Validation of the unit converter

Each conversion the tool offers is worked out here by hand from a published definition, and set against the number the tool actually returns. The working is shown in full so that the result can be checked without running anything.

Subject: The unit converter at /calc/units/, and the engine module it runs on, src/engine/units.ts.
Report date: 2026-09-17
Source revision: 09c0347
Archived source: 10.5281/zenodo.22794265
Checks performed: 345
Acceptance criterion: Relative difference below 1e-12, and a refusal where a refusal is required.
Largest difference found: 1.455e-16 (C-04)
Result: All checks pass.

Scope

This report covers the unit converter only. It covers every unit the converter offers, in every quantity, together with the two steps the converter presents as physics rather than as conversion, and the cases in which it is required to refuse an answer. It does not cover the other calculators on this site; each of those needs its own report, and the validation index records which ones have one.

What is being established is narrow and worth stating plainly: that the number the tool returns is the number that follows from the published definition of the units involved. Whether the quantity you selected is the quantity you meant is not something a converter can check, and the closing section sets out the rest of what this exercise does not establish.

Method

Independence

A calculation tool can always be made to agree with itself. The only useful comparison is against something derived without reference to the tool, so the expected values in this report are written out from the sources listed under Definitions used and are held in a file that imports no value from the calculation engine. Where arithmetic was needed to reach a factor, it was performed on the numbers printed in those sources, which is why each derivation below names the published statement it starts from. A check in the build refuses to publish if that file ever acquires an import from the engine's constants.

Acceptance criterion

Agreement is judged by relative difference, against a threshold of 1e-12. Double-precision binary arithmetic resolves about 2.2 × 10⁻¹⁶, so a threshold four orders of magnitude above that is loose enough never to fail on rounding and tight enough to catch any error of substance. The largest difference actually observed across every check in this report is stated in the header and repeated in the coverage table; it is at the resolution of the arithmetic itself.

Exactness and representability

Every one of the 44 factors below is exact by definition. That is not the same as being representable: one sixtieth, which the disintegration per minute rests on, has no finite binary expansion, and neither does 0.037. Those factors are therefore carried as the nearest double-precision value, which differs from the exact quantity by less than one part in 10¹⁶. The one constant in this tool that is not exact is the mean energy per ion pair in air used in the crossing to air kerma, which is a measured quantity carrying a stated uncertainty.

The implementation under test

The converter holds, for each quantity, one multiplier per unit taking that unit to the quantity's base unit, and performs every conversion as a single expression:

return (value * t[from]) / t[to];

There is no table of pairs and no special case. It follows that verifying each unit's multiplier against its published definition verifies every conversion within that quantity, since each is the ratio of two verified multipliers. The coverage table records that this argument was not left as an argument: all 276 ordered pairs were computed and compared.

The t lookup is per quantity, so a unit belonging to another quantity is absent rather than wrong, and the function returns NaN. The refusals below exercise that path.

Definitions used

Every expected value in this report traces to one of the statements below, each read from the source named rather than from memory or from a secondary compilation.

