The atomic mass of 14C14C is 14.00324214.003242 u. Show that the β−\$$beta^{-}$$ decay of 14C14C is energetically possible, and calculate the energy released in the decay.

Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 15

Chapter 43, Problem 15

The atomic mass of $14 C$ is $14.003242$ u. Show that the $β^{-}$ decay of $14 C$ is energetically possible, and calculate the energy released in the decay.

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Determine the nuclear reaction for the beta-minus (β⁻) decay of 14C. In β⁻ decay, a neutron in the nucleus converts into a proton, emitting an electron (β⁻ particle) and an antineutrino. The reaction is:

C^{14} → N^{14} + β⁻ + ν̅.

Look up the atomic mass of the daughter nucleus, nitrogen-14 (14N). The atomic mass of 14N is 14.003074 u. Note that the atomic masses include the mass of electrons, so we do not need to separately account for the emitted β⁻ particle's mass.

Calculate the mass difference (Δm) between the parent nucleus (14C) and the daughter nucleus (14N). Use the formula:

Δm = m_{C 14} - m_{N 14}

. Substitute the given values:

= 14.003242 \, \(\text{u}\) - 14.003074 \, \(\text{u}\)

Convert the mass difference (Δm) into energy using Einstein's mass-energy equivalence formula:

E = Δm \, c^{2}

. Here,

c

is the speed of light. To convert atomic mass units (u) to energy, use the conversion factor:

1 \, \(\text{u}\) = 931.5 \, \(\text{MeV/c}\)^2

. Thus,

= Δm \, imes \, 931.5 \, \(\text{MeV}\)

Interpret the result: If the calculated energy (E) is positive, the β⁻ decay of 14C is energetically possible. The positive energy value represents the energy released during the decay process.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Beta Decay

Beta decay is a type of radioactive decay in which a beta particle (an electron or positron) is emitted from an atomic nucleus. In the case of carbon-14 (14C), it undergoes beta decay to transform into nitrogen-14 (14N) by converting a neutron into a proton, which increases the atomic number by one while keeping the mass number the same.

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In nuclear reactions, the difference in mass between the reactants and products can be converted into energy, which is released during processes like beta decay. This principle is crucial for calculating the energy released in the decay of 14C.

Q-Value of a Decay

The Q-value of a decay is the amount of energy released during a nuclear reaction, calculated as the difference in mass between the initial and final states, multiplied by c². For the beta decay of 14C, the Q-value can be determined by comparing the atomic masses of 14C and 14N, along with the emitted beta particle, allowing us to quantify the energy released in the process.

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The atomic mass of 14C14C is 14.00324214.003242 u. Show that the β−\(\beta\)^{-} decay of 14C14C is energetically possible, and calculate the energy released in the decay.

Verified video answer for a similar problem:

Key Concepts

Beta Decay

Mass-Energy Equivalence

Q-Value of a Decay

The atomic mass of $14 C$ is $14.003242$ u. Show that the $β^{-}$ decay of $14 C$ is energetically possible, and calculate the energy released in the decay.