Author: Jun Cai cai-j12@mails.tsinghua.edu.cn
Date: 2019-11-10

# Humidity Measures

#### Jun Cai cai-j12@mails.tsinghua.edu.cn

#### 2019-11-10

## Saturation Vapor Pressure \(e_s\)

Saturation vapor pressure \(e_s\) is calculated from a given temperature \(T\) (in \(K\)) by using the Clausius-Clapeyron relation. \[\begin{equation} e_s(T) = e_s(T_0)\times \exp \left(\frac{L}{R_w}\left(\frac{1}{T_0} - \frac{1}{T}\right)\right) \tag{1} \end{equation}\] where \(e_s(T_0) = 6.11 hPa\) is the saturation vapor pressure at a reference temperature \(T_0 = 273.15 K\), \(L = 2.5 \times 10^6 J/kg\) is the latent heat of evaporation for water, and \(R_w = \frac{1000R}{M_w} = 461.52 J/(kg K)\) is the specific gas constant for water vapor (where \(R = 8.3144621 J / (mol K)\) is the molar gas constant and \(M_w = 18.01528 g/mol\) is the molar mass of water vapor). More details refer to Shaman and Kohn (2009).

An alternative way to calculate saturation vapor pressure \(e_s\) is per the equation proposed by Murray (1967). \[\begin{equation} e_s = 6.1078\exp{\left[\frac{a(T - 273.16)}{T - b}\right]} \end{equation}\] where \(\begin{cases} a = 21.8745584 \\ b = 7.66 \end{cases}\) over ice; \(\begin{cases} a = 17.2693882 \\ b = 35.86 \end{cases}\) over water.

The resulting \(e_s\) is in hectopascal (\(hPa\)) or millibar (\(mb\)).

## Vapor Pressure \(e\)

When given dew point \(T_d\) (in \(K\)), the actual vapor pressure \(e\) can be computed by plugging \(T_d\) in place of \(T\) into equation (1). The resulting \(e\) is in millibar (\(mb\)).

## Relative Humidity \(\psi\)

Relative humidity \(\psi\) is defined as the ratio of the partial water vapor pressure \(e\) to the saturation vapor pressure \(e_s\) at a given temperature \(T\), which is usually expressed in \(\%\) as follows \[\begin{equation} \psi = \frac{e}{e_s}\times 100 \tag{2} \end{equation}\]

Therefore, when given the saturation vapor pressure \(e_s\) and relative humidity \(\psi\), the partial water vapor pressure \(e\) can also be easily calculated per equation (2). \[ e = \psi e_s \] The resulting \(e\) is in \(Pa\).

## Absolute Humidity \(\rho_w\)

Absolute humidity \(\rho_w\) is the total amount of water vapor \(m_w\) present in a given volume of air \(V\). The definition of absolute humidity can be described as follows \[ \rho_w = \frac{m_w}{V} \]

Water vapor can be regarded as ideal gas in the normal atmospheric temperature and atmospheric pressure. Its equation of state is \[\begin{equation} e = \rho_w R_w T \tag{3} \end{equation}\]

Absolute humidity \(\rho_w\) is derived by solving equation (3). \[ \rho_w = \frac{e}{R_w T} \] The resulting \(\rho_w\) is in \(kg/m^3\).

## Mixing Ratio \(\omega\)

Mixing ratio \(\omega\) is the ratio of water vapor mass \(m_w\) to dry air mass \(m_d\), expressed in equation as follows \[ \omega = \frac{m_w}{m_d} \]

The resulting \(\omega\) is in \(kg/kg\).

## Specific Humidity \(q\)

Specific humidity \(q\) is the ratio of water vapor mass \(m_w\) to the total (i.e., including dry) air mass \(m\) (namely, \(m = m_w + m_d\)). The definition is described as \[ q = \frac{m_w}{m} = \frac{m_w}{m_w + m_d} = \frac{\omega}{\omega + 1} \]

Specific humidity can also be expressed in following way. \[ \begin{equation} q = \frac{\frac{M_w}{M_d}e}{p - (1 - \frac{M_w}{M_d})e} \tag{4} \end{equation} \] where \(M_d = 28.9634 g/mol\) is the molar mass of dry air; \(p\) represents atmospheric pressure and the standard atmospheric pressure is equal to \(101,325 Pa\). The details of formula derivation refer to Wikipedia.

Substitute \(\frac{M_w}{M_d} \approx 0.622\) into equation (4) and simplify the formula. \[ q \approx \frac{0.622e}{p - 0.378e} \tag{5} \] The resulting \(q\) is in \(kg/kg\).

Hence, by solving equation (5) we can obtain the equation for calculating the partial water vapor pressure \(e\) given the specific humidity \(q\) and atmospheric pressure \(p\).

\[ e \approx \frac{qp}{0.622 + 0.378q} \tag{6} \] Substituting equations (1) and (6) into equation (2), we can get the equation for converting specific humidity \(q\) into relative humidity \(\psi\) at a given temperature \(T\) and under atmospheric pressure \(p\).

## References

Murray, F. W. 1967. “On the Computation of Saturation Vapor Pressure.” J. Appl. Meteor. 6 (1): 203–4. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2.

Shaman, J., and M. Kohn. 2009. “Absolute Humidity Modulates Influenza Survival, Transmission, and Seasonality.” PNAS 106 (9): 3243–8. https://doi.org/10.1073/pnas.0806852106.