Statement	Source
The becquerel, symbol Bq, is the special name for the SI unit of activity, equal to one reciprocal second.	15th CGPM (1975), Resolution 8
The gray, symbol Gy, is the special name for the SI unit of absorbed dose, equal to one joule per kilogram.	15th CGPM (1975), Resolution 9
The sievert, symbol Sv, is the coherent SI unit of dose equivalent, expressed in SI base units as m² s⁻², that is, joule per kilogram.	SI Brochure, Table 4
The coherent SI unit of exposure (x- and γ-rays) is the coulomb per kilogram, C kg⁻¹.	SI Brochure, Table 5
The curie is retained, outside the SI, as a unit of activity with the value 3.7 × 10¹⁰ s⁻¹; the symbol for this unit is Ci. A note records that 1 Ci = 3.7 × 10¹⁰ Bq.	12th CGPM (1964), Resolution 7
curie (Ci) → becquerel (Bq): 3.7 E+10, printed in boldface, which the table defines as meaning the factor is exact.	NIST SP 811, Appendix B.9, radiology
rad → gray (Gy): 1.0 E−02, printed in boldface, that is, exact.	NIST SP 811, Appendix B.9, radiology
rem → sievert (Sv): 1.0 E−02, printed in boldface, that is, exact.	NIST SP 811, Appendix B.9, radiology
roentgen (R) → coulomb per kilogram (C/kg): 2.58 E−04, printed in boldface, that is, exact.	NIST SP 811, Appendix B.9, radiology
SI prefixes: kilo (k) 10³, mega (M) 10⁶, giga (G) 10⁹, tera (T) 10¹², centi (c) 10⁻², milli (m) 10⁻³, micro (µ) 10⁻⁶, nano (n) 10⁻⁹, pico (p) 10⁻¹².	SI Brochure, Table 7
1 min = 60 s.	SI Brochure, Table 8
1 l = 1 L = 1 dm³ = 10⁻³ m³.	SI Brochure, Table 8
W_air is the mean energy expended in air per ion pair formed, more usually expressed as W_air/e. For dry air the value is taken to be 33.97 J/C, with a standard uncertainty estimated at 0.2%.	IAEA TRS-398, Appendix II.2.2
Recommended half-lives with standard uncertainties, evaluated independently of the ENSDF file that this site's nuclear data are taken from. Each sheet quotes the value in the form T1/2 = value (uncertainty on the last digits), with the comma as the decimal separator and 'a' for the year.	BIPM Monographie-5, per-nuclide data sheets
Gamma transition energies with transition probabilities Pγ+ce and internal conversion coefficients αT, and for transitions above 1.022 MeV an internal pair creation coefficient απ. The photon emission probability is not tabulated directly; it follows as Pγ = Pγ+ce / (1 + αT + απ).	BIPM Monographie-5, per-nuclide sheets, section 2.2
The recommended data comprise half-lives, decay modes, alpha, beta, gamma, X-ray and electron emissions, and the characteristics of the transitions. No gamma-ray constant of any kind is tabulated: the words kerma, exposure and dose rate constant do not appear anywhere in the volume. The evaluation stops at the emissions, which is where the conventions end and the application-specific ones begin.	BIPM Monographie-5, Volume 1 (2004), introduction and full text
The tabulated constant is the unshielded gamma-ray dose-equivalent rate at 1 m, computed as Γ = (1/4πR²) Σ Sᵢ D(Eᵢ) with D(E) the fluence-to-dose-rate conversion function of ANSI/ANS-6.1.1-1977. It is therefore a dose-equivalent rate constant, not an air kerma rate constant.	ORNL/RSIC-45, page 1, equations (1) and (2)
The evaluated atomic mass of the neutral atom, in micro-atomic-mass-units, with its standard uncertainty. The same line also carries the mass excess in keV, from which the atomic mass follows as m = A + Δ/c²; the two columns are independent renderings of the same adjustment.	AME2020, file mass_1.mas20, column ATOMIC MASS (micro-u)
Carbon-12 is listed with an atomic mass of exactly 12 000 000 micro-u and zero uncertainty, which is the definition of the unified atomic mass unit.	AME2020, file mass_1.mas20, carbon-12 line
Isomer excitation energy in keV, with its uncertainty, for each excited state identified by the isomer index i in the four-character ZZZi field.	NUBASE2020, file nubase_4.mas20, columns 43-54
molar mass constant: 1.000 000 001 05 e-3 kg mol^-1, with a standard uncertainty of 0.000 000 000 31 e-3 kg mol^-1.	CODATA 2022, molar mass constant
atomic mass constant energy equivalent in MeV: 931.494 103 72, with a standard uncertainty of 0.000 000 29 MeV.	CODATA 2022, atomic mass constant energy equivalent in MeV
Avogadro constant: 6.022 140 76 e23 mol^-1, marked (exact).	CODATA 2022, Avogadro constant
CSDA range: a very close approximation to the average path length traveled by a charged particle as it slows down to rest, calculated in the continuous-slowing-down approximation. In this approximation, the rate of energy loss at every point along the track is assumed to be equal to the total stopping power. Energy-loss fluctuations are neglected. The CSDA range is obtained by integrating the reciprocal of the total stopping power with respect to energy.	NIST ESTAR, Appendix, definition of CSDA range
Projected range: average value of the depth to which a charged particle will penetrate in the course of slowing down to rest. This depth is measured along the initial direction of the particle. Detour factor: ratio of the projected range to the CSDA range. As the result of multiple scattering, the trajectory of the particle is wiggly rather than straight, and the detour factor is always smaller than unity.	NIST ESTAR, Appendix, definitions of projected range and detour factor
Radiation yield: average fraction of the initial kinetic energy of an electron that is converted to bremsstrahlung energy as a particle slows down to rest, calculated in the continuous-slowing-down approximation. Important only for electrons.	NIST ESTAR, Appendix, definition of radiation yield
L_C = k√(2B) and L_D = k² + 2k√(2B), where L_C is the critical level in counts, L_D is the detection limit in counts, k is the Poisson probability sum for alpha and beta (assuming alpha and beta are equal), and B is the number of background counts expected to occur while performing an actual measurement. If values of 0.05 for both alpha and beta are selected as acceptable, then k = 1.645 and these can be written as L_C = 2.33√B and L_D = 3 + 4.65√B.	MARSSIM, page 6-34, equations (6-5) and (6-6)
Note: In Currie's derivation, the constant factor of 3 in the L_D formula was stated as being 2.71, but since that time it has been shown (Brodsky 1992) and generally accepted that a constant factor of 3 is more appropriate. If the sample count times and background count times are different, a slightly different formulation is used.	MARSSIM, page 6-34, note under equation (6-6)
Currie assumed “paired blanks” when deriving the above stated relationships (Currie 1968), which is interpreted to mean that the sample and background count times are the same.	MARSSIM, page 6-34
Values of d′ for selected true positive and false positive proportions. At a false positive proportion of 0.60 and a true positive proportion of 0.95, d′ = 1.38. At a false positive proportion of 0.25 and the same true positive proportion, d′ = 2.32.	MARSSIM, page 6-40, Table 6.5
The minimum detectable number of net source counts in the interval is given by s_i = d′√(b_i), and MDCR = s_i × (60/i), where b_i is the average number of background counts in the observation interval i.	MARSSIM, pages 6-40 and 6-41, equations (6-8) and (6-9)
Scanning is divided into two stages. At the first stage a high rate of correct detections is required (e.g., 95%) and a correspondingly high rate of false positives (e.g., 60%) will be tolerated, giving d′ = 1.38. At the second stage the required rate of true positives remains high (e.g., 95%) but fewer false positives (e.g., 20%) can be tolerated, such that d′ is now 2.48. The greater value of MDCR from each of the scan stages is used.	MARSSIM, pages 6-40 to 6-42
Scan MDC = MDCR / (√p · ε_i · ε_s · (probe area / 100 cm²)), where MDCR is the minimum detectable count rate, ε_i is the instrument efficiency, ε_s is the surface efficiency and p is the surveyor efficiency.	MARSSIM, page 6-43, equation (6-10)
Collision kerma in air and exposure are related by K_col = X · (W_air/e). Total air kerma is larger by the factor 1/(1 − g), where g is the fraction of the charged-particle energy lost to bremsstrahlung.	IAEA STI/PUB/1196, Section 2.7.4, Eqs (2.29) and (2.30)

Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition (2019), V4.01, June 2026. doi:10.59161/AUEZ1291. Published by the BIPM under CC BY 4.0.
12th Conférence Générale des Poids et Mesures (1964), Resolution 7 — Curie. Reproduced in the SI Brochure, 9th edition, Appendix 1. Published by the BIPM under CC BY 4.0.
15th Conférence Générale des Poids et Mesures (1975), Resolutions 8 and 9 — SI units for ionizing radiation (becquerel and gray). Reproduced in the SI Brochure, 9th edition, Appendix 1. Published by the BIPM under CC BY 4.0.
Thompson, A. and Taylor, B. N., Guide for the Use of the International System of Units (SI), NIST Special Publication 811, 2008 edition, Appendix B.9 (factors printed in boldface are exact). Freely published by NIST.
International Atomic Energy Agency, Absorbed Dose Determination in External Beam Radiotherapy, Technical Reports Series No. 398, IAEA, Vienna (2000), Appendix II.2.2. Freely published by the IAEA.
Table of Radionuclides, Monographie BIPM-5, Bureau International des Poids et Mesures — the Decay Data Evaluation Project evaluation, maintained in per-nuclide form by the Laboratoire National Henri Becquerel (LNE-LNHB). Freely published by the BIPM; the current per-nuclide sheets are at lnhb.fr/nuclides/.
Unger, L. M. and Trubey, D. K., Specific Gamma-Ray Dose Constants for Nuclides Important to Dosimetry and Radiological Assessment, ORNL/RSIC-45, Oak Ridge National Laboratory (1981). Freely published by the US Department of Energy through OSTI.
Hubbell, J. H. and Seltzer, S. M., Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients, NIST Standard Reference Database 126, doi:10.18434/T4D01F. Published by NIST as a Standard Reference Database; not a public-domain work. Used here to compute, not redistributed as a dataset.
Wang, M., Huang, W. J., Kondev, F. G., Audi, G. and Naimi, S., The AME2020 atomic mass evaluation (II). Tables, graphs and references, Chinese Physics C 45, 030003 (2021). Data file mass_1.mas20, Atomic Mass Data Center, dated 3 March 2021. The evaluation and its data files are distributed without charge by the Atomic Mass Data Center through the IAEA Nuclear Data Section.
Kondev, F. G., Wang, M., Huang, W. J., Naimi, S. and Audi, G., The NUBASE2020 evaluation of nuclear physics properties, Chinese Physics C 45, 030001 (2021). Data file nubase_4.mas20. Distributed without charge by the Atomic Mass Data Center through the IAEA Nuclear Data Section.
Mohr, P. J., Newell, D. B., Taylor, B. N. and Tiesinga, E., CODATA recommended values of the fundamental physical constants: 2022, as published by NIST in the Fundamental Physical Constants complete listing. Freely published by NIST.
Berger, M. J., Coursey, J. S., Zucker, M. A. and Chang, J., ESTAR: Stopping Powers and Range Tables for Electrons, NIST Standard Reference Database 124, National Institute of Standards and Technology. The method description cites ICRU (1984) for the treatment of mean excitation energies. Freely published by NIST as an online database.
Katz, L. and Penfold, A. S., Range-Energy Relations for Electrons and the Determination of Beta-Ray End-Point Energies by Absorption, Reviews of Modern Physics 24, 28-44 (1952). doi:10.1103/RevModPhys.24.28. Behind a subscription. The bibliographic record was confirmed through Crossref; the full text was not opened for this report, and what that limits is stated in the report itself.
Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM), NUREG-1575 Revision 1 / EPA 402-R-97-016 Rev. 1 / DOE/EH-0624 Rev. 1, August 2000. Chapter 6, Field Measurement Methods and Instrumentation. Published without charge by the US NRC, EPA and DOE. Revision 1 is the final issue; the Revision 2 draft is marked not for citation and is not used here.
Currie, L. A., Limits for Qualitative Detection and Quantitative Determination: Application to Radiochemistry, Analytical Chemistry 40, 586-593 (1968). Behind a subscription. It is cited here through MARSSIM's restatement of its results, which is quoted verbatim rather than paraphrased.
Podgorsak, E. B. (ed.), Radiation Oncology Physics: A Handbook for Teachers and Students, IAEA, Vienna (2005), STI/PUB/1196, Section 2.7.4. Freely published by the IAEA.

Conversion factors

One subsection per quantity. Each table gives the factor taking one of that unit to the quantity's base unit: the Hand column is the value derived below, the Tool column is what the converter returns, and the last column is the relative difference between them. The derivations are given in full below each table.

Activity

Base unit Bq. 12 units, 132 ordered conversions between them.

Unit	Hand	Tool
Bq U-ACT-01	1×10⁰	1×10⁰
kBq U-ACT-02	1×10³	1×10³
MBq U-ACT-03	1×10⁶	1×10⁶
GBq U-ACT-04	1×10⁹	1×10⁹
TBq U-ACT-05	1×10¹²	1×10¹²
Ci U-ACT-06	3.7×10¹⁰	3.7×10¹⁰
mCi U-ACT-07	3.7×10⁷	3.7×10⁷
µCi U-ACT-08	3.7×10⁴	3.7×10⁴
nCi U-ACT-09	3.7×10¹	3.7×10¹
pCi U-ACT-10	3.7×10⁻²	3.7×10⁻²
dps U-ACT-11	1×10⁰	1×10⁰
dpm U-ACT-12	1.66666666667×10⁻²	1.66666666667×10⁻²

Hand calculations for activity

U-ACT-01 — one Bq in Bq

The becquerel is the SI unit of activity, equal to one reciprocal second.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: 15th CGPM (1975), Resolution 8

U-ACT-02 — one kBq in Bq

The SI prefix kilo (k) denotes 10³.
1 kBq = 10³ Bq.
Factor to base = 1 × 10³.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 7

U-ACT-03 — one MBq in Bq

The SI prefix mega (M) denotes 10⁶.
1 MBq = 10⁶ Bq.
Factor to base = 1 × 10⁶.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 7

U-ACT-04 — one GBq in Bq

The SI prefix giga (G) denotes 10⁹.
1 GBq = 10⁹ Bq.
Factor to base = 1 × 10⁹.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 7

U-ACT-05 — one TBq in Bq

The SI prefix tera (T) denotes 10¹².
1 TBq = 10¹² Bq.
Factor to base = 1 × 10¹².

