1+Q_2+Q_3+Q_4+Q_5+Q_6 \\ =1.95Mev+1.19Mev+7.55Mev+7.30Mev+1/73Mev+4.97Mev \\ =24.7Mev \end{gathered}$$

The Q value is same for both cycles.

Therefore, it is verified that the two cycles have the same Q value.