Magnus equation
An empirical curve fit giving saturation vapor pressure from temperature alone. Not derived from first principles: its constants come from fitting measured vapor pressure.
Saturation vapor pressure Saturation vapor pressure (SVP) The ceiling on how much water vapor the atmosphere can hold at a given temperature, in kPa. It climbs steeply as a room warms, so the same VPD needs a higher relative humidity in a warmer room. is the ceiling on how much water vapor air can hold before it begins to condense, and that ceiling depends almost entirely on temperature. The Magnus equation is how you get it from a thermometer reading. It is not derived from first principles: it is a formula fitted to measured vapor pressure across a temperature range, which is why its constants are fit parameters rather than physical quantities.
The equation
SVP = 0.61094 x exp(17.625 x T / (T + 243.04))
T is in °C, and it is the only input the equation takes. Saturation vapor pressure depends on temperature and nothing else. The 0.61094 is the saturation vapor pressure at freezing, which is the one constant with a physical meaning. The other two are fit parameters and mean nothing on their own.
Where it comes from
Heinrich Gustav Magnus (1802 to 1870) was a German experimental scientist who ran one of the most influential physics laboratories in Europe, at the University of Berlin. Outside meteorology he is better known for the Magnus effect, the sideways deflection that makes a spinning ball curve. Between 1844 and 1854 he measured the vapor pressure of water and of various solutions, and it is the 1844 work this equation traces to.
He was an experimenter rather than a theoretician, which is the whole reason the equation has the character it does. It does not fall out of thermodynamics. It is a shape chosen because it tracks what a thermometer and a pressure gauge actually reported, and that is also why its constants can be refitted against better data without it becoming a different equation.
The name is a convention rather than a claim of priority. The same functional form was published independently by August in 1828, and again by Roche, so it also appears in the literature as the August-Roche-Magnus formula. Atmospheric science settled on Magnus because of the measurements.
The three constants above are not Magnus's own. They come from Alduchov and Eskridge (1996), who refitted the Magnus form against modern reference data and produced the version now standard in meteorology, accurate to within about 0.4 percent from -40 to 140 °F (-40 to 60 °C). Lawrence (2005) is where most people outside meteorology met them.
Getting to relative humidity
Vapor pressure deficit Vapor pressure deficit The gap between how much moisture the atmosphere holds and how much it could hold. When VPD collapses, transpiration stalls and calcium stops reaching developing tissue. is the gap between that ceiling and what the air is actually carrying, which is the gradient pulling water out of the leaf. Once you know saturation vapor pressure, the conversion to relative humidity Relative humidity The share of moisture air is holding against the most it could hold at that temperature. Because it moves with temperature, cooling air raises RH with no water added. is one line:
RH = 1 - (VPD / SVP)
Worked through at a flower setpoint: at 84 °F (29 °C) the equation returns an SVP of 3.97 kPa, so holding VPD at 1.3 kPa puts the room at 1 - (1.3 / 3.97), or 67 percent. Push the target to 1.4 kPa and it falls to 65 percent.
Why the curve matters
| Canopy temperature | RH at 0.8 kPa | RH at 1.0 kPa | RH at 1.2 kPa | RH at 1.4 kPa |
|---|---|---|---|---|
| 68 °F (20 °C) | 66% | 57% | 49% | 40% |
| 71 °F (22 °C) | 69% | 61% | 54% | 46% |
| 74 °F (23 °C) | 72% | 65% | 58% | 51% |
| 77 °F (25 °C) | 75% | 68% | 62% | 56% |
| 80 °F (27 °C) | 77% | 71% | 66% | 60% |
| 83 °F (28 °C) | 79% | 74% | 69% | 64% |
| 86 °F (30 °C) | 81% | 76% | 72% | 67% |
The exponential is the part worth carrying around. SVP at 84 °F (29 °C) is roughly 26 percent higher than at 77 °F (25 °C), so the same VPD target needs a meaningfully higher humidity in a warmer room. Read a single row and the numbers look arbitrary; read a column downward and the curve is the whole story.
What it does not tell you
The equation describes air, not plants. A transpiring Transpiration Water moving up through the plant and evaporating out through the stomata. It drives nutrient uptake, and carries calcium, which moves almost entirely by that flow. leaf sits below the temperature of the air around it, so the real VPD at the leaf surface is lower than the number a wall sensor implies, and that offset shrinks under CO₂ enrichment. It also says nothing about where condensation forms. For that you want dew point Dew point The temperature at which saturation occurs and water condenses. Sizing dehumidification to a target dew point is more reliable than targeting a humidity percentage. , the temperature at which this same air would reach saturation, which is what the setpoint tables in dew point is the setpoint are built on.
References
- Magnus, G. (1844). Versuche über die Spannkräfte des Wasserdampfs. Annalen der Physik, 137(2), 225–247. https://doi.org/10.1002/andp.18441370202
- Alduchov, O. A., & Eskridge, R. E. (1996). Improved Magnus form approximation of saturation vapor pressure. Journal of Applied Meteorology, 35(4), 601–609. https://doi.org/10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2
- Lawrence, M. G. (2005). The relationship between relative humidity and the dewpoint temperature in moist air: A simple conversion and applications. Bulletin of the American Meteorological Society, 86(2), 225–234. https://doi.org/10.1175/BAMS-86-2-225