Playground
Interactive nuclear reaction energy calculator: adjust reactant and product masses to see the Q-value and whether the reaction is exothermic or endothermic.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Reaction energyoutput Net energy released (Q > 0) or absorbed (Q < 0) | J | M·L²·T⁻² | -1e-11 – 1e-11 | |
| Initial mass Total rest mass of reactants | kg | M | 0 – 1e-25 | |
| Final mass Total rest mass of products | kg | M | 0 – 1e-25 | |
| Speed of light Speed of light in vacuum | m/s | L·T⁻¹ | 299792458 – 299792458 |
Deep dive
Derivation
By conservation of total relativistic energy in a nuclear reaction: Σ(m_reactants·c² + KE_reactants) = Σ(m_products·c² + KE_products). Rearranging: Q = Σ KE_products − Σ KE_reactants = (Σ m_reactants − Σ m_products)·c² = (mᵢ − m_f)·c². Positive Q means kinetic energy increased (exothermic); negative Q means kinetic energy decreased (endothermic).
Experimental verification
Cockcroft and Walton (1932) verified Q-values in the first artificial nuclear reaction: ⁷Li + p → 2 ⁴He, measuring Q = 17.3 MeV in agreement with mass tables. Modern accelerator experiments routinely confirm Q-values to keV precision using magnetic spectrometers.
Common misconceptions
- Q-value is NOT the kinetic energy of any single product — it is the total kinetic energy released (or absorbed) across ALL products minus all reactant kinetic energies.
- A positive Q does not mean the reaction happens spontaneously — there may be a Coulomb barrier or other activation energy to overcome.
- Q-value calculations using atomic masses automatically include electron masses, which cancel for most nuclear reactions but must be handled carefully for beta decay.
Real-world applications
- Fusion reactor design (D-T fusion: Q = 17.6 MeV, the most energetically favorable fusion reaction)
- Nuclear fission energy budgets (²³⁵U fission: Q ≈ 200 MeV per event)
- Astrophysical nucleosynthesis models (predicting which reactions power different stellar burning stages)
- Medical isotope production planning (determining energy requirements for transmutation reactions)
Worked examples
D-T fusion Q-value
Given:
- mᵢ_u:
- 5.03016
- m_f_u:
- 5.01127
- c:
- 299792458
Find: Q
Solution
Q = (5.03016 − 5.01127) × 931.494 = 17.6 MeV
Lithium-7 proton capture
Given:
- mᵢ_u:
- 8.02383
- m_f_u:
- 8.00521
- c:
- 299792458
Find: Q
Solution
Q = (8.02383 − 8.00521) × 931.494 = 17.3 MeV
Scenarios
What if…
- scenario:
- What if Q is exactly zero?
- answer:
- The reaction is at threshold — products have zero kinetic energy in the center-of-mass frame. In the lab frame, a minimum projectile kinetic energy (threshold energy) is still needed to conserve momentum.
- scenario:
- What if Q is negative and we supply insufficient kinetic energy?
- answer:
- The reaction cannot proceed. The minimum lab-frame kinetic energy required is KE_threshold = |Q| × (1 + m_projectile/m_target) to conserve both energy and momentum.
- scenario:
- What if we use atomic masses instead of nuclear masses?
- answer:
- Electron masses cancel in most reactions since the same number of electrons appear on each side. Exception: beta decay, where one electron is created or absorbed, requiring explicit correction.
Limiting cases
- condition:
- mᵢ = m_f
- result:
- Q = 0
- explanation:
- No mass difference means no energy is released or absorbed — the reaction is at the threshold of being possible.
- condition:
- mᵢ > m_f
- result:
- Q > 0 (exothermic)
- explanation:
- Products are lighter; the missing mass appears as kinetic energy of products. Fusion of light nuclei and fission of heavy nuclei are exothermic.
- condition:
- mᵢ < m_f
- result:
- Q < 0 (endothermic)
- explanation:
- Products are heavier; kinetic energy must be supplied to create the extra mass. The reaction has a threshold energy.
Context
Nuclear physics community · 1932
Q-values became standard after Cockcroft and Walton's first artificial nuclear reaction confirmed Einstein's mass-energy equivalence experimentally.
Hook
How do we know a reaction will release energy before running it?
In the reaction D + T → ⁴He + n, the mass difference is 0.01889 u. Find Q in MeV.
Dimensions:
- lhs:
- Q → [M·L²·T⁻²]
- rhs:
- (mᵢ − m_f)·c² → [M]·[L·T⁻¹]² = [M·L²·T⁻²]
- check:
- Both sides are [M·L²·T⁻²] = Joules. ✓
Validity: Valid for any nuclear reaction where rest masses of all reactants and products are known. Assumes the calculation is done in the center-of-mass frame or that kinetic energies are properly accounted for. Does not include energy carried away by neutrinos unless explicitly included in the mass/energy balance.