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 7

U-ACT-06 — one Ci in Bq

The 12th CGPM retained the curie outside the SI with the value 3.7 × 10¹⁰ s⁻¹.
One reciprocal second is one becquerel, so 1 Ci = 3.7 × 10¹⁰ Bq.
NIST SP 811 prints this factor in boldface, which its table defines as exact.
Factor to base = 3.7 × 10¹⁰.

From: 12th CGPM (1964), Resolution 7 · NIST SP 811, Appendix B.9, radiology

U-ACT-07 — one mCi in Bq

1 Ci = 3.7 × 10¹⁰ Bq, and the prefix milli (m) denotes 10⁻³.
1 mCi = 10⁻³ × 3.7 × 10¹⁰ Bq = 3.7 × 10⁷ Bq.
Factor to base = 3.7 × 10⁷.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

U-ACT-08 — one µCi in Bq

1 Ci = 3.7 × 10¹⁰ Bq, and the prefix micro (µ) denotes 10⁻⁶.
1 µCi = 10⁻⁶ × 3.7 × 10¹⁰ Bq = 3.7 × 10⁴ Bq.
Factor to base = 3.7 × 10⁴, that is, 37 kBq.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

U-ACT-09 — one nCi in Bq

1 Ci = 3.7 × 10¹⁰ Bq, and the prefix nano (n) denotes 10⁻⁹.
1 nCi = 10⁻⁹ × 3.7 × 10¹⁰ Bq = 3.7 × 10¹ Bq.
Factor to base = 37.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

U-ACT-10 — one pCi in Bq

1 Ci = 3.7 × 10¹⁰ Bq, and the prefix pico (p) denotes 10⁻¹².
1 pCi = 10⁻¹² × 3.7 × 10¹⁰ Bq = 3.7 × 10⁻² Bq.
Factor to base = 0.037.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

U-ACT-11 — one dps in Bq

A disintegration per second is one nuclear transformation per second.
The becquerel is defined as one reciprocal second, so the two coincide.
Factor to base = 1 exactly.

From: 15th CGPM (1975), Resolution 8

U-ACT-12 — one dpm in Bq

A disintegration per minute is one transformation per minute.
The SI Brochure gives 1 min = 60 s, so one transformation per minute is 1/60 per second.
The becquerel is one reciprocal second, so 1 dpm = (1/60) Bq.
Factor to base = 1/60 ≈ 1.6666667 × 10⁻², the reciprocal of 1 Bq = 60 dpm.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 8

Absorbed dose

Base unit Gy. 5 units, 20 ordered conversions between them.

Unit	Hand	Tool
Gy U-ABS-01	1×10⁰	1×10⁰
mGy U-ABS-02	1×10⁻³	1×10⁻³
µGy U-ABS-03	1×10⁻⁶	1×10⁻⁶
rad U-ABS-04	1×10⁻²	1×10⁻²
mrad U-ABS-05	1×10⁻⁵	1×10⁻⁵

Hand calculations for absorbed dose

U-ABS-01 — one Gy in Gy

The gray is the SI unit of absorbed dose, equal to one joule per kilogram.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: 15th CGPM (1975), Resolution 9

U-ABS-02 — one mGy in Gy

The prefix milli (m) denotes 10⁻³.
1 mGy = 10⁻³ Gy.
Factor to base = 1 × 10⁻³.

From: 15th CGPM (1975), Resolution 9 · SI Brochure, Table 7

U-ABS-03 — one µGy in Gy

The prefix micro (µ) denotes 10⁻⁶.
1 µGy = 10⁻⁶ Gy.
Factor to base = 1 × 10⁻⁶.

From: 15th CGPM (1975), Resolution 9 · SI Brochure, Table 7

U-ABS-04 — one rad in Gy

NIST SP 811 gives rad → gray as 1.0 E−02 in boldface, that is, exact.
1 rad = 10⁻² Gy = 1 cGy.
Factor to base = 0.01.

From: NIST SP 811, Appendix B.9, radiology

U-ABS-05 — one mrad in Gy

1 rad = 10⁻² Gy, and the prefix milli (m) denotes 10⁻³.
1 mrad = 10⁻³ × 10⁻² Gy = 10⁻⁵ Gy = 10 µGy.
Factor to base = 1 × 10⁻⁵.

From: NIST SP 811, Appendix B.9, radiology · SI Brochure, Table 7

Dose equivalent

Base unit Sv. 5 units, 20 ordered conversions between them.

Unit	Hand	Tool
Sv U-EQV-01	1×10⁰	1×10⁰
mSv U-EQV-02	1×10⁻³	1×10⁻³
µSv U-EQV-03	1×10⁻⁶	1×10⁻⁶
rem U-EQV-04	1×10⁻²	1×10⁻²
mrem U-EQV-05	1×10⁻⁵	1×10⁻⁵

Hand calculations for dose equivalent

U-EQV-01 — one Sv in Sv

The sievert is the coherent SI unit of dose equivalent, joule per kilogram.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: SI Brochure, Table 4

U-EQV-02 — one mSv in Sv

The prefix milli (m) denotes 10⁻³.
1 mSv = 10⁻³ Sv.
Factor to base = 1 × 10⁻³.

From: SI Brochure, Table 4 · SI Brochure, Table 7

U-EQV-03 — one µSv in Sv

The prefix micro (µ) denotes 10⁻⁶.
1 µSv = 10⁻⁶ Sv.
Factor to base = 1 × 10⁻⁶.

From: SI Brochure, Table 4 · SI Brochure, Table 7

U-EQV-04 — one rem in Sv

NIST SP 811 gives rem → sievert as 1.0 E−02 in boldface, that is, exact.
1 rem = 10⁻² Sv = 1 cSv.
Factor to base = 0.01.

