# Q value (nuclear science)

In nuclear physics and chemistry, the Q value for a nuclear reaction is the amount of energy absorbed or released during the reaction. The value relates to the enthalpy of a chemical reaction or the energy of radioactive decay products. It can be determined from the masses of reactants and products. Q values affect reaction rates. In general, the larger the positive Q value for the reaction, the faster the reaction proceeds, and the more likely the reaction is to "favor" the products.
    $Q=(\,m_{\text{r}}-m_{\text{p}}\,)\times {\text{0.9315 GeV }}$
where the masses are in atomic mass units. Also, both $\;m_{\text{r}}\;$ and $\;m_{\text{p}}\;$ are the sums of the reactant and product masses respectively.

## Definition

The conservation of energy, between the initial and final energy of a nuclear process ${\text{( }}E_{\text{i}}=E_{\text{f}}{\text{ ),}}$ 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:
    $Q=K_{\text{f}}-K_{\text{i}}=(\,m_{\text{i}}-m_{\text{f}}\,)\,c^{2}~$
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. 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 $\Delta M$ of the nuclear species as:
    $Q=\Delta M_{\text{i}}-\Delta M_{\text{f}}~$

### Proof

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

## Applications

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 Q$^5$. 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:
    $Q=(m_{\text{n}}-m_{\text{p}}-m_{\mathrm {\overline {\nu }} }-m_{\text{e}})c^{2}=K_{\text{p}}+K_{\text{e}}+K_{\overline {\nu }}={\text{0.782 MeV ,}}$
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.