Every physics and chemistry course has a dozen constants that students look up again and again: epsilon naught for Coulomb's law, mu naught for magnetic fields, the proton and neutron masses for nuclear physics, Boltzmann's constant and the gas constant for gases, Avogadro's number for moles, Planck's constant for photons. This page puts them in one table with the values from the 2018 CODATA adjustment (the ones in force since the 2019 redefinition of the SI), the units, the forms you meet in exams (kilograms, atomic mass units and MeV for the masses; four different unit sets for R), and the formula each appears in. The printable is the same table on one sheet with a page of the formulas and worked uses.
Download the physical constants sheet (PDF) (the constants with their 2018 CODATA values, units and symbols; the particle masses in kg, u and MeV; R in six unit sets; the formula each constant appears in; and ten practice questions with answers; two pages, free).
The table
| Constant | Symbol | Value (2018 CODATA) | Units | Where it appears |
|---|---|---|---|---|
| Permittivity of free space (epsilon naught) | ε₀ | 8.8541878128 × 10⁻¹² | F/m = C²/(N·m²) | Coulomb's law, capacitance, Gauss's law |
| Coulomb's constant | k_e = 1/(4πε₀) | 8.9875517923 × 10⁹ | N·m²/C² | F = k q₁q₂/r² |
| Permeability of free space (mu naught) | μ₀ | 1.25663706212 × 10⁻⁶ (= 4π × 10⁻⁷) | N/A² = H/m = T·m/A | fields of wires, coils, solenoids |
| Speed of light in vacuum | c | 299,792,458 (exact) | m/s | E = mc², c = fλ, c² = 1/(μ₀ε₀) |
| Elementary charge | e | 1.602176634 × 10⁻¹⁹ (exact) | C | charge of a proton; 1 eV = 1.602 × 10⁻¹⁹ J |
| Planck's constant | h | 6.62607015 × 10⁻³⁴ (exact) | J·s | E = hf, photons, de Broglie λ = h/p |
| Reduced Planck constant | ħ = h/2π | 1.054571817 × 10⁻³⁴ | J·s | quantum mechanics, uncertainty |
| Boltzmann's constant | k, k_B | 1.380649 × 10⁻²³ (exact) | J/K | KE = (3/2)kT, entropy S = k ln W |
| Avogadro's number | N_A | 6.02214076 × 10²³ (exact) | mol⁻¹ | particles in a mole |
| Gas constant | R = N_A k | 8.314462618 | J/(mol·K) | PV = nRT |
| Faraday constant | F = N_A e | 96,485.33212 | C/mol | electrolysis, Q = nF |
| Atomic mass unit | u (Da) | 1.66053906660 × 10⁻²⁷ | kg | = 931.494 MeV/c²; 1/12 of a carbon-12 atom |
| Proton mass | m_p | 1.67262192369 × 10⁻²⁷ | kg | 1.007276 u; 938.272 MeV/c² |
| Neutron mass | m_n | 1.67492749804 × 10⁻²⁷ | kg | 1.008665 u; 939.565 MeV/c² |
| Electron mass | m_e | 9.1093837015 × 10⁻³¹ | kg | 0.000549 u; 0.510999 MeV/c² |
| Gravitational constant | G | 6.67430 × 10⁻¹¹ | N·m²/kg² | F = Gm₁m₂/r² |
| Standard gravity | g | 9.80665 (defined) | m/s² | weight = mg (9.8 or 9.81 in exams) |
| Stefan-Boltzmann constant | σ | 5.670374419 × 10⁻⁸ | W/(m²·K⁴) | P = σAT⁴ |
| Rydberg constant | R∞ | 1.0973731568 × 10⁷ | m⁻¹ | hydrogen spectrum 1/λ = R(1/n₁² − 1/n₂²) |
| Bohr radius | a₀ | 5.29177210903 × 10⁻¹¹ | m | size of the hydrogen atom |
| Fine-structure constant | α | 7.2973525693 × 10⁻³ (1/137.036) | none | strength of the electromagnetic force |
| Wien's displacement constant | b | 2.897771955 × 10⁻³ | m·K | λ_max = b/T |
| Electronvolt | eV | 1.602176634 × 10⁻¹⁹ (exact) | J | atomic and nuclear energies |
| Standard atmosphere | atm | 101,325 (exact) | Pa | pressure conversions |
| Absolute zero | −273.15 | °C | K = °C + 273.15 |
Since the 2019 redefinition of the SI, e, h, k and N_A have exact defined values and the kilogram, ampere, kelvin and mole are defined from them. μ₀ and ε₀ were exact before 2019 and are now measured (through the fine-structure constant), which is why μ₀ is written 1.25663706212 × 10⁻⁶ rather than exactly 4π × 10⁻⁷; the difference is in the tenth significant figure and no school exam cares.
