Q value (nuclear science)

In nuclear physics and chemistry, the Q value for a reaction is the amount of energy absorbed or released during the nuclear 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.

${\displaystyle Q=(m_{r}-m_{p})\cdot 931\;{\text{MeV}}}$

where the masses are in atomic mass units. Also both ${\displaystyle m_{r}}$ and ${\displaystyle m_{p}}$ 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 (${\displaystyle E_{\text{i}}=E_{\text{f}}}$), 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 binding energies of the nuclear species as:

${\displaystyle Q=B_{f}-B_{i}}$

Proof: Note that the count of nucleons is conserved in a nuclear reaction. Hence, total reactant protons = total product protons and total reactant neutrons = total product neutrons. Hence, the sum of binding energy and rest mass energy of nuclei of both reactant and products are same. In other words, ${\displaystyle B_{i}+m_{i}c^{2}=B_{f}+m_{f}c^{2}}$. Using the first relation (${\displaystyle Q=(m_{\text{i}}-m_{\text{f}})c^{2}}$), this proof is complete.

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 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]

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

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 2 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.

See also

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

Notes and references

1. K.S. Krane (1988). Introductory Nuclear Physics. John Wiley & Sons. p. 381. ISBN 978-0-471-80553-3.
2. B.R. Martin and G. Shaw (2007). Particle Physics. John Wiley & Sons. p. 34. ISBN 978-0-471-97285-3.

External links

• Nuclear Structure and Decay Data – IAEA with query on decays' Q-values
• will demonstrate simply the mass-energy equivalence