Humidity: RH, dew point, vapour pressure
Convert any one humidity quantity to the others at a stated temperature and pressure: relative humidity, dew point, frost point, vapour pressure, mixing ratio, absolute humidity and Assmann psychrometer wet bulb. Everything goes through the vapour pressure, with Hardy’s ITS-90 formulation by default and WMO Magnus as an approximate option. Below 0 °C, choose RH with respect to water (the WMO convention) or ice; the enhancement factor for moist air is an option.
Humidity · RH, dew point, vapour pressure
v0.1.0 · Hardy 1998 · WMO-No. 8Dew point10.44 °CHardy 1998 (ITS-90), reference · e′ = 12.65 hPa at 1013.25 hPa
| Quantity | Value |
|---|---|
| RH with respect to water (as chosen) | 45.00 % |
| RH with respect to ice (≤ 0.01 °C) | — |
| Dew point | 10.44 °C |
| Frost point | — |
| Vapour pressure e′ | 12.65 hPa |
| Saturation vapour pressure at t | 28.11 hPa |
| Mixing ratio | 7.86 g/kg |
| Absolute humidity | 9.26 g/m³ |
| Wet bulb (Assmann) | 15.54 °C |
| Enhancement factor f at t | 1 (not applied) |
- No frost point: the vapour pressure is above that of ice at its melting point, so this air would form dew, not frost.
- No enhancement factor (f = 1): pure-phase saturation vapour pressure. WMO-No. 8 (4.A.11–12) puts the error of this at 0.5 % or less in the meteorological range; it grows with pressure.
- Mixing ratio and absolute humidity treat the gas as air and ideal.
- Results are full precision. Round for display or a certificate separately.
Everything goes through e′: RH = 100 e′ / (f e_s(t)) · f e_w(T_d) = e′ · r = 0.621 98 e′ / (p − e′) · A = 2.166 79 e′ / T · Hardy water −100 to 100 °C, ice −100 to 0.01 °C · Magnus water −45 to 60 °C, ice −65 to 0 °C
Test vectors for this tool
The reference cases this calculator is tested against on every change. Use them to check your own software. All test vectors · Download JSON
| ID | Inputs | Expected | Tolerance | Source |
|---|---|---|---|---|
| HUM-01 | t_C = 0, phase = water, formulation = hardy | 611.213 Pa | ± 0.001 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 611.213 Pa at 0 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-02 | t_C = 19.995045005414188, phase = water, formulation = hardy | 2338.54 Pa | ± 0.01 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 2338.54 Pa at 20 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-03 | t_C = 39.98977266978722, phase = water, formulation = hardy | 7381.27 Pa | ± 0.01 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 7381.27 Pa at 40 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-04 | t_C = 59.984416225439944, phase = water, formulation = hardy | 19933.09 Pa | ± 0.01 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 19933.09 Pa at 60 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-05 | t_C = 79.97920708638615, phase = water, formulation = hardy | 47374.85 Pa | ± 0.01 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 47374.85 Pa at 80 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-06 | t_C = 99.97435922712069, phase = water, formulation = hardy | 101324.99 Pa | ± 0.01 | Wexler 1976, J. Res. NBS 80A, 775, table 7 (IPTS-68, nvlpubs.nist.gov): 101324.99 Pa at 100 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.1). |
| HUM-07 | t_C = -9.997710864551737, phase = ice, formulation = hardy | 259.922 Pa | ± 0.001 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 259.922 Pa at -10 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-08 | t_C = -19.99558105053768, phase = ice, formulation = hardy | 103.276 Pa | ± 0.001 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 103.276 Pa at -20 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-09 | t_C = -39.99188623565758, phase = ice, formulation = hardy | 12.8486 Pa | ± 0.0001 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 12.8486 Pa at -40 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-10 | t_C = -59.989058338541795, phase = ice, formulation = hardy | 1.08203 Pa | ± 0.00001 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 1.08203 Pa at -60 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-11 | t_C = -79.98718282567725, phase = ice, formulation = hardy | 0.0548067 Pa | ± 1e-7 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 0.0548067 Pa at -80 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-12 | t_C = -99.9862799580986, phase = ice, formulation = hardy | 0.0014058 Pa | ± 1e-8 | Wexler 1977, J. Res. NBS 81A, 5, table 6 (IPTS-68, nvlpubs.nist.gov): 0.0014058 Pa at -100 °C (IPTS-68). Hardy fitted his ITS-90 coefficients to Wexler’s equation (§3.2). |
| HUM-13 | t_C = 0.01, phase = water, formulation = hardy | 611.657 Pa | ± 0.0005 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §3.1: g2 and g3 were adjusted to hold the triple point of water at 611.657 Pa. |
| HUM-14 | t_C = 0.01, phase = ice, formulation = hardy | 611.657 Pa | ± 0.0005 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §3.2: fit constrained to 611.657 Pa at the triple point. |
| HUM-15 | t_C = -60 | -60 °C | ± 0.0003 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.1: the dewpoint formula eq (4) agrees with eq (2) to better than 0.3 mK from −100 to 100 °C. |
| HUM-16 | t_C = 0 | 0 °C | ± 0.0003 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.1: the dewpoint formula eq (4) agrees with eq (2) to better than 0.3 mK from −100 to 100 °C. |
| HUM-17 | t_C = 25 | 25 °C | ± 0.0003 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.1: the dewpoint formula eq (4) agrees with eq (2) to better than 0.3 mK from −100 to 100 °C. |
| HUM-18 | t_C = 90 | 90 °C | ± 0.0003 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.1: the dewpoint formula eq (4) agrees with eq (2) to better than 0.3 mK from −100 to 100 °C. |
| HUM-19 | t_C = -80 | -80 °C | ± 0.0001 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.2: the frostpoint formula eq (5) agrees with eq (3) to better than 0.1 mK. |
| HUM-20 | t_C = -30 | -30 °C | ± 0.0001 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.2: the frostpoint formula eq (5) agrees with eq (3) to better than 0.1 mK. |
| HUM-21 | t_C = -1 | -1 °C | ± 0.0001 | Hardy 1998, ITS-90 formulations (Proc. 3rd Int. Symp. on Humidity & Moisture), §4.2: the frostpoint formula eq (5) agrees with eq (3) to better than 0.1 mK. |
| HUM-22 | t_C = 0, p_Pa = 100000, phase = water | 1.0039 | ± 0.0002 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.0039 for water at 0 °C, 1 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.011 % of Hyland at this temperature. |
| HUM-23 | t_C = 10, p_Pa = 100000, phase = water | 1.00388 | ± 0.00015 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.00388 for water at 10 °C, 1 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.014 % of Hyland at this temperature. |
| HUM-24 | t_C = 20, p_Pa = 100000, phase = water | 1.004 | ± 0.00015 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.004 for water at 20 °C, 1 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.013 % of Hyland at this temperature. |
| HUM-25 | t_C = 40, p_Pa = 100000, phase = water | 1.00467 | ± 0.0001 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.00467 for water at 40 °C, 1 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.004 % of Hyland at this temperature. |
| HUM-26 | t_C = 60, p_Pa = 100000, phase = water | 1.00571 | ± 0.0001 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.00571 for water at 60 °C, 1 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.003 % of Hyland at this temperature. |
| HUM-27 | t_C = 20, p_Pa = 500000, phase = water | 1.0158 | ± 0.0002 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.0158 for water at 20 °C, 5 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.013 % of Hyland at this temperature. |