From: NIST SP 811, Appendix B.9, radiology

U-EQV-05 — one mrem in Sv

1 rem = 10⁻² Sv, and the prefix milli (m) denotes 10⁻³.
1 mrem = 10⁻³ × 10⁻² Sv = 10⁻⁵ Sv = 10 µSv.
Factor to base = 1 × 10⁻⁵.

From: NIST SP 811, Appendix B.9, radiology · SI Brochure, Table 7

Exposure

Base unit C/kg. 4 units, 12 ordered conversions between them.

Unit	Hand	Tool
C/kg U-EXP-01	1×10⁰	1×10⁰
R U-EXP-02	2.58×10⁻⁴	2.58×10⁻⁴
mR U-EXP-03	2.58×10⁻⁷	2.58×10⁻⁷
µR U-EXP-04	2.58×10⁻¹⁰	2.58×10⁻¹⁰

Hand calculations for exposure

U-EXP-01 — one C/kg in C/kg

The coherent SI unit of exposure is the coulomb per kilogram.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: SI Brochure, Table 5

U-EXP-02 — one R in C/kg

NIST SP 811 gives roentgen → coulomb per kilogram as 2.58 E−04 in boldface, that is, exact.
1 R = 2.58 × 10⁻⁴ C/kg.
Factor to base = 2.58 × 10⁻⁴.

From: NIST SP 811, Appendix B.9, radiology

U-EXP-03 — one mR in C/kg

1 R = 2.58 × 10⁻⁴ C/kg, and the prefix milli (m) denotes 10⁻³.
1 mR = 10⁻³ × 2.58 × 10⁻⁴ C/kg = 2.58 × 10⁻⁷ C/kg.
Factor to base = 2.58 × 10⁻⁷.

From: NIST SP 811, Appendix B.9, radiology · SI Brochure, Table 7

U-EXP-04 — one µR in C/kg

1 R = 2.58 × 10⁻⁴ C/kg, and the prefix micro (µ) denotes 10⁻⁶.
1 µR = 10⁻⁶ × 2.58 × 10⁻⁴ C/kg = 2.58 × 10⁻¹⁰ C/kg.
Factor to base = 2.58 × 10⁻¹⁰.

From: NIST SP 811, Appendix B.9, radiology · SI Brochure, Table 7

Surface activity

Base unit Bq/cm². 6 units, 30 ordered conversions between them.

Unit	Hand	Tool
Bq/cm² U-SUR-01	1×10⁰	1×10⁰
Bq/m² U-SUR-02	1×10⁻⁴	1×10⁻⁴
kBq/m² U-SUR-03	1×10⁻¹	1×10⁻¹
dpm/100cm² U-SUR-04	1.66666666667×10⁻⁴	1.66666666667×10⁻⁴
dpm/cm² U-SUR-05	1.66666666667×10⁻²	1.66666666667×10⁻²
µCi/cm² U-SUR-06	3.7×10⁴	3.7×10⁴

Hand calculations for surface activity

U-SUR-01 — one Bq/cm² in Bq/cm²

Activity per unit area, with the becquerel over the square centimetre.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 7

U-SUR-02 — one Bq/m² in Bq/cm²

The prefix centi (c) denotes 10⁻², so 1 cm = 10⁻² m and 1 m = 10² cm.
Squaring: 1 m² = (10² cm)² = 10⁴ cm².
Spreading 1 Bq over 1 m² is spreading it over 10⁴ cm², so 1 Bq/m² = 10⁻⁴ Bq/cm².
Factor to base = 1 × 10⁻⁴.

From: SI Brochure, Table 7

U-SUR-03 — one kBq/m² in Bq/cm²

1 kBq = 10³ Bq and 1 m² = 10⁴ cm².
1 kBq/m² = 10³ Bq / 10⁴ cm² = 10⁻¹ Bq/cm².
Factor to base = 0.1.

From: SI Brochure, Table 7

U-SUR-04 — one dpm/100cm² in Bq/cm²

1 dpm = (1/60) Bq, because 1 min = 60 s and 1 Bq = 1 s⁻¹.
The denominator is 100 cm², so divide again by 100.
1 dpm/100 cm² = (1/60) Bq / 100 cm² = 1/6000 Bq/cm².
Factor to base = 1/6000 ≈ 1.6666667 × 10⁻⁴, the reciprocal of 6000 dpm/100 cm² = 1 Bq/cm².

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 8

U-SUR-05 — one dpm/cm² in Bq/cm²

1 dpm = (1/60) Bq, and the denominator is already the square centimetre.
1 dpm/cm² = (1/60) Bq/cm².
Factor to base = 1/60 ≈ 1.6666667 × 10⁻².

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 8

U-SUR-06 — one µCi/cm² in Bq/cm²

1 µCi = 10⁻⁶ × 3.7 × 10¹⁰ Bq = 3.7 × 10⁴ Bq.
The denominator is already the square centimetre.
1 µCi/cm² = 3.7 × 10⁴ Bq/cm².
Factor to base = 3.7 × 10⁴.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

Mass concentration

Base unit Bq/g. 5 units, 20 ordered conversions between them.

Unit	Hand	Tool
Bq/g U-MAS-01	1×10⁰	1×10⁰
Bq/kg U-MAS-02	1×10⁻³	1×10⁻³
kBq/kg U-MAS-03	1×10⁰	1×10⁰
MBq/kg U-MAS-04	1×10³	1×10³
pCi/g U-MAS-05	3.7×10⁻²	3.7×10⁻²

Hand calculations for mass concentration

U-MAS-01 — one Bq/g in Bq/g

Activity per unit mass, with the becquerel over the gram.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: 15th CGPM (1975), Resolution 8

U-MAS-02 — one Bq/kg in Bq/g

The prefix kilo (k) denotes 10³, so 1 kg = 10³ g.
Spreading 1 Bq over 1 kg is spreading it over 10³ g, so 1 Bq/kg = 10⁻³ Bq/g.
Factor to base = 1 × 10⁻³.

