If your weather station displays barometric pressure, temperature and relative humidity like the one pictured above, you can calculate the amount of water vapor in the air expressed either as grams of water vapor per kilogram of dry air (known as Mixing Ratio) or as grams of water vapor per kilogram of vapor-containing air (known as Specific Humidity). The two measures are very similar for cooler air; differences only become apparent for warmer air.

**Formulas for calculating Mixing Ratio and Specific Humidity**

In the formulas below, barometric pressure P is expressed in hectopascals (hPa), temperature T is expressed in degrees Celsius, relative humidity rh is expressed in %, and e is the Euler number 2.71828 [raised to the power of the contents of the square brackets]. In developing these formulas, the following textbook was consulted: Atmospheric Thermodynamics by Grant W. Petty, Sundog Publishing, Madison Wisconsin. ISBN-10: 0-9729033-2-1

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**Derivation of Mixing Ratio formula**

Mixing Ratio is defined as the mass of water vapor in grams mixed into a kilogram of dry air.

Both water vapor and dry air exert pressures, so by the law of partial pressures

Barometric pressure (P) = Pressure of dry air (Pdry) + Pressure of water vapor (Pvap)

and therefore

Pdry = P – Pvap (1)

The ratio Pvap/Pdry also expresses the molar ratio of water vapor to dry air since by the ideal gas law PV = nRT

Pvap/Pdry = n_{vap}RTV^-1 / n_{dry}RTV^-1 = moles vapor / moles dry air (RTV^-1 cancels)

The molar mass of water is 0.0180153 kg/mol (ref: Petty, see above)

The molar mass of dry air is 0.0289655 kg/mol (ref: Petty, see above)

Therefore

Pvap × 0.0180153 / Pdry x 0.0289655 = kg vapor / kg dry air

Pvap × 0.622 / Pdry = kg vapor / kg dry air

Substituting Pdry by equation 1

Pvap × 0.622 / (P – Pvap) = kg vapor / kg dry air

Therefore

Mixing Ratio = Pvap × 622 / (P – Pvap) units: g vapor / kg dry air (2)

Pvap can be calculated from temperature T and relativity humidity rh using the formula originated and published by CarnotCycle blog

Pvap = 6.112 × e^[(17.67 × T)/(T+243.5)] × (rh/100)

where T is expressed in degrees Celsius and rh is expressed in %.

Substituting the formula for Pvap into equation 2 completes the derivation.

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**Derivation of Specific Humidity formula**

Specific Humidity is defined as the mass of water vapor in grams per kilogram of moist air.

By the law of partial pressures

Barometric pressure (Pmoist) = Pressure of dry air (Pdry) + Pressure of water vapor (Pvap)

The ratio Pvap/(Pdry + Pvap) expresses the molar ratio of water vapor to moist air since by the ideal gas law PV = nRT

Pvap/(Pdry + Pvap) = n_{vap}RTV^-1 / (n_{dry}RTV^-1 + n_{vap}RTV^-1) = moles vapor / moles moist air

Pvap/(Pdry + Pvap) = n_{vap} / (n_{dry} + n_{vap}) = moles vapor / moles moist air (RTV^-1 cancels)

Let M_{1} = molar mass of water vapor = 0.0180153 kg

Let M_{2} = molar mass of dry air = 0.0289655 kg

M_{1}Pvap / (M_{2}Pdry + M_{1}Pvap) = kg vapor / kg moist air (3)

Substitute Pdry = Pmoist – Pvap in equation 3

M_{1}Pvap / (M_{2}Pmoist – M_{2}Pvap + M_{1}Pvap) = kg vapor / kg moist air

M_{1}Pvap / (M_{2}Pmoist – (M_{2} – M_{1})Pvap) = kg vapor / kg moist air

Divide by M_{2}

(M_{1}/M_{2})Pvap / (Pmoist – (1 – M_{1}/M_{2})Pvap) = kg vapor / kg moist air

0.622 × Pvap / (Pmoist – 0.378 × Pvap) = kg vapor / kg moist air

Therefore

Specific Humidity = Pvap × 622/ (Pmoist – 0.378 × Pvap) units: g vapor / kg moist air (4)

Pvap can be calculated from temperature T and relativity humidity rh using the formula originated and published by CarnotCycle blog

Pvap = 6.112 × e^[(17.67 × T)/(T+243.5)] × (rh/100)

where T is expressed in degrees Celsius and rh is expressed in %.

Substituting the formula for Pvap into equation 4 completes the derivation.

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P Mander June 2020, derivations added February 2023