John Dalton first introduced a method for reliably measuring the dewpoint, accomplished by determining the highest temperature of water in a glass container that condenses water on its exterior (Dalton 1802b,a). This temperature was originally referred to as the “vapor point” or “point of condensation,” but the term “dewpoint” came into use shortly thereafter (Howard 1818; Dalton 1824). In the years since, many different analytic expressions have been proposed for the dewpoint in terms of temperature and relative humidity (e.g., Mitchell 1967; Wanielista et al. 1997; Sargent 1980; Urbank et al. 2001; Lawrence 2005), but only one has been derived theoretically and that one, as we will see, has errors that can exceed 1 K. This short note derives, from first principles, analytic expressions for the dewpoint and frost point that are accurate to within hundredths of a degree.

*T*

_{d}is the temperature to which air must be cooled at constant pressure to reach saturation with respect to liquid water. At the dewpoint, the vapor pressure

*p*

_{υ}equals the saturation vapor pressure with respect to liquid

*l*reminds us that this is with respect to a planar surface of liquid. In the absence of phase changes, the partial pressure of water vapor does not change as the air is cooled isobarically, so we can write

*p*

_{υ}as

*T*is the original temperature of the air and RH

_{l}is the original relative humidity of the air with respect to liquid water. Therefore,

_{l}as an explicit function of temperature

*T*and dewpoint

*T*

_{d}. Likewise, given the frost point

*T*

_{f}, we can calculate the relative humidity with respect to solid water RH

_{s}as

Given *T* and RH_{l} (or RH_{s}), the most accurate value of the dew (or frost) point is obtained by using an empirical expression for *T*_{d} (or *T*_{f}) using a numerical root solver. Our goal here, however, is to derive expressions for *T*_{d} and *T*_{f} that are not only accurate, but also explicit and analytic in terms of temperature and relative humidity. Therefore, we must invert

To this end, we will use the approximations of Rankine (1866) and Kirchhoff (1858), which assume an ideal gas, constant heat capacities, and zero specific volume of condensates. Following Romps (2021, manuscript submitted to *Quart. J. Roy. Meteor. Soc.*), we may refer to this trio of assumptions as the Rankine–Kirchhoff (RK) approximations, which have been used to derive accurate, explicit, and analytic expressions for the equivalent potential temperature (Romps and Kuang 2010; Hauf and Höller 1987; Emanuel 1994), the quantity conserved by an adiabatically lifted parcel (Romps 2015), and the lifting condensation level (Romps 2017). Here, we aim to add to that list the dewpoint and frost point.

*R*

_{υ}is the specific gas constant for water vapor;

*c*

_{υυ},

*c*

_{υl},

*c*

_{υs}are the specific heat capacities at constant volume of water vapor, liquid water, and solid water, respectively;

*c*

_{pυ}=

*c*

_{υυ}+

*R*

_{υ}is the specific heat capacity of water vapor at constant pressure;

*p*

_{trip}and

*T*

_{trip}are the triple-point vapor pressure and temperature, respectively;

*E*

_{0υ}is the difference in specific internal energy between water vapor and liquid water at the triple point; and

*E*

_{0s}is the difference in specific internal energy between liquid water and solid water at the triple point. Following Romps (2017), a set of optimized values for these constants are

*c*

_{υυ}= 1418 J kg

^{−1}K

^{−1},

*p*

_{trip}= 611.65 Pa,

*T*

_{trip}= 273.16 K,

*E*

_{0υ}= 2.3740 MJ kg

^{−1},

*E*

_{0s}= 0.3337 MJ kg

^{−1},

*R*

_{υ}= 461 J kg

^{−1}K

^{−1},

*c*

_{υl}= 4119 J kg

^{−1}K

^{−1}, and

*c*

_{υs}= 1861 J kg

^{−1}K

^{−1}.

*T*

_{d},

*W*

_{−1}is the lower branch of the Lambert W function. Likewise, using Eq. (4), Eq. (2) can be inverted to give the frost point

*T*

_{f},

*W*

_{0}is the principal branch of the Lambert

*W*function. The reason the dewpoint and frost point use different branches is that

*c*

_{υl}−

*c*

_{pυ}and

*c*

_{υs}−

*c*

_{pυ}have different signs for the choice of heat capacities used here. If different heat capacities were used for which

*c*

_{υs}>

*c*

_{pυ}, then the lower branch would be used for the frost point instead.

*e*

^{c}can be very large and can cause the argument of

*W*

_{0}to exceed the largest number that can be represented using the IEEE double-precision floating-point format, which is about 1.8 × 10

^{308}. If the logarithm of the argument exceeds 709 ≈ log(10

^{308}), then we can use an approximation for

*W*

_{0},

*L*

_{1}= log(

*x*) and

*L*

_{2}= log(

*L*

_{1}) (Corless et al. 1996). When

*L*

_{1}exceeds 709, this approximation replicates

*W*

_{0}with a relative error that is smaller than double-precision rounding error.

_{s}, but relative humidity is most commonly stated with respect to liquid water in the field of atmospheric science. To calculate

*T*

_{f}from RH

_{l}, we can note that

_{l}to RH

_{s}or vice versa, enabling the calculation of both

*T*

_{d}and

*T*

_{f}no matter which relative humidity is given.