From: SI Brochure, Table 7

U-MAS-03 — one kBq/kg in Bq/g

1 kBq = 10³ Bq and 1 kg = 10³ g.
1 kBq/kg = 10³ Bq / 10³ g = 1 Bq/g. The two prefixes cancel.
Factor to base = 1 exactly.

From: SI Brochure, Table 7

U-MAS-04 — one MBq/kg in Bq/g

1 MBq = 10⁶ Bq and 1 kg = 10³ g.
1 MBq/kg = 10⁶ Bq / 10³ g = 10³ Bq/g.
Factor to base = 1 × 10³.

From: SI Brochure, Table 7

U-MAS-05 — one pCi/g in Bq/g

1 pCi = 10⁻¹² × 3.7 × 10¹⁰ Bq = 3.7 × 10⁻² Bq.
The denominator is already the gram.
1 pCi/g = 0.037 Bq/g, equivalently 37 Bq/kg.
Factor to base = 0.037.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

Volume concentration

Base unit Bq/L. 7 units, 42 ordered conversions between them.

Unit	Hand	Tool
Bq/L U-VOL-01	1×10⁰	1×10⁰
kBq/L U-VOL-02	1×10³	1×10³
MBq/L U-VOL-03	1×10⁶	1×10⁶
Bq/mL U-VOL-04	1×10³	1×10³
Bq/m³ U-VOL-05	1×10⁻³	1×10⁻³
pCi/L U-VOL-06	3.7×10⁻²	3.7×10⁻²
µCi/mL U-VOL-07	3.7×10⁷	3.7×10⁷

Hand calculations for volume concentration

U-VOL-01 — one Bq/L in Bq/L

Activity per unit volume, with the becquerel over the litre.
It is the base unit of this group, so its factor to the base is 1 exactly.

From: 15th CGPM (1975), Resolution 8 · SI Brochure, Table 8

U-VOL-02 — one kBq/L in Bq/L

The prefix kilo (k) denotes 10³.
1 kBq/L = 10³ Bq/L.
Factor to base = 1 × 10³.

From: SI Brochure, Table 7

U-VOL-03 — one MBq/L in Bq/L

The prefix mega (M) denotes 10⁶.
1 MBq/L = 10⁶ Bq/L.
Factor to base = 1 × 10⁶.

From: SI Brochure, Table 7

U-VOL-04 — one Bq/mL in Bq/L

The prefix milli (m) denotes 10⁻³, so 1 mL = 10⁻³ L and 1 L = 10³ mL.
Concentrating 1 Bq into 1 mL is 10³ times the concentration of 1 Bq in 1 L.
1 Bq/mL = 10³ Bq/L.
Factor to base = 1 × 10³.

From: SI Brochure, Table 7 · SI Brochure, Table 8

U-VOL-05 — one Bq/m³ in Bq/L

The SI Brochure gives 1 L = 1 dm³ = 10⁻³ m³, so 1 m³ = 10³ L.
Spreading 1 Bq over 1 m³ is spreading it over 10³ L, so 1 Bq/m³ = 10⁻³ Bq/L.
Factor to base = 1 × 10⁻³.

From: SI Brochure, Table 8

U-VOL-06 — one pCi/L in Bq/L

1 pCi = 10⁻¹² × 3.7 × 10¹⁰ Bq = 3.7 × 10⁻² Bq.
The denominator is already the litre.
1 pCi/L = 0.037 Bq/L.
Factor to base = 0.037.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7

U-VOL-07 — one µCi/mL in Bq/L

1 µCi = 10⁻⁶ × 3.7 × 10¹⁰ Bq = 3.7 × 10⁴ Bq.
1 L = 10³ mL, so a concentration per millilitre is 10³ times the same number per litre.
1 µCi/mL = 3.7 × 10⁴ × 10³ Bq/L = 3.7 × 10⁷ Bq/L = 37 MBq/L.
Factor to base = 3.7 × 10⁷.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 7 · SI Brochure, Table 8

Composed conversions

The factors above are verified one unit at a time against the base unit. These are worked through end to end, with a value carried through, in the pairs that actually appear on survey records and source certificates. They are the same arithmetic the single-expression argument above predicts, performed independently so that the prediction is tested rather than assumed.

Conversion	Hand	Tool	Difference
5 µCi → Bq C-01	1.85×10⁵	1.85×10⁵	0
1 Ci → GBq C-02	3.7×10¹	3.7×10¹	0
2.5 kBq → dpm C-03	1.5×10⁵	1.5×10⁵	0
1 Gy → mrad C-04	1×10⁵	1×10⁵	1.46×10⁻¹⁶
350 mrad → mGy C-05	3.5×10⁰	3.5×10⁰	0
1 mSv → mrem C-06	1×10²	1×10²	0
15 mR → C/kg C-07	3.87×10⁻⁶	3.87×10⁻⁶	0
6000 dpm/100cm² → Bq/cm² C-08	1×10⁰	1×10⁰	0
1 µCi/cm² → kBq/m² C-09	3.7×10⁵	3.7×10⁵	0
1 pCi/g → Bq/kg C-10	3.7×10¹	3.7×10¹	0
5 pCi/L → Bq/L C-11	1.85×10⁻¹	1.85×10⁻¹	0
1 µCi/mL → MBq/L C-12	3.7×10¹	3.7×10¹	0

C-01 — 5 µCi to Bq

1 µCi = 3.7 × 10⁴ Bq (U-ACT-08).
5 µCi = 5 × 3.7 × 10⁴ Bq
= 1.85 × 10⁵ Bq.

Why this pair: Sealed check sources are labelled in microcuries and logged in becquerel.

C-02 — 1 Ci to GBq

1 Ci = 3.7 × 10¹⁰ Bq (U-ACT-06).
1 GBq = 10⁹ Bq (U-ACT-04).
1 Ci = 3.7 × 10¹⁰ / 10⁹ GBq
= 37 GBq.

Why this pair: The single most quoted equivalence in the field.

