PHYS 304 Physics of the Human Body 73 Chapter 8 The properties of water

“From water does all life begin.” the cross section for a collision between two such —OC Bible, 457 Kalima. molecules is therefore about four times as large, −15 2 σ − ≈ 10 cm . H2O H2O The mean free path of a molecule (that is, the average distance it can travel between collisions) 1. Static properties is therefore The chemical formula of water is H O. Its molecu- 2 1 lar structure is roughly as shown below: λ = ≈ 2.5 Å . n σ

We can estimate the average inter-molecular spac- ing in liquid water by computing the volume per molecule (often called the specific volume):

df 1 4π ω = = r3 n 3 0 giving r ≈ 2 Å , which means the inter-molecu- The density of water, at standard temperature and 0 = pressure is, by definition lar spacing is 2r0 4 Å . Thus the molecules in water are not actually touching each other (which ρ = ⁄ 3 = 3 ⁄ 3 1 gm cm 10 kg m . is why it is a liquid rather than a solid) but they are Since the molecular weight of water (ignoring very close together. The mean free path is smaller isotopes like 18O and 2H) is 18, the number density than the average inter-molecular spacing. of molecules in liquid water is

N − n = A ≈ 3×1022 cm 3 . 18 2. Solvent properties Water has a very large dielectric constant, about 80 (times the dielectric constant of vacuum). Thus The spacing between the hydrogen and oxygen the Coulomb potential between—say—the atoms atoms in the molecule is about of the NaCl molecule (salt) is modified to df − − 2 r ≈ 1 Å = 10 8 cm . = e VH O 2 ε r The interatomic potential looks like an attractive We may estimate the collision cross-section of a Coulomb potential at long distances, but has a water molecule (with a point-like particle) as its short-ranged repulsive part, so the total looks geometric cross section: something like the figure below: σ ≈ π r2 ; 74 Physics of the Human Body Solubility of H2O in H2O

^ y vx →

∂vx Fx = ηA → ∂y

The lower plate is fixed, the upper one is dragged

at constant speed vx in the x-direction. The force required to drag it is proportional to the area of the The actual binding energy of the NaCl molecule is plates, the viscosity η of the liquid, and to the about 4.3 eV, but when the molecule is immersed gradient of the relative velocity in the direction in water and the Coulomb potential is reduced by normal to the plates. a factor 80, the minimum of the potential is now of the order of the thermal energy at room tem- Objects moving at small speeds through liquid experience a viscous drag force proportional to and perature (T ≈ 300 oK). That is, the molecule is opposing their velocity. The force has the form essentially no longer bound. → → F = −Γ a v , 3. Solubility of H2O in H2O where Γ is a geometric factor and a is the linear The title of this section sounds almost paradoxi- dimension. Thus for example, a sphere of radius r cal—how can something dissolve itself? The inter- experiences the drag force (Stokes’s Law) esting thing is that water does just that. The energy required to remove a hydrogen ion from a free F = −6πrηv . water molecule is of the order of several eV. If that removal energy were the same for a water molecule in the liquid phase as for a gas molecule, the We previously introduced the idea of Reynolds’s Boltzmann distribution would then predict that number: an object of volume Ω and linear dimen- the fraction of H+ ions in liquid water would be sion r, moving through liquid of density ρ, has to = Ω ρ −∆ ⁄