| HUM-28 | t_C = 20, p_Pa = 1000000, phase = water | 1.0308 | ± 0.0002 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.0308 for water at 20 °C, 10 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.013 % of Hyland at this temperature. |
| HUM-29 | t_C = 20, p_Pa = 2000000, phase = water | 1.0615 | ± 0.0003 | Hyland 1975, J. Res. NBS 79A, 551, table 8 (predicted enhancement factors): f = 1.0615 for water at 20 °C, 20 bar. Greenspan 1976 (J. Res. NBS 80A, 41), table 2: his equation is within 0.013 % of Hyland at this temperature. |
| HUM-30 | t_C = 20, phase = water, formulation = magnus | 2332.596022097807 Pa | ± 0.000002332596022097807 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO-No. 8 (2018), Volume I, Annex 4.B, eq 4.B.1, e_w = 6.112 exp(17.62 t/(243.12 + t)) hPa. |
| HUM-31 | t_C = -40, phase = water, formulation = magnus | 19.021201211131032 Pa | ± 1.9021201211131032e-8 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO-No. 8 (2018), Volume I, Annex 4.B, eq 4.B.1, e_w = 6.112 exp(17.62 t/(243.12 + t)) hPa. |
| HUM-32 | t_C = -20, phase = ice, formulation = magnus | 103.26096299134647 Pa | ± 1.0326096299134649e-7 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO-No. 8 (2018), Volume I, Annex 4.B, eq 4.B.3, e_i = 6.112 exp(22.46 t/(272.62 + t)) hPa. |
| HUM-33 | temperature = 40, pressure = 99800, from = vapourPressure, value = 7375, output = mixingRatio | 49.63 g/kg | ± 0.005 | Vaisala, Humidity Conversion Formulas B210973 rev. F (Japanese edition, vaisala.com), §5 example: dew point 40 °C (Pw = 73.75 hPa) at 998 hPa gives X = 49.63 g/kg. |
| HUM-34 | temperature = 20, from = vapourPressure, value = 1870, output = absoluteHumidity | 13.82 g/m³ | ± 0.005 | Vaisala, Humidity Conversion Formulas B210973 rev. F (Japanese edition, vaisala.com), §7 example: 20 °C, Pw = 1870 Pa gives A = 13.82 g/m³ (C = 2.16679 g·K/J). |
| HUM-35 | temperature = 40, from = rh, value = 50, output = dewPoint | 27.6 °C | ± 0.05 | Vaisala, Humidity Conversion Formulas B210973 rev. F (Japanese edition, vaisala.com), §3 example: 40 °C and 50 %RH give Td = 27.6 °C. |
| HUM-36 | temperature = 23, from = rh, value = 45, output = vapourPressure | 1264.9679147078928 Pa | ± 0.0000012649679147078928 | Hand-derived with 50-digit arithmetic (Python mpmath) from Hardy eq (2): e′ = (U/100) e_w(t), no enhancement factor (f = 1). |
| HUM-37 | temperature = 23, from = rh, value = 45, enhancement = greenspan, output = vapourPressure | 1270.099943918575 Pa | ± 0.000001270099943918575 | Hand-derived with 50-digit arithmetic (Python mpmath) from Hardy eqs (2), (7)–(9): e′ = (U/100) f(p,t) e_w(t) at 101 325 Pa. |
| HUM-38 | temperature = -10, from = rh, value = 80, output = rhIce | 88.20350708001148 % | ± 1e-9 | Hand-derived with 50-digit arithmetic (Python mpmath) from Hardy eqs (2) and (3): U_i = 100 e′/e_i(t) with e′ = 0.8 e_w(t). |
| HUM-39 | temperature = -10, from = rh, value = 80, rhReference = ice, output = rhWater | 72.55947310796167 % | ± 1e-9 | Hand-derived with 50-digit arithmetic (Python mpmath) from Hardy eqs (2) and (3): U_w = 100 e′/e_w(t) with e′ = 0.8 e_i(t). |
| HUM-40 | temperature = 20, from = rh, value = 50, output = mixingRatio | 7.263586777129947 g/kg | ± 1e-9 | Hand-derived with 50-digit arithmetic (Python mpmath) from r = 0.621 98 e′/(p − e′) (WMO-No. 8 Annex 4.A, 4.A.13) with Hardy e_w at 101 325 Pa. |
| HUM-41 | temperature = 20, from = wetBulb, value = 15, output = vapourPressure | 1370.212131947822 Pa | ± 0.000001370212131947822 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO-No. 8 Annex 4.B eq 4.B.8 (Assmann psychrometer) with Hardy e_w(t_w), f = 1, p = 101 325 Pa. |
| HUM-42 | temperature = 20, from = wetBulb, value = 15, output = rh | 58.57453761370856 % | ± 1e-9 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO 4.B.8 and 4.B.10 with Hardy e_w. |
| HUM-43 | temperature = -5, from = wetBulb, value = -7, bulb = ice, output = vapourPressure | 221.64127259947915 Pa | ± 2.2164127259947923e-7 | Hand-derived with 50-digit arithmetic (Python mpmath) from WMO-No. 8 Annex 4.B eq 4.B.9 (ice bulb) with Hardy e_i(t_i), f = 1, p = 101 325 Pa. |
| HUM-44 | temperature = 23, rh = 45 | 45 % | ± 0.002 | Round trip RH → dew point → RH. For Hardy the dew point inverse (eq (4)) agrees with eq (2) to 0.3 mK (§4.1), which is under 0.002 %RH here (the library refines it to the exact inverse of eq (2)); the Magnus inverse 4.B.6 is exact. |
| HUM-45 | temperature = -20, rh = 70 | 70 % | ± 0.002 | Round trip RH → dew point → RH. For Hardy the dew point inverse (eq (4)) agrees with eq (2) to 0.3 mK (§4.1), which is under 0.002 %RH here (the library refines it to the exact inverse of eq (2)); the Magnus inverse 4.B.6 is exact. |
| HUM-46 | temperature = 80, rh = 10, enhancement = greenspan | 10 % | ± 0.002 | Round trip RH → dew point → RH. For Hardy the dew point inverse (eq (4)) agrees with eq (2) to 0.3 mK (§4.1), which is under 0.002 %RH here (the library refines it to the exact inverse of eq (2)); the Magnus inverse 4.B.6 is exact. |
| HUM-47 | temperature = 20, rh = 50, formulation = magnus | 50 % | ± 1e-9 | Round trip RH → dew point → RH. For Hardy the dew point inverse (eq (4)) agrees with eq (2) to 0.3 mK (§4.1), which is under 0.002 %RH here (the library refines it to the exact inverse of eq (2)); the Magnus inverse 4.B.6 is exact. |
| HUM-48 | temperature = 25, pressure = 85000, mixingRatio = 12.5 | 12.5 g/kg | ± 1e-9 | Round trip mixing ratio → vapour pressure → RH → mixing ratio. Closed-form steps, so it must close to rounding error. |
| HUM-49 | t_C = 100.5, phase = water, formulation = hardy | OutOfRangeError | exact | Hardy’s formulations cover −100 to +100 °C (paper title). |
| HUM-50 | t_C = -100.1, phase = water, formulation = hardy | OutOfRangeError | exact | Hardy’s formulations cover −100 to +100 °C. |
| HUM-51 | t_C = 0.02, phase = ice, formulation = hardy | OutOfRangeError | exact | Ice exists only up to the triple point; Hardy eq (3) covers −100 to 0.01 °C (§3.2). |
| HUM-52 | t_C = 60.1, phase = water, formulation = magnus | OutOfRangeError | exact | WMO-No. 8 Annex 4.B: 4.B.1 is for water −45 to 60 °C. |
| HUM-53 | t_C = -45.1, phase = water, formulation = magnus | OutOfRangeError | exact | WMO-No. 8 Annex 4.B: not suitable at all below −45 °C for liquid water. |
| HUM-54 | t_C = -65.1, phase = ice, formulation = magnus | OutOfRangeError | exact | WMO-No. 8 Annex 4.B: 4.B.3 is for ice −65 to 0 °C. |
| HUM-55 | t_C = 20, p_Pa = 90000, phase = water | OutOfRangeError | exact | Greenspan 1976: equations cover 0.1 to 2 MPa. |
| HUM-56 | t_C = 20, p_Pa = 2100000, phase = water | OutOfRangeError | exact | Greenspan 1976: equations cover 0.1 to 2 MPa. |
| HUM-57 | t_C = -51, p_Pa = 101325, phase = water | OutOfRangeError | exact | Greenspan 1976 / Hardy §5.1: water −50 to 100 °C. |
| HUM-58 | temperature = 20, from = rh, value = 101, output = vapourPressure | OutOfRangeError | exact | RH with respect to water is 0 to 100 %. |
| HUM-59 | temperature = 20, from = rh, value = 0, output = vapourPressure | OutOfRangeError | exact | RH must be greater than 0 % for a dew point to exist. |
| HUM-60 | temperature = 20, from = dewPoint, value = 25, output = rh | RangeError | exact | A dew point above the air temperature is supersaturation. |
| HUM-61 | temperature = 20, from = wetBulb, value = 21, output = rh | RangeError | exact | The wet bulb cannot read above the dry bulb. |
| HUM-62 | temperature = 20, pressure = 1000, from = vapourPressure, value = 1200, output = rh | RangeError | exact | The partial pressure of water vapour must be below the total pressure. |
| HUM-63 | temperature = 70, from = rh, value = 50, formulation = magnus, output = dewPoint | OutOfRangeError | exact | WMO-No. 8 Annex 4.B: Magnus over water −45 to 60 °C. |
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