C-03 — 2.5 kBq to dpm

2.5 kBq = 2.5 × 10³ Bq = 2500 Bq (U-ACT-02).
1 dpm = (1/60) Bq (U-ACT-12), so 1 Bq = 60 dpm.
2500 Bq = 2500 × 60 dpm
= 1.5 × 10⁵ dpm.

Why this pair: Counting rooms report in dpm; source certificates are in becquerel.

C-04 — 1 Gy to mrad

1 mrad = 10⁻⁵ Gy (U-ABS-05).
1 Gy = 1 / 10⁻⁵ mrad
= 1 × 10⁵ mrad.

Why this pair: Legacy instruments read in millirad while procedures are written in gray.

C-05 — 350 mrad to mGy

1 mrad = 10⁻⁵ Gy (U-ABS-05).
350 mrad = 350 × 10⁻⁵ Gy = 3.5 × 10⁻³ Gy.
1 mGy = 10⁻³ Gy (U-ABS-02).
3.5 × 10⁻³ / 10⁻³ = 3.5 mGy.

Why this pair: A reading taken on an older survey meter, recorded in SI.

C-06 — 1 mSv to mrem

1 mSv = 10⁻³ Sv (U-EQV-02).
1 mrem = 10⁻⁵ Sv (U-EQV-05).
1 mSv = 10⁻³ / 10⁻⁵ mrem
= 100 mrem.

Why this pair: Dose records cross this boundary whenever two countries share a worker.

C-07 — 15 mR to C/kg

1 mR = 2.58 × 10⁻⁷ C/kg (U-EXP-03).
15 mR = 15 × 2.58 × 10⁻⁷ C/kg
= 38.7 × 10⁻⁷ C/kg = 3.87 × 10⁻⁶ C/kg.

Why this pair: Ion chamber readings in milliroentgen, reported in SI.

C-08 — 6000 dpm/100cm² to Bq/cm²

1 dpm/100 cm² = 1/6000 Bq/cm² (U-SUR-04).
6000 dpm/100 cm² = 6000 × (1/6000) Bq/cm²
= 1 Bq/cm² exactly.

Why this pair: The equivalence that appears on almost every contamination survey record.

C-09 — 1 µCi/cm² to kBq/m²

1 µCi/cm² = 3.7 × 10⁴ Bq/cm² (U-SUR-06).
1 kBq/m² = 0.1 Bq/cm² (U-SUR-03).
3.7 × 10⁴ / 0.1 = 3.7 × 10⁵ kBq/m².

Why this pair: Crosses both the curie and the area prefix in one step, where two errors can cancel.

C-10 — 1 pCi/g to Bq/kg

1 pCi/g = 0.037 Bq/g (U-MAS-05).
1 Bq/kg = 10⁻³ Bq/g (U-MAS-02).
0.037 / 10⁻³ = 37 Bq/kg.

Why this pair: Soil and waste clearance levels are quoted in Bq/kg; US data arrive in pCi/g.

C-11 — 5 pCi/L to Bq/L

1 pCi/L = 0.037 Bq/L (U-VOL-06).
5 pCi/L = 5 × 0.037 Bq/L
= 0.185 Bq/L.

Why this pair: A drinking-water figure quoted in pCi/L, the case the tool's own prose warns about.

C-12 — 1 µCi/mL to MBq/L

1 µCi/mL = 3.7 × 10⁷ Bq/L (U-VOL-07).
1 MBq/L = 10⁶ Bq/L (U-VOL-03).
3.7 × 10⁷ / 10⁶ = 37 MBq/L.

Why this pair: Stock solutions and radiopharmacy vials are labelled this way.

Steps that are not conversions

The converter offers two crossings that a table of factors cannot make. Both are labelled on the tool itself as physics rather than conversion, and both are validated here on the same terms as the rest, with one difference: the constant in the first is measured, not defined, so the result inherits its uncertainty. The value used is 33.97 J/C.

Step	Hand	Tool
1 R to air kerma in mGy (air) B-01	8.76426×10⁰	8.76426×10⁰
1 Bq/L of water, per kilogram in Bq/kg B-02	1×10⁰	1×10⁰
5 pCi/L of water, per kilogram in Bq/kg B-03	1.85×10⁻¹	1.85×10⁻¹
A lighter liquid, at 0.80 g/mL in Bq/kg B-04	1.25×10⁰	1.25×10⁰
1 Bq/g of water, per litre in Bq/L B-05	1×10³	1×10³
A denser solid, at 1.20 g/mL in Bq/L B-06	1.2×10³	1.2×10³

B-01 — 1 R to air kerma

1 R = 2.58 × 10⁻⁴ C/kg (U-EXP-02).
Collision air kerma and exposure are related by K_col = X · (W_air/e).
W_air/e for dry air is 33.97 J/C.
K_col = 2.58 × 10⁻⁴ C/kg × 33.97 J/C = 8.76426 × 10⁻³ J/kg
= 8.76426 × 10⁻³ Gy = 8.76426 mGy.

Not a unit conversion. W_air/e is a measured constant carrying a standard uncertainty of 0.2%, so this result is not exact in the way the table above is. The product is the collision air kerma; total air kerma is larger by 1/(1 − g), with g the fraction of electron energy lost to bremsstrahlung, which is small but not zero at photon energies of interest.

From: NIST SP 811, Appendix B.9, radiology · IAEA TRS-398, Appendix II.2.2 · IAEA STI/PUB/1196, Section 2.7.4, Eqs (2.29) and (2.30)

B-02 — 1 Bq/L of water, per kilogram

1 L = 10³ mL (U-VOL-04).
At a density of 1.00 g/mL, one litre has a mass of 10³ mL × 1.00 g/mL = 1000 g = 1 kg.
So 1 Bq spread through 1 L is 1 Bq in 1 kg.
= 1 Bq/kg, equivalently 1 × 10⁻³ Bq/g.

