These are the core relationships of school and undergraduate science, implemented so that the units are unambiguous and the constants are the current internationally fixed values rather than textbook approximations. Every calculator here works in SI internally and converts on the way out, which removes the single largest source of error in physics and chemistry problems — mixing unit systems halfway through a calculation and getting an answer that is wrong by a factor of 32, or 1,000, or 3,600.
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The 2019 revision of the SI fixed the Planck constant, the elementary charge, the Boltzmann constant and the Avogadro constant at exact numerical values. The speed of light had already been fixed in 1983. This means several quantities that used to carry measurement uncertainty are now exact by definition: the gas constant R is exactly 8.31446261815324 J/(mol·K), the mole is exactly 6.02214076 × 10²³ entities, and the metre is derived from the second and the speed of light.
Practically, nothing changed for anyone doing ordinary calculations — the numbers are the same to more decimal places than any laboratory measurement reaches. What changed is where the uncertainty sits. It has moved from the constants into the measurement apparatus, which is philosophically tidier and occasionally matters in metrology.
The relationships that surprise people are the non-linear ones. Kinetic energy goes as velocity squared, so a 40% speed increase raises impact energy by 96%. Decay is exponential, so a substance at 3% remaining is five half-lives in, not thirty-three. Photon energy scales with frequency, so amplitude has nothing to do with whether radiation ionises. In each case linear intuition gives an answer that is not slightly wrong but categorically wrong.
Chemistry has its own version in the mole. Reactions balance by particle count, not by mass, so stoichiometry always routes through moles even when every measurement you take is a mass or a volume. Skipping that step is why an otherwise careful calculation ends up short of reagent.
Most of these formulas are idealisations with a stated domain of validity. The ideal gas law assumes point particles with no mutual attraction and holds within about 1% at ambient conditions, drifting badly above roughly 10 atmospheres or near condensation. ½mv² is the low-speed limit of a relativistic expression and is fine below about a tenth of light speed. Density treated as a material constant ignores thermal expansion, which is negligible for solids and significant for gases.
Knowing the boundary is more useful than memorising a correction. If your conditions sit inside the domain, the simple formula is not an approximation worth apologising for — it is the right tool. If they sit outside, no amount of extra decimal places will help and you need a different equation of state.
For the inputs, yes — kilograms, metres, seconds, pascals, kelvin and moles. Every calculator converts the output into common alternatives such as atmospheres, litres, Celsius, pounds-force and electronvolts, so you only need to convert on the way in.
Gas laws and thermodynamic relationships are proportional to absolute temperature. Celsius has an arbitrary zero point, so using it produces results that are wrong and sometimes negative where they cannot be.
They follow the IUPAC standard atomic weights, which are natural-abundance averages. Revisions land in the third or fourth decimal place and never change a practical laboratory weighing.
Yes, and the methodology and worked-example sections show the intermediate steps so you can reproduce the arithmetic by hand rather than submitting an unexplained number.