Four functions for computing boiling temperature are tested and the results are compared to data from the *CRC Handbook of Physics and Chemistry*.

*e*

_{s}is the saturation vapor pressure (Pa),

*e*

_{0}is the vapor pressure at the triple point of water (611.12 Pa),

*l*

_{υ}is the latent heat of vaporization (equal to 2.5008 × 10

^{6}J kg

^{−1}at 0°C, decreasing by about 10% as temperature increases to 100°C),

*R*

_{υ}is the individual gas constant for vapor (461.2 J kg

^{−1}K

^{−1}),

*T*

_{0}is the temperature at the triple point (273.16 K), and

*T*is the temperature (K) (Miller 2015). This equation can be recast to show the relationship between total atmospheric pressure and the boiling temperature of water by

*p*

_{B}is the total atmospheric pressure at which the water is boiled (Pa) and

*T*

_{B}is the boiling temperature of water (K) at pressure

*p*

_{B}. By inverting this equation, we can then compute the boiling temperature at any given pressure. After about five steps, we arrive at

*l*

_{υ}is

*not*a constant (e.g., Court 1985; Henderson-Sellers 1984, 1985; Rogers and Yau 1989). It varies with temperature (in this case

*T*

_{B}), which implies that we would need to compute

*l*

_{υ}before computing

*T*

_{B}. To do that, we need to know

*T*

_{B}first. There are numerical methods for closing this loop, but another method that yields results with a relatively small error is to simply use an average value of

*l*

_{υ}in the known range of temperatures. The purposes and motivation of the research described in this paper were 1) to try three variations on the latent heat of vaporization (one a constant and two different functions of temperature) and apply these toward computing boiling temperature as a function of the total atmospheric pressure in the Clausius–Clapeyron equation and 2) to determine a direct function of total atmospheric pressure for computing boiling temperature. For the purposes of this research, the boiling temperatures listed in Lide (2006) were considered correct.

## METHODS AND RESULTS.

### Method 1: Constant l_{υ}.

In the first method, the boiling temperature of water was computed for the pressures between mean sea level pressure (MSLP; 1,013.25 hPa) and the lower pressure (high elevation) limit shown on most skew *T*–log*p* diagrams (100 hPa) using (3) and the value of *l*_{υ} at 50°C (2.3893 × 10^{6} J kg^{−1}) from Table 1 (Tsonis 2007). This value of *l*_{υ} was chosen for two reasons: 1) its corresponding temperature is midway between the known boiling temperatures at MSLP and at pressures near the top of the stratosphere and 2) doing so made it possible to determine the accuracy of the results when only a rough approximation of the parameter is used. The results are shown in Table 2.

Latent heat of vaporization for water. Some values have been interpolated from available data (Tsonis 2007).

Comparison of boiling temperature values computed from the Clausius–Clapeyron equation with constant *l*_{υ} (method 1) to values from Lide (2006). Elevations correspond to the *U.S. Standard Atmosphere* (NASA 1962, 1966, 1976).

These results indicate that, to within less than half a percentage point (mean error 0.33%) and about 1.15°C (mean bias), the Clausius–Clapeyron equation can be used to estimate the boiling point temperature of water in pressures typical of Earth’s lower atmosphere, even when using a rough estimate of the value of the latent heat of vaporization.

### Method 2: Linearly varying l_{υ}.

*l*

_{υ}, and then used the computed value of

*l*

_{υ}in (3) in an error-reduction loop to compute boiling temperature as a function of total atmospheric pressure. The linear function is

*l*

_{υ}is the latent heat of vaporization at temperature

*T*(J kg

^{−1}),

*l*

_{υ}

_{0}is its value at 0°C (2.5008 × 10

^{6}J kg

^{−1}),

*l*

_{υ}

_{1}is the slope of a linear function (−2,369), and

*T*is the temperature (°C) (Miller 2015). This equation is described in greater detail in Rogers and Yau (1989). In this method,

*l*

_{υ}

_{0}and

*l*

_{υ}

_{1}were considered fixed constants, not variables to be determined. An error-reduction loop involving variable coefficients was employed in the third method, described below.

In the error-reduction loop for method 2, a first-guess temperature was used to estimate the value of *l*_{υ} with (5), and the inverted Clausius–Clapeyron equation (3) was then used with the estimated *l*_{υ} to compute the boiling temperature at a selected pressure. The resulting temperature was then substituted into the latent heat relationship (5), yielding an updated value of *l*_{υ}, and the process was repeated. This was continued for each selected pressure level until the resulting boiling temperature from (3) and the guess temperature used for *l*_{υ} in (5) were within 0.01°C. This method was used to compute boiling temperatures for the same pressures listed in Table 2, then compared to boiling point temperatures from Lide (2006). The results are shown in Table 3 and indicate a mean bias of about 6.45°C and a mean error of about 1.78% in the applicable range of pressures. Both of these are larger than the results described in Table 2. That is, by substituting this functional value of *l*_{υ} for the fixed value (in an attempt to improve the prediction of boiling point temperature), the results got worse, not better.

