On July 22 2010 we posted
Anthony F. Fucaloro Professor of Chemistry Joint Science Department Claremont Colleges California
sent us the following thoughtful e-mail and summary which he has permitted to be posted.
Dear Dr. Pielke:
I read Dr. Massen’s method for computing moist enthalpy on your blog, in which he proposes a linear empirical relationship between latent heat and temperature. Since latent heat is not even approximately linear with respect to temperature, this is not a good approach. His high r squared value is a consequence of the near constancy of the latent heat with temperature and the limited temperature range employed. On the attached pdf file, I develop what I believe to be a better approach. I’m fairly sure I’m not the first to suggest this general approach.
Anthony F. Fucaloro
Professor of Chemistry
Joint Science Department
Here is Professor Fucaloro’ guest post.
The derivative form of the Clausius-Clapeyron (CC) equation may be written as
d(ln p)/d(1/T) = -L/R (eq. 1)
where p is the equilibrium vapor pressure of water in contact with its liquid and R is the universal gas constant (8.3145 J/mole K). R is Boltzmann’s constant on a per mole basis. L is in units of J/mole. (Please see the comment at the end.) The only assumptions made to this point are that the vapor is a perfect (formerly, ideal) gas and that the molar volume of the liquid is negligible. Both are good assumptions for this temperature-pressure regime. The integrated form of the CC equation usually assumes L is a constant which, as mentioned by Dr. Massen, is a poor assumption.
As you undoubtedly know there are reliable tables for the vapor pressure of water as a function of temperature. One may easily plot ln p vs. 1/T and perform a polynomial regression giving
ln p= a + b(1/T) + c(1/T)^2 + e(1/T)^3 … (eq. 2)
L = -R[d(ln p)/d(1/T)] = -R [b + 2c(1/T) + 3e(1/T)^2 …], (eq. 3)
where eq. 2 has been differentiated in order to get eq. 3. This regression is considerably more stable than L vs. T and, if memory serves, is reliable for the entire temperature range of liquid water.
Thus we now have fairly reliable expressions for L and p as a function of temperature.
Comment: This is not meant as a criticism but rather an acknowledgement of differences in nomenclature and practices between two disciplines. I am a chemist and not well acquainted with climate science. I find the use of the units g and Kg as appears common in your field rather than moles to be awkward and inefficient. For example, the specific humidity (q), as I understand it, is the ratio of the mass of water vapor to the mass of dry air. Thus,
q = m[water]/m[dry air]
q = (moles[water]/moles[dry air]) (molar mass[water]/molar mass[dry air]),
where the masses have been converted to moles using the molar mass. The molar mass of dry air [(grams of dry air)/(mole of dry air)] is the mole average of dry air for nitrogen and oxygen. In any event, the ratio of the molar masses is ca. 0.622 but, more importantly, the ratio of the moles is equal to the ratio of the partial pressures by Dalton’s Law. Of course, this result is included in Dr. Massen’s text but requires more work to tease out than is necessary.