TB: Humidity, Dew Point, and Vapor Pressure

Overview

Three quantities — air temperature $T$, relative humidity $\mathrm{RH}$, and dew point $T_d$ — are tied together by a single physical object: the saturation vapor pressure of water, $e_s(T)$. Once $e_s(T)$ is in hand, every common humidity quantity is a simple algebraic combination of $T$, the actual water-vapor partial pressure $e$, and $e_s$. The page collects the definitions, the exact relation, and the leading-order Clausius–Clapeyron consequences in one place.

Core relation

With $e$ the partial pressure of water vapor in the air and $e_s(T)$ the saturation vapor pressure at temperature $T$:

Relative humidity: $\mathrm{RH} = e / e_s(T)$.
Dew point $T_d$: the temperature at which the current vapor pressure would equal saturation, i.e. $e_s(T_d) \equiv e$.

Combining these:

\[ \boxed{\ \mathrm{RH} = \dfrac{e_s(T_d)}{e_s(T)}\ } \]

with $T_d \le T$ always, and $T_d = T \iff \mathrm{RH}=100\%$.

Physical interpretation:

$T_d$ is a proxy for absolute moisture content — it is unchanged by warming or cooling the parcel (no condensation, no mixing).
$\mathrm{RH}$ depends on both moisture and temperature. Heat the air without adding water and $\mathrm{RH}$ drops; cool it and $\mathrm{RH}$ rises until $T = T_d$ and condensation begins.

Saturation vapor pressure (Clausius–Clapeyron)

Liquid–vapor equilibrium gives

\[ \frac{d e_s}{d T} = \frac{L_v\, e_s}{R_v\, T^2} \]

with $L_v$ the latent heat of vaporization and $R_v$ the specific gas constant for water vapor. Integrating with $L_v$ taken as roughly constant yields an exponential dependence; the Magnus form is a standard practical fit (with $T$ in °C):

\[ e_s(T) \approx 6.112\ \exp\!\left(\frac{17.62\, T}{243.12 + T}\right)\ \text{hPa} \]

Near room temperature this gives roughly 7% increase in $e_s$ per °C. This is why a small temperature change shifts $\mathrm{RH}$ a lot but barely moves $T_d$ — and why warmer air can hold dramatically more water.

Useful approximation

For $\mathrm{RH} \gtrsim 50\%$ near room temperature:

\[ T - T_d \;\approx\; \frac{100 - \mathrm{RH}\,(\%)}{5}\ \text{°C} \]

So 80% RH ⇒ dew point ~4 °C below air temperature; 50% RH ⇒ ~10 °C below.

Partial pressure — what $e$ actually means

For an ideal gas mixture the partial pressure of species $i$ is the pressure that component would exert alone in the same volume at the same temperature:

\[ p_i = \frac{n_i\, R\, T}{V} = x_i\, P \]

where $x_i$ is the mole fraction and $P$ the total pressure. Dalton's law: $P = \sum_i p_i$.

Microscopically, pressure is momentum flux into the walls; in an ideal gas the species don't interact, so each delivers its own flux independently. For real gases the species do interact and the clean replacement for partial pressure in chemical-potential expressions is the fugacity $f_i$ — but for atmospheric water vapor at ambient conditions the ideal-gas approximation is good to a fraction of a percent.

Why $e$ is the right humidity variable: liquid–vapor equilibrium is set by equality of chemical potentials, and the vapor's $\mu$ depends on $T$ and on $e$ alone — not on $P$ or on the partial pressures of N$₂$/O$_2$. So saturation is a condition on $e$ alone (at given $T$), and $e_s(T)$ and the dew point are well-defined independent of altitude and total pressure to leading order.

Sanity checks

Cooling air at fixed water content lowers $e_s$ but leaves $e$ alone, so $\mathrm{RH} = e/e_s$ rises. At $T = T_d$, $e_s = e$ and $\mathrm{RH} = 1$.
Mixing air parcels conserves $e$ to leading order; mixing two equally-moist parcels at different $T$ raises the cooler parcel's $\mathrm{RH}$.
The 7%/°C Clausius–Clapeyron slope shows up in climate science as the leading-order scaling of atmospheric water vapor with surface warming.

Related Concepts

Physics and the Nature of Reality — saturation as an equality of chemical potentials is a clean instance of how thermodynamic equilibrium conditions look much simpler than the underlying microscopic dynamics.
Climate and Energy — Clausius–Clapeyron's 7%/°C is the basic reason a warmer atmosphere holds more water and amplifies precipitation extremes.

Provenance

First-principles physics primer (no book source). Written 2026-04-28 from a conversation working through the dew-point / RH / partial-pressure relations.