Comparison of boiling temperature values computed with linear-function *l*_{υ} (method 2) to values from Lide (2006). Bias and error are as defined in text.

### Method 3: Second-order polynomial function for l_{υ}.

*variable*values of

*l*

_{υ}) to the saturation vapor pressures derived for the same temperature using an advanced form of the equation (which assumes a

*fixed*value of

*l*

_{υ}, called

*l*

_{υ}

_{0}):

*e*

_{0}is the reference pressure (611.12 Pa),

*l*

_{υ}

_{0}is the latent heat of vaporization at 0°C (2.5008 × 10

^{6}J kg

^{−1}),

*R*

_{υ}is the individual gas constant for water vapor (461.2 J kg

^{−1}K

^{−1}),

*T*

_{0}is the reference temperature at the triple point of water (273.16 K),

*T*is the in situ temperature (K),

*c*is the specific heat of liquid water at 0°C (4,215 J kg

^{−1}K

^{−1}), and

^{−1}K

^{−1}) (Miller 2015). Equation (6) can be derived analytically by referring to Rogers and Yau (1989), combining Eqs. (2.10) and (2.14) in their text, and integrating the result. It is also listed in Brock and Richardson (2001) in a slightly different form.

The values of *l*_{υ} as a function of temperature between 0° and 100°C (in 1°C increments) were estimated by

computing saturation vapor pressure with (1), which uses the temperature-dependent variable value of

*l*_{υ}, starting with a first-guess value of*l*_{υ};computing saturation vapor pressure with (6), which uses the fixed value of

*l*_{υ}_{0}; andadjusting the variable value of

*l*_{υ}used in (1) to systematically minimize the difference between the two vapor pressures.

*l*

_{υ}as a function of

*T*. With this example in mind, a second-order polynomial (

*R*

^{2}≈ 1.0000 and

*σ*= 49.11 J kg

^{−1}) was fitted to the results of the error-reduction calculations described above, taking the form

*l*

_{υ}is the latent heat of vaporization (J kg

^{−1});

*l*

_{υ}

_{0}is the new zeroth-order coefficient (2.5007 × 10

^{6}),

*l*

_{υ}

_{1}is the new first-order coefficient (−1,173.7723),

*l*

_{υ}

_{2}is the second-order coefficient (1.1315), and

*T*is the temperature (°C). [A first-order fit, similar to (5), yielded an

*R*

^{2}of 0.9992.] Figure 1 shows the shape of the nearly linear function, and Table 4 summarizes the comparison between the computed values of

*l*

_{υ}and those listed in Tsonis (2007). The mean bias between 0° and 50°C is 0.0309 × 10

^{6}(J kg

^{−1}) (meaning the computed values are slightly high), and the mean error is 1.28%, indicating that (7) yields values of

*l*

_{υ}that are probably serviceable for most meteorological applications (e.g., calculations of water vapor mixing ratio and integrated precipitable water). The rate of error growth in the available range of data suggests that the error at 100°C is about 5%, which is also probably sufficient for most meteorological applications.

From here, a second error-reduction loop was used to compute the boiling temperature as a function of pressure. In this loop, a first-guess temperature was used to compute the latent heat term using (7), and (3) was then used to compute the boiling temperature at a selected pressure. The resulting temperature was then substituted into the latent heat relationship (7), and the process was repeated. This was continued until the resulting boiling temperature from (3) and the guess temperature used for *l*_{υ} in (7) were within 0.01°C. This method was used to compute boiling temperatures for pressures between 50 and 1,080 hPa, in 1-hPa increments. The results were stored in a file with two columns (one containing pressure and the other boiling temperature) and are plotted in Fig. 2.

Boiling temperature as a function of pressure, computed by method 3.

Citation: Bulletin of the American Meteorological Society 98, 7; 10.1175/BAMS-D-16-0174.1

Boiling temperature as a function of pressure, computed by method 3.

Citation: Bulletin of the American Meteorological Society 98, 7; 10.1175/BAMS-D-16-0174.1

Boiling temperature as a function of pressure, computed by method 3.

Citation: Bulletin of the American Meteorological Society 98, 7; 10.1175/BAMS-D-16-0174.1

Sample values of the boiling temperature computed with (7) in the error-reduction loop were compared to boiling point temperatures taken from Lide (2006), which is summarized in Table 5. The bias and error values shown in columns 4 and 5 of Table 5 indicate the results are still warm relative to the Lide (2006) values, but to a smaller degree than the results of the calculations that used the fixed value of *l*_{υ} (Table 2), and to a much lesser degree than the calculations using the linear-functional *l*_{υ} (Table 3). The mean bias is 0.31°C, and the mean error in the range of pressures shown is 0.08%.