The numerical equality holds only because water is 1.00 g/mL. It is a coincidence of the material, not a property of the units.

From: SI Brochure, Table 8 · SI Brochure, Table 7

B-03 — 5 pCi/L of water, per kilogram

5 pCi/L = 0.185 Bq/L (C-11).
At 1.00 g/mL one litre has a mass of 1 kg.
0.185 Bq in 1 kg = 0.185 Bq/kg.
Note what this is not: 5 pCi/L is 0.185 Bq/kg, not 185, and not 0.000185 unless the denominator is the gram.

A factor of 1000 hides here whenever the gram is written where the kilogram was meant.

From: 12th CGPM (1964), Resolution 7 · SI Brochure, Table 8

B-04 — A lighter liquid, at 0.80 g/mL

At 0.80 g/mL, one litre has a mass of 10³ mL × 0.80 g/mL = 800 g = 0.8 kg.
1 Bq in 0.8 kg = 1 / 0.8 Bq/kg
= 1.25 Bq/kg.

The same activity per litre is a higher activity per kilogram in a lighter material. Any tool that crosses this boundary with a fixed factor is wrong for everything but water.

From: SI Brochure, Table 8

B-05 — 1 Bq/g of water, per litre

1 Bq/g = 10³ Bq/kg (U-MAS-02 read backwards).
At 1.00 g/mL one litre has a mass of 1 kg.
So the activity in one litre is 10³ Bq.
= 1000 Bq/L.

The reverse of B-02, through the same density.

From: SI Brochure, Table 8 · SI Brochure, Table 7

B-06 — A denser solid, at 1.20 g/mL

At 1.20 g/mL, one litre has a mass of 10³ mL × 1.20 g/mL = 1200 g.
1 Bq/g × 1200 g = 1200 Bq in that litre.
= 1200 Bq/L.

Density enters as a multiplication here and as a division in B-04. Reversing the two is the usual way this step is got wrong.

From: SI Brochure, Table 8

Required refusals

A converter that answers a question it cannot answer is worse than one that answers nothing, because the number it produces carries no sign of being wrong. These are the cases where the correct output is no output. Each returns NaN from the engine. Most of them cannot be reached from the interface at all, because a unit belonging to another quantity is simply absent from the list, and that absence is the first line of defence; the refusal underneath it is the second, for anything that reaches the engine by another route. Where a NaN does reach the display it is drawn as a dash, never as a number.

Asked	Why it must refuse	Returned
Bq/L asked for as Bq/g R-01	Per litre and per kilogram are different quantities. An earlier version of this tool answered 1, which was wrong by a factor of 1000 for anything but water, and wrong by 1000 even for water when the gram was meant.	NaN
pCi/g asked for as pCi/L R-02	The same refusal in the other direction. Crossing needs a density, which the number does not carry.	NaN
Gray asked for as sievert R-03	Absorbed dose to dose equivalent needs a radiation weighting factor, which depends on the radiation type and ranges from 1 to 20. There is no fixed factor to apply.	NaN
Sievert asked for as gray R-04	The reverse is no more defined than the forward direction.	NaN
Millisievert asked for as becquerel R-05	Activity to dose depends on the nuclide, the geometry and the distance. It is a calculation, not a conversion, and a different tool on this site answers it.	NaN
A unit that does not exist R-06	An unrecognised unit must not fall through to a silent zero or to the first unit in the group.	NaN
A density of zero R-07	Dividing by a zero density would yield an infinity that looks like an answer. The step refuses instead.	NaN

The number on the screen

Everything above validates the calculation. What a person reads is the calculation after the interface has rounded it for display, and a report that stopped at the engine would not have looked at what anyone actually sees. A separate check drives the published page in a browser, selects each quantity in turn, enters a value, reads every rendered figure out of the conversion table and compares each with the hand calculation above, allowing only the declared rounding. It runs on every change alongside the checks in this report.

The rounding is not the same everywhere, which is worth stating because it was found by making this check rather than by reading the code. The conversion table is drawn to six significant figures; the two derived figures under it, the air kerma and the crossed concentration, are drawn to four. Both are more than any decision here turns on, but neither is the full double-precision result, and a check written to one of those two tolerances would silently accept or wrongly reject the other. The tolerances are therefore set separately.

Writing the check this way also answers the question of whether the calculation and the display can be validated together. They cannot. With the engine left untouched and only the direction of the table's lookup reversed, every one of the 345 checks above still passed, while the browser check failed on nearly every figure it read.

Coverage and result

Check	Count	Result
Units offered by the tool	44	—
Units with a hand-derived factor	44	complete
Ordered conversions within a quantity	276	all checked
Composed conversions worked end to end	12	pass
Steps that are not conversions	6	pass
Required refusals	7	pass
Largest relative difference observed	1.455e-16	within criterion

Coverage is complete in a checkable sense rather than a claimed one: the count of units the tool offers is read from the tool, the count of units with a derivation is read from this report, and the build fails if they differ. Adding a unit to the converter without adding its derivation here is therefore not possible.

What this does not establish

That the quantity you chose is the quantity you meant. Selecting absorbed dose and reading it as dose equivalent produces a correct conversion of the wrong thing.
That the value you entered is the value you measured. Nothing here checks transcription.
Anything about the other calculators on this site. Their numbers rest on nuclear data and on physical models, not only on definitions, and each needs its own report.
That the definitions cited are current. They are the editions read on the report date, named so that the reading can be repeated; the SI Brochure and NIST SP 811 are revised from time to time.
Fitness for a regulatory submission. This site does not replace an accredited calibration or test, and a validated conversion does not change that.

Reporting a disagreement

If a figure here disagrees with a source you hold, the disagreement is worth sending. Several values in this project's test suite exist because a check found an error during development, including one in this very converter, where activity per litre was treated as activity per kilogram and answers were wrong by a factor of a thousand for every material but water. Contact details are on the contact page, and the method behind the rest of RadCalc is on the methods page.