The particle masses in all three units
| Particle | kg | u (atomic mass units) | MeV/c² |
|---|---|---|---|
| Proton | 1.67262 × 10⁻²⁷ | 1.007276 | 938.272 |
| Neutron | 1.67493 × 10⁻²⁷ | 1.008665 | 939.565 |
| Electron | 9.10938 × 10⁻³¹ | 0.000549 | 0.510999 |
| Hydrogen-1 atom | 1.67353 × 10⁻²⁷ | 1.007825 | 938.783 |
| Alpha particle (He-4 nucleus) | 6.64466 × 10⁻²⁷ | 4.001506 | 3,727.38 |
| Atomic mass unit itself | 1.66054 × 10⁻²⁷ | 1 (exactly) | 931.494 |
Converting: multiply u by 1.66054 × 10⁻²⁷ to get kg; multiply u by 931.494 to get MeV/c². The proton is 1,836 times the electron's mass; the neutron is heavier than the proton by 1.29 MeV, which is why a free neutron can decay into a proton, an electron and an antineutrino. Mass defect questions use the u and MeV columns: the helium-4 nucleus (4.001506 u) is lighter than two protons plus two neutrons (4.031882 u) by 0.030376 u, a binding energy of 0.030376 × 931.494 = 28.3 MeV.
The gas constant R in every unit
The same constant takes a different number in each unit set, and a PV = nRT question only works when R's pressure and volume units match the question's.
| Pressure in | Volume in | R |
|---|---|---|
| Pa (N/m²) | m³ | 8.314 J/(mol·K) |
| kPa | L | 8.314 kPa·L/(mol·K) |
| atm | L | 0.08206 L·atm/(mol·K) |
| Torr or mmHg | L | 62.36 L·Torr/(mol·K) |
| bar | L | 0.08314 L·bar/(mol·K) |
| in calories (thermochemistry) | 1.987 cal/(mol·K) |
Example: 2 mol of gas at 1.5 atm and 300 K: V = nRT/P = 2 × 0.08206 × 300 ÷ 1.5 = 32.8 L. The same in SI: P = 151,988 Pa, V = 2 × 8.314 × 300 ÷ 151,988 = 0.0328 m³ = 32.8 L.
What the electromagnetic pair are for
Epsilon naught, ε₀ sets how strongly charges attract and repel in empty space:
- Coulomb's law: F = q₁q₂ ÷ (4πε₀r²). Two charges of 1 μC each, 10 cm apart: F = 8.988 × 10⁹ × (10⁻⁶)² ÷ 0.01 = 0.9 N.
- Parallel-plate capacitor: C = ε₀A ÷ d. Plates of 0.01 m² separated by 1 mm: C = 8.854 × 10⁻¹² × 0.01 ÷ 0.001 = 8.85 × 10⁻¹¹ F, 88.5 pF.
- Gauss's law: the flux through a closed surface is Q ÷ ε₀.
Mu naught, μ₀ sets how strongly currents make magnetic fields:
- A long straight wire: B = μ₀I ÷ (2πr). 10 A at 5 cm: B = 4π × 10⁻⁷ × 10 ÷ (2π × 0.05) = 4 × 10⁻⁵ T, about the Earth's field.
- A solenoid: B = μ₀nI, with n the turns per metre.
- Force between parallel wires: F/L = μ₀I₁I₂ ÷ (2πd).