Before we calculate the errors in expressions for the dew and frost points, let us first estimate the uncertainties from modern laboratory measurements. We can estimate the empirical uncertainty in the dewpoint by taking the reported relative uncertainty in laboratory measurements of the vapor pressure *L*_{e}/(*R*_{υ}*T*^{2}), where *L*_{e} is the latent heat of evaporation. (This estimation is used here as an alternative to propagating errors in the standard way because it is not known to what degree the reported uncertainties in the vapor pressures at *T* and *T*_{d} are correlated.) Similarly, we can estimate the uncertainty in the frost point using the *L*_{e} + *L*_{m})/(*R*_{υ}*T*^{2}), where *L*_{m} is the latent heat of melting. This procedure gives an uncertainty of ~0.003 K at temperatures around 300 K [using

*c*

_{pυ}=

*c*

_{υl}=

*c*

_{υs}. In that case, Eq. (3) reduces to (e.g., Callendar 1911; Murphy and Koop 2005)

*L*

_{e0}=

*E*

_{0υ}+

*R*

_{υ}

*T*

_{trip}. Using

*T*

_{d}, we get (Lawrence 2005)

A similar expression may be obtained for *T*_{f}, replacing RH_{l} with RH_{s} and replacing the latent heat of evaporation with the approximated sum of the latent heats of evaporation and sublimation, *L*_{e0} *+ L*_{m0} = *E*_{0υ} + *E*_{0s} + *R*_{υ}. When calculating these expressions, the values used for *E*_{0υ}, *E*_{0s}, and *R*_{υ} are the same as those given above.

The errors in these equal-heat-capacity dewpoint and frost-point expressions can be calculated by subtracting from them the dewpoints and frost points obtained (by use of a root solver) from laboratory measurements of the saturation vapor pressure. For this purpose, we will use Eq. (10) of Murphy and Koop (2005) for *T* such that both *T* and *T*_{d} lie between 180 and 273 K. For the dewpoint, the curves are plotted for values of *T* such that both *T* and *T*_{d} lie between 230 and 330 K.

The error in the equal-heat-capacity expression for the frost point is less than or equal to 0.05 K, which is the same order of magnitude as the uncertainty from laboratory measurements. For the dewpoint, however, the assumption of *c*_{pυ} = *c*_{υl} generates errors that far exceed the uncertainties from empirical data. For low relative humidities (less than or equal to about 0.2) or for high air temperatures (exceeding about 310 K, 37°C, or 100°F), the dewpoint error can approach and exceed 0.5°–1.0°C or 1°–2°F.

Repeating this calculation for the Rankine–Kirchhoff expressions in Eqs. (5) and (7) gives the solid curves in Fig. 1. We see that the errors in the RK dewpoint and RK frost point are less than 0.04 and 0.07 K, respectively. These errors are the same order of magnitude as the uncertainties from the laboratory measurements. With maximum errors measured in hundredths of a kelvin, the Rankine–Kirchhoff expressions for the dewpoint and frost point are sufficiently accurate for most, if not all, atmospheric applications.

To give a graphical summary of the Rankine–Kirchhoff dewpoint and frost-point depressions, Fig. 2 plots min[*T* − *T*_{d}(*T*, RH), *T* − *T*_{f}(*T*, RH)] (i.e., the smaller of either the dewpoint depression or the frost-point depression) as a function of temperature and relative humidity (with respect to the saturation vapor pressure over liquid for *T* ≥ 273.16 K and over solid for *T* < 273.16 K). The kink in the curves at a temperature of 273.16 K marks the transition from relative humidity with respect to solid (below the kink) and with respect to liquid (above the kink). A second locus of kinks, marked by the dashed line, occurs where the dewpoint depression and frost-point depression are equal. Above the dashed line, dew forms before frost as the air is cooled, and vice versa below the dashed line, assuming readily available surfaces for both condensation and deposition. Code to evaluate Eqs. (5) and (7) are available on the author’s website in R, Python, FORTRAN, and MATLAB.

Minimum of the dewpoint depression and frost-point depression plotted as a function of temperature and relative humidity (with respect to the saturation vapor pressure over liquid for *T* ≥ 273.16 K and over solid for *T* < 273.16 K). The dashed line separates the regions where, as the air is cooled, dew forms first (red) and frost forms first (blue).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0301.1

Minimum of the dewpoint depression and frost-point depression plotted as a function of temperature and relative humidity (with respect to the saturation vapor pressure over liquid for *T* ≥ 273.16 K and over solid for *T* < 273.16 K). The dashed line separates the regions where, as the air is cooled, dew forms first (red) and frost forms first (blue).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0301.1

Minimum of the dewpoint depression and frost-point depression plotted as a function of temperature and relative humidity (with respect to the saturation vapor pressure over liquid for *T* ≥ 273.16 K and over solid for *T* < 273.16 K). The dashed line separates the regions where, as the air is cooled, dew forms first (red) and frost forms first (blue).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0301.1

## Acknowledgments

This work was supported by the U.S. Department of Energy’s (DOE) Atmospheric System Research (ASR), an Office of Science, Office of Biological and Environmental Research program; Lawrence Berkeley National Laboratory is operated for the DOE by the University of California under Contract DE-AC02-05CH11231. Code to calculate the Rankine–Kirchhoff dewpoint and frost point may be downloaded from https://romps.berkeley.edu/papers/pubs-2020-dewpoint.html.

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