Comparison of boiling temperature values computed with second-order function *l*_{υ} (method 3) to values from Lide (2006).

### Method 4: Polynomial fits to method 3 results.

*R*

^{2}= 0.9998 and

*σ*= 0.2377°C, to the boiling pressure and temperature data in the file derived with the third method, using pressure as the independent variable and eliminating latent heat altogether:

*T*

_{B}is the boiling temperature (°C),

*p*

_{B}is the pressure (hPa), and

*a–f*are coefficients shown in Table 6. Lower-order polynomials yielded lower values of

*R*

^{2}, which is to be expected. A first-order fit yielded an

*R*

^{2}value of 0.9236, and a second-order fit had a value of 0.9874. The fifth-order fit was judged a reasonable balance between quality of fit and usability, although additional work could certainly be done to fit higher-order polynomials. Different functions (a natural log function would be the logical choice) could also be tried in some future work.

Coefficients for fifth-order polynomial fit for boiling temperature as a function of pressure (method 4).

Sample values of the boiling temperature computed with (8) were compared to boiling point temperatures taken from Lide (2006) and are summarized in Table 7. The bias and error values shown in columns 4 and 5 of Table 7 indicate this function is also slightly warm relative to the Lide (2006) values. The mean bias is 0.25°C, and the mean error in the range of pressures shown is 0.09%.

Comparison of boiling temperature values computed with fifth-order polynomial (method 4) to values from Lide (2006).

## SUMMARY AND CONCLUSIONS.

Boiling is an extreme form of evaporation that occurs when the saturation vapor pressure is equal to the total atmospheric pressure (Glickman 2000). The Clausius–Clapeyron equation (1) was recast to describe the boiling point (2) and solved for boiling temperature (3). Since one term in the equation is the latent heat of vaporization *l*_{υ}, which is a function of temperature, one can either use an approximation of *l*_{υ} to compute boiling temperature *T*_{B} at a given pressure *P*_{B} or use a temperature-dependent functional expression of *l*_{υ} and proceed through an error-reduction loop. The purposes and motivation of this research were 1) to test one constant value and two temperature-dependent functional expressions for *l*_{υ} in the Clausius–Clapeyron equation and 2) to derive a simple polynomial function, with atmospheric pressure as the independent variable, to compute boiling temperature. Values of boiling temperature as a function of pressure as reported in Lide (2006) were used as the standard by which all four methods were judged. Results are summarized in Table 8.

Comparison of mean bias and error between MSLP and 100 hPa for methods tested to compute boiling temperature.

The first method used the value of *l*_{υ} valid for 50°C and obtained results that were correct to within a mean error of 0.33% and a mean bias of 1.15°C for pressures typical of Earth’s troposphere and lower stratosphere. The second method used the linear expression for *l*_{υ} described by (5) (Rogers and Yau 1989), and the result was an even greater disagreement between computed values of *T*_{B} from (3) and those listed in Lide (2006), with a mean bias of about 6.45°C between MSLP and 100 hPa (indicating computed values of the boiling temperature were too warm) and a mean error of about 1.78%.

The third method began by deriving a new, second-order, temperature-dependent polynomial (7) for *l*_{υ}. The new function for *l*_{υ} yields a mean latent heat error of 1.28% between 0° and 50°C when compared to those listed in Tsonis (2007), and a probable error at 100°C of about 5%. An error-reduction loop was used to compute the boiling temperature as a function of pressure, wherein a first-guess temperature was used to compute the latent heat coefficient using the second-order polynomial shown in (7), and (3) was then used to compute the boiling temperature at a selected pressure. The resulting temperature from (3) was then substituted into the latent heat relationship (7), and the process was repeated until the resulting boiling temperature from (3) and the guess temperature used for *l*_{υ} in (7) were within 0.01°C. This method was used to compute boiling temperatures for pressures between 50 and 1,080 hPa, in 1-hPa increments. This method for computing *T*_{B} showed a warm bias (mean value 0.31°C between MSLP and 100 hPa) compared to Lide (2006) and a mean error about 4 times smaller than those associated with computed values of *T*_{B} that used the constant value of *l*_{υ}.

The fourth method fitted a fifth-order polynomial (eliminating *l*_{υ} and making *P*_{B} the sole independent variable) to the boiling temperatures resulting from the third method (8). The polynomial shows an *R*^{2} value of 0.9998 and fit standard deviation of 0.2377°C. Computed values of *T*_{B} using the polynomial were associated with a mean bias of 0.25°C and a mean error of 0.09% when compared to Lide (2006).

## ACKNOWLEDGMENTS

I wish to thank my colleagues R. Bruce Telfeyan, Don S. Harper III, and Stephen Augustyn, as well as three anonymous reviewers, for their valuable feedback. I would also like to thank my colleague Jason Cordeira for his assistance.

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