Question

Verify that the total number of nucleons, total charge, and electron family number are conserved for each of the fusion reactions in the carbon cycle given in the above problem. (List the value of each of the conserved quantities before and after each of the reactions.)

Short answer

The number of nucleons and total charge for each reaction is –

\(\begin{array}{*{20}{l}}{^{12}C{ + ^1}H}&{{ \to ^{13}}N + \gamma ,{\rm{ }}C = 7,N = 13}\\{^{13}N}&{{ \to ^{13}}C + {e^ + } + {v_e},{\rm{ }}C = 7,N = 13}\\{^{13}C{ + ^1}H}&{{ \to ^{14}}N + \gamma ,{\rm{ }}C = 7,N = 14}\\{^{14}N{ + ^1}H}&{{ \to ^{15}}O + \gamma ,{\rm{ }}C = 8,N = 15}\\{^{15}O}&{{ \to ^{15}}N + {e^ + } + {v_e},{\rm{ }}C = 8,N = 15}\\{^{15}N{ + ^1}H}&{{ \to ^{12}}C{ + ^4}He.{\rm{ }}C = 8,N = 16}\end{array}\)

Only for reaction two and five the lepton family electron is conserved with position and electron neutrino. All the quantities are conserved in each reaction.

Step-by-step solution

Concept Introduction

The power kept in the centre of an atom is known as nuclear energy. All stuff in the cosmos is made up of tiny particles called atoms. The centre of an atom's nucleus is typically where most of its mass is concentrated. Neutrons and protons are the two subatomic particles that make up the nucleus. Atomic bonds that hold them together carry a lot of energy.

Information Provided

• Reactions from the carbon cycle are: \(\begin{array}{*{20}{l}}{^{12}C{ + ^1}H}&{{ \to ^{13}}N + \gamma ,}\\{^{13}N}&{{ \to ^{13}}C + {e^ + } + {v_e},}\\{^{13}C{ + ^1}H}&{{ \to ^{14}}N + \gamma ,}\\{^{14}N{ + ^1}H}&{{ \to ^{15}}O + \gamma ,}\\{^{15}O}&{{ \to ^{15}}N + {e^ + } + {v_e},}\\{^{15}N{ + ^1}H}&{{ \to ^{12}}C{ + ^4}He.}\end{array}\).

Nucleon, Charge and Electron family for the first reaction

The lepton family of electrons, would only be valued for reaction number two and reaction number five. In reaction two-electron family number is conserved with position and electron neutrino, and the same case is worth forreaction five.

For the first reaction conservation of charge and nucleons is following –

\(^{12}C{ + ^1}H{ \to ^{13}}N + \gamma \)

	Nucleons	Charges
\(^{12}C\)	\(12\)	\(6\)
\(^1H\)	\(1\)	\(1\)
\(^{13}N\)	\(13\)	\(7\)
\(\gamma \)	\(0\)	\(0\)

Adding the numbers on the left and right we can see for nucleons we have \(12 + 1 = 13\) and for charges, there is \(6 + 1 = 7\).

Therefore, the total nucleons and total charge is \(13\) and \(7\)respectively. The lepton family of electrons is not conserved in this reaction.

Nucleon, Charge and Electron family for the second reaction

For the second reaction conservation of charge and nucleons is following –

\(^{13}N{ \to ^{13}}C + {e^ + } + {v_e}\)

	Nucleons	Charges
\(^{13}C\)	\(13\)	\(6\)
\({e^ + }\)	\(0\)	\(1\)
\(^{13}N\)	\(13\)	\(7\)
\({v_e}\)	\(0\)	\(0\)

It can be seen that there are\(13\) nucleons on the left and right, and seven positive charges on the left and right.

Therefore, the total nucleons and total charge is \(13\) and \(7\)respectively. The lepton family of electrons is conservedwith position and electron neutrino in this reaction.

Nucleon, Charge and Electron family for the third reaction

For the third reaction conservation of charge and nucleons is following –

\(^{{\rm{13}}}{\rm{C}}{{\rm{ + }}^{\rm{1}}}{\rm{H}}{ \to ^{{\rm{14}}}}{\rm{N + }}\gamma \)

	Nucleons	Charges
\(^{13}C\)	\(13\)	\(6\)
\(^1H\)	\(1\)	\(1\)
\(^{14}N\)	\(13\)	\(7\)

It can be seen that there are\(14\) nucleons on the left and right, and seven positive charges on the left and right.

Therefore, the total nucleons and total charge is \(14\) and \(7\)respectively. The lepton family of electrons is not conserved in this reaction.

Nucleon, Charge and Electron family for the fourth reaction

For the fourth reaction conservation of charge and nucleons is following –

\(^{14}N{ + ^1}H{ \to ^{15}}O + \gamma \)

	Nucleons	Charges
\(^{15}O\)	\(15\)	\(8\)
\(^1H\)	\(1\)	\(1\)
\(^{14}N\)	\(13\)	\(7\)

It can be seen that there are\(15\) nucleons on the left and right, and eight positive charges on the left and right.

Therefore, the total nucleons and total charge is \(15\) and \(8\)respectively. The lepton family of electrons is not conserved in this reaction.

Nucleon, Charge and Electron family for the fifth reaction

For the fifth reaction conservation of charge and nucleons is following –

\(^{15}O{ \to ^{15}}N + {e^ + } + {v_e}\)

	Nucleons	Charges
\(^{15}O\)	\(15\)	\(8\)
\({e^ + }\)	\(0\)	\(1\)
\(^{15}N\)	\(13\)	\(7\)
\({v_e}\)	\(0\)	\(0\)

It can be seen that there are\(15\) nucleons on the left and right, and eight positive charges on the left and right.

Therefore, the total nucleons and total charge is \(15\) and \(8\)respectively. The lepton family of electrons is conserved with position and electron neutrino in this reaction.

Nucleon, Charge and Electron family for the sixth reaction

For the sixth reaction conservation of charge and nucleons is following –

\(^{15}N{ + ^1}H{ \to ^{12}}C{ + ^4}He\)

	Nucleons	Charges
\(^{12}C\)	\(12\)	\(6\)
\(^1H\)	\(1\)	\(1\)
\(^{15}N\)	\(15\)	\(7\)
\(^4He\)	\(4\)	\(2\)

It can be seen that there are\(16\) nucleons on the left and right, and eight positive charges on the left and right.

Therefore, the total nucleons and total charge is \(16\) and \(8\)respectively. The lepton family of electrons is not conserved in this reaction.