Q value (nuclear science)

The conservation of energy, between the initial and final energy of a nuclear process ${\displaystyle {\text{( }}E_{\text{i}}=E_{\text{f}}{\text{ ),}}}$ enables the general definition of Q based on the mass–energy equivalence. For any radioactive particle decay, the kinetic energy difference will be given by:

${\displaystyle Q=K_{\text{f}}-K_{\text{i}}=(\,m_{\text{i}}-m_{\text{f}}\,)\,c^{2}~}$

where K denotes the kinetic energy of the mass m . A reaction with a positive Q value is exothermic, i.e. has a net release of energy, since the kinetic energy of the final state is greater than the kinetic energy of the initial state. A reaction with a negative Q value is endothermic, i.e. requires a net energy input, since the kinetic energy of the final state is less than the kinetic energy of the initial state.[1] Observe that a chemical reaction is exothermic when it has a negative enthalpy of reaction, in contrast a positive Q value in a nuclear reaction.

The Q value can also be expressed in terms of the Mass excess ${\displaystyle \Delta M}$ of the nuclear species as:

${\displaystyle Q=\Delta M_{\text{i}}-\Delta M_{\text{f}}~}$

Proof

The mass of a nucleus can be written as ${\displaystyle M=Au+\Delta M,~}$ where ${\displaystyle A~}$ is the mass number (sum of number of protons and neutrons) and ${\displaystyle u=^{12}\!\!C/12=931.494}$MeV/c${\displaystyle ^{2}~}$. Note that the count of nucleons is conserved in a nuclear reaction. Hence, ${\displaystyle A_{f}=A_{i}~}$ and ${\displaystyle Q=\Delta M_{\text{i}}-\Delta M_{\text{f}}~}$.

Chemical Q values are measurement in calorimetry. Exothermic chemical reactions tend to be more spontaneous and can emit light or heat, resulting in runaway feedback(i.e. explosions).

Q values are also featured in particle physics. For example, Sargent's rule states that weak reaction rates are proportional to Q⁵. The Q value is the kinetic energy released in the decay at rest. For neutron decay, some mass disappears as neutrons convert to a proton, electron and antineutrino:[2]

${\displaystyle Q=(m_{\text{n}}-m_{\text{p}}-m_{\mathrm {\overline {\nu }} }-m_{\text{e}})c^{2}=K_{\text{p}}+K_{\text{e}}+K_{\overline {\nu }}={\text{0.782 MeV ,}}}$

where m_n is the mass of the neutron, m_p is the mass of the proton, m_ν is the mass of the electron antineutrino, and m_e is the mass of the electron; and the K are the corresponding kinetic energies. The neutron has no initial kinetic energy since it is at rest. In beta decay, a typical Q is around 1 MeV.

The decay energy is divided among the products in a continuous distribution for more than two products. Measuring this spectrum allows one to find the mass of a product. Experiments are studying emission spectrums to search for neutrinoless decay and neutrino mass; this is the principle of the ongoing KATRIN experiment.

• Binding energy
• Calorimeter (particle physics)
• Decay energy
• Fusion energy gain factor
• Pandemonium effect

• "Query input form". Nuclear Structure and Decay Data. IAEA. – interactive query form for Q-value of requested decay.
• Schuster, Eugenio (Fall 2020). "Nuclear energy release; fusion reactions" (PDF). Mechanical Engineering 362 – Nuclear Fusion and Radiation. Bethlehem, PA: Lehigh University. ME 362 Lecture 1. Retrieved 5 March 2021. – demonstrates simply the mass-energy equivalence.