And the two together give the speed of light: c = 1 ÷ √(μ₀ε₀) = 1 ÷ √(1.2566 × 10⁻⁶ × 8.854 × 10⁻¹²) = 2.998 × 10⁸ m/s, which is how Maxwell knew light was an electromagnetic wave.
Boltzmann, Avogadro and R: three views of one constant
R = N_A × k: the gas constant is Boltzmann's constant per mole. Boltzmann's constant is the per-molecule version: the average translational kinetic energy of a gas molecule is (3/2)kT, so at 300 K it is 1.5 × 1.380649 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J. Multiply by Avogadro's number and the energy per mole is (3/2)RT = 3,741 J/mol. The mole itself is 6.02214076 × 10²³ particles; one mole of carbon-12 has a mass of 12 g (to within a few parts in a billion since 2019), which is where the atomic mass unit comes from.
Where it sits in school
Avogadro's number and R arrive in grade 10 and 11 chemistry everywhere (moles, the ideal gas law); Coulomb's law with k or ε₀ in AP Physics 2 and C, A level, IB and CBSE Class 12 (Electrostatics); μ₀ in the same courses' magnetism units; the particle masses, u and MeV in A level, IB and Class 12 nuclear physics and AP Physics 2; Boltzmann's constant and Planck's constant in A level, IB and AP Physics 2 thermal and quantum units. Exam boards print their own constants sheet; the values above agree with every board's to the precision the board prints.
Practice: ten questions
- Write Coulomb's constant k in terms of ε₀ and compute it.
- Two charges of +2 μC and −3 μC are 0.2 m apart. Find the force.
- What is the magnetic field 2 cm from a wire carrying 5 A?
- Convert the proton mass 1.007276 u into kg and into MeV/c².
- Find the binding energy of a nucleus whose mass defect is 0.1 u.
- 0.5 mol of gas at 2 atm and 27 °C: what volume in litres?
- The same gas in SI units: convert and check.
- Find the average kinetic energy of a gas molecule at 500 K.
- How many molecules are in 9 g of water (molar mass 18 g/mol)?
- Show from μ₀ and ε₀ that c is about 3 × 10⁸ m/s.
Answers: 1) k = 1/(4πε₀) = 1/(4π × 8.854 × 10⁻¹²) = 8.99 × 10⁹ N·m²/C². 2) F = 8.99 × 10⁹ × 2 × 10⁻⁶ × 3 × 10⁻⁶ ÷ 0.04 = 1.35 N, attractive. 3) B = 4π × 10⁻⁷ × 5 ÷ (2π × 0.02) = 5 × 10⁻⁵ T. 4) 1.007276 × 1.66054 × 10⁻²⁷ = 1.6726 × 10⁻²⁷ kg; 1.007276 × 931.494 = 938.27 MeV/c². 5) 0.1 × 931.494 = 93.1 MeV. 6) V = 0.5 × 0.08206 × 300 ÷ 2 = 6.15 L. 7) P = 202,650 Pa; V = 0.5 × 8.314 × 300 ÷ 202,650 = 0.00615 m³ = 6.15 L. 8) 1.5 × 1.380649 × 10⁻²³ × 500 = 1.04 × 10⁻²⁰ J. 9) 0.5 mol × 6.022 × 10²³ = 3.01 × 10²³. 10) 1/√(1.2566 × 10⁻⁶ × 8.854 × 10⁻¹²) = 1/√(1.1127 × 10⁻¹⁷) = 2.998 × 10⁸ m/s.
Download the physical constants sheet (PDF)
Where to get help
A 1-on-1 online physics or chemistry class with a tutor who holds a master's degree in science, for AP Physics, A level, IB, the Australian Year 11 and 12 courses and CBSE Class 11 and 12: the constants used in worked problems on the shared screen with the units carried through every line, then the exam questions by topic. The recording and notes after every class, the same tutor every week, US$10 to $13 an hour. The physics page has the courses; the free 30-minute first class is a real lesson, usually on the homework the student brings.
Questions parents ask
1What is the value of epsilon naught?
2What is the value of mu naught?
3What is the mass of a proton?
4What is the value of the gas constant R?
5What is Boltzmann's constant?
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