Evaporation and perspiration; calculations

Principle

Evaporation

With evaporation, the opposite process of condensation, the atoms or molecules of the liquid gain sufficient energy to enter the gaseous state. It is exclusively a surface phenomenon that occurs at all temperatures and should not be confused with boiling. Even at cool temperatures, a liquid can still evaporate, but only a few particles would escape over a long period of time. Boiling occurs throughout a liquid (Fig. 1) and is characterized by the boiling point (e.g. for H₂0 at 1 bar 100 ^oC). For a liquid to boil its vapor pressure must equal the ambient pressure and bubbles are generated in the liquid.

For particles of a liquid to evaporate, they must be located near the surface, and moving in the proper direction, and have sufficient kinetic energy to overcome the surface tension. Only a very small proportion of the molecules meet these criteria, so the rate of evaporation is limited. Since the average kinetic energy of the molecules rises with temperature and a larger fraction of molecules reaches the requested velocity, evaporation proceeds more quickly at higher temperature. As the faster-moving molecules escape, the remaining molecules have lower average kinetic energy, and the temperature of the liquid thus decreases. This phenomenon is also called evaporative cooling.

For simplicity, from now the liquid is considered water and the gas air with some water vapor.

Fig. 1 Evaporation and boiling

Factors influencing rate of evaporation of water

· Evaporation is proportional with the difference in the partial water pressure (p_H2O) given by the temperature of the water and p_H2O in the air.

· Evaporation increases with the temperature of the water.

· The higher the speed of air compared to the water surface, the faster the evaporation, which is due to constantly lowering the vapor pressure in the boundary layer. Wind speeds < 0.15 m/s are neglected.

· Other substances, especially polar ones (e.g. salts) decrease evaporation (Raoult's law). When a (monomolecular layer of) surfactant (a lipid-polar substance, see surface tension) or a non-polar liquid with a smaller density (oil etc.) covers the water surface, evaporation is very strongly reduced.

The vapor pressure of saturation increases about exponential with the temperature, see Fig. 2.

Fig. 2 Vapor pressure of equilibrium, p_H2O, as function of temperature. 760 Torr = 1atm = 1.013 bar. At 0 ^oC p_H2O is 6.2 mbar and at 100 ^oC 1001.3 mbar ( = 1001.3 hPa).

Perspiration

Perspiration is the process of water evaporation from the human skin.

The process of evaporative cooling is the reason why evaporating sweat cools the human body. The cooling effect of flowing air of the (human) body is caused by convection but also by evaporation. Evaporation and so cooling happens also by expiration. At a body temperature of 37 ^oC, the alveoli, always, so irrespective the ambient pressure, have a p_H2O of 47 mm Hg (= 62.6 mbar = 6260 Pa).

One square meter of a physiological salt solution evaporates about 6 L per day (see More Info). Assuming that the human skin is covered by a thin layer of physiological salt and that its surface is 1.9 m², then the perspiration is 1.9 m² x 6.0 L∙m^-2∙day^-1 = 11.4 L/day. Actually, the loss is only 0.75 L/day (see Body heat dissipation and related water loss). The reason is that the skin is generally not covered by a layer of physiological salt. Skin is not well permeable for water. Evaporation mainly occurs in the sweat glands, which cover only a small fraction of the skin. However, with extensive sweating (heavy endurance sport, sauna) the skin is covered with a thin layer of liquid and perspiration is some 0.5 L/hour. Under these conditions, total sweat production is higher, but much drips off.

More Info

Evaporation is a state change from liquid to gas, and as gas has less order than liquid matter, the entropy of the system is increased, which always requires energy input. This means that the enthalpy change for evaporation (ΔH_liquid) and the standard enthalpy change of vaporization or heat of evaporization (ΔH⁰_liquid) is always positive, making it an endothermic process and subsequently, a cooling process.

If the evaporation takes place in a closed vessel, the escaping molecules accumulate as a vapor above the liquid. Many molecules return to the liquid and more molecules return as the density and pressure of the vapor increases. When the process of escape and return reaches equilibrium, the vapor is said to be "saturated," and no further change in either vapor pressure and density or liquid temperature will occur. For a system consisting of vapor and liquid of a pure substance, this equilibrium state is directly related to the vapor pressure of the substance, as given by the Clausius-Clapeyron relation:

ln p*_x = − ΔĤ_x/RT + B, where (1)

p^*_x is the vapor pressure (bar)
ΔĤ_x the heat of vaporization of liquid x (kJ/mole)
R is the gas constant (8.315 J/(mol∙K)
T is the temperature (K)
B is a variable based on the substance and the system parameters.

ΔĤ_H2O varies from 45.05 kJ/mol at 0 ^oC to 40.66 kJ/mol H₂0 at 100 ^oC. This gives a strong dependency of p* in eq 1 from temperature.

Rewriting equation (1) for water, p*_H2O is approximated by:

p*_H2O = 1.27∙10⁶ e^-5219/T (bar). (2)

The rate of evaporation in an open system is related to the vapor pressure found in a closed system. If a liquid is heated with the vapor pressure reaching the ambient pressure, the liquid will boil.

The underlying physics of (1) is not too hard, but calculating the amount of liquid mass that is evaporated is another thing. The idea is that the fastest water molecules escape from a monomolecular layer of water, which accounts for the surface tension. The solution is an approximation, actually only for the state of equilibrium but for an open system, not in equilibrium, it works rather well.

The number of evaporating particles per second per unit area (ref. 1) is equal to:

N_e = (1/A)(v/d)e^−W/kT, where (3)

· A is the cross sectional area of the particle (m², water molecule 0.057∙10⁻¹⁸ m²),

· d the thickness of a monomolecular layer of the particles (m, water molecule ca. 0.27∙10⁻⁹ m),

· k Boltzmann’s constant (1.3805∙10⁻²³ J/K).

· v the root mean square velocity of the liquid particle (m/s), a function of T^0.5: It can be calculated from Einsteins equation (ref. 2) of the Brownian motion (random movement of particles suspended in a fluid): l² = (2kT/(3πηd)∙t where η is the dynamic viscosity coefficient, for water η • 0.001 Pa∙s, at 37 ^oC, l the free travel distance (estimated in water at 7 nm, nearly ten times less than the 66 nm in air) and t the time of the free path length. Rewriting gives V = (2kT/(3πηd)l ⁻¹. This yields v =0.908 m/s).

· W is the energy needed to evaporate. ΔĤ_water = 2428 kJ/kg at 37 ^oC, and a rough estimate of physiological salt is ΔĤ_{physiological salt} = 2437 kJ/kg at 37 ^oC. Knowing Avogadro’s number being 6.0225∙10²³ , W_water = 7263∙10⁻²³ J/molecule),

· The factor e^−W/kT is the fraction of particles that have enough velocity to escape, so it presents also the probability (water at 310 K gives 42.6∙10⁻⁹, physiological salt 40.0∙10⁻⁹).

Finally, for water at 37 ^oC an evaporation of 7.48 mg∙s⁻¹m⁻² is found. This means lowering the surface with 6.4 mm per day or an evaporation of about 6.4 L/m² per day (physiological salt 6.0 L/m² per day). At 37 ^oC, experimentally (no air convections, humidity low) a surface lowering of 1.04 mm/day was found whereas 1.07 mm/day is predicted.

References

1. Feynman R.P., Leighton R.B. and Sands M. The Feynman lectures on Physics. Addison-Wesley, Reading, Mass., 1963.

2. Kronig R. (ed.). Leerboek der Natuurkunde, Scheltema & Holkema NV, Amsterdam 1966.