Evaporation (E) can be defined using Fick’s law (see textbook Chapter 3):

Web Equation 9.4.1  

where E is the evaporation rate; e_i and e_a are the partial pressures of water inside the leaf and in the ambient air, respectively; P is the atmospheric pressure; and r is the stomatal resistance. The ratio (e_i–e_a)/P is the difference of water vapor between the inside and the outside of the leaf (in mole fraction; see Web Topic 9.3).

The concept of a resistance to gas diffusion is intuitively obvious but has the disadvantage that evaporation is inversely related to r. However, one can define conductance to water vapor (g) as the inverse of r. In contrast to r, g is directly related to E, and it is the parameter usually used in gas exchange analysis. Web Equation 9.4.1 can thus be rearranged as follows:

Web Equation 9.4.2  

and Web Equation 9.4.2 can be used to solve for g:

Web Equation 9.4.3  

The rate of evaporation from a leaf can be determined with devices that measure the amount of water leaving the leaf, and the vapor pressure of water in the ambient air can also be measured. The air in the intercellular spaces of the leaf is assumed to be at 100% humidity, and the vapor pressure of water at 100% relative humidity is a function of temperature. Therefore, we determine e_i by measuring the leaf temperature.

Since CO₂ diffuses along the same pathway as water, we can write an equation defining photosynthetic CO₂ assimilation (A) based on Web Equation 9.4.2:

Web Equation 9.4.4  

where C_a and C_i are the partial pressures of CO₂ in ambient air and in the intercellular spaces of the leaf, respectively. The CO₂ molecule is larger than H₂O and therefore has a smaller diffusion coefficient; the difference between the two diffusion coefficients has been empirically determined to be 1.6 (von Caemmerer and Farquhar 1981). Use of this correction factor in Web Equation 9.4.4 allows us to convert the measured resistance to water vapor diffusion into a resistance to CO₂ diffusion.

From Web Equation 9.4.4 we can calculate the partial pressure of CO₂ in the intercellular spaces of the leaf:

Web Equation 9.4.5  

Since all the variables on the right-hand side of Web Equation 9.4.5 can be either measured or calculated, this equation is the basis for the calculation of C_i in experiments in which A is measured as a function of C_i (von Caemmerer and Farquhar 1981).

Water use efficiency can be quantified by a combination of Web Equations 9.4.2 and 9.4.4:

Web Equation 9.4.6  

Let's assume a leaf temperature of 30°C and an atmosphere with relative humidity of 50%. For a C₃ plant, C_a – C_i is typically 10 Pa. At a leaf temperature of 30°C and a relative humidity of 50%, e_i – e_a = 2.1 × 10³ Pa. From Web Equation 9.4.6, we compute that A/E = 0.003 mol CO₂ (mol H₂O)^–1, or a transpiration ratio (E/A) of 336 (see textbook Chapter 4). If water availability is reduced, stomata close and evaporation decreases, leading to improved water use efficiency. Total CO₂ assimilation also decreases, but the plant conserves water and increases its chances of survival.

These equations have been used very successfully to advance our understanding of photosynthesis and the responses of plants to their environments. However, every technique has limitations, and these equations assume uniform stomatal behavior across the leaf surface. Problems might arise when some stomata are closed and others are open, especially in leaves in which discrete regions of mesophyll are sealed off from each other by leaf veins. Under some conditions, in one mesophyll region the stomata are open and there is a correspondingly high photosynthetic rate, whereas in an immediately adjacent region the stomata are partially closed and the mesophyll has a low photosynthetic rate. This phenomenon is termed patchy stomata and might arise under low fluence rates of illumination, or when leaves are stressed (Terashima 1992). Although patchy stomata may reflect normal interactions between stomata and leaf mesophyll (Siebke and Weis 1995), patchiness can lead to inaccurate determinations of C_i. In other experimental systems, patchiness has been shown not to affect calculations of intercellular carbon dioxide by conventional gas exchange methods (Kueppers et al. 1999).