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 $$ M = A u + \Delta M, ~$$ 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 mn is the mass of the neutron, $m$p is the mass of the proton, $m$$\overline{$&nu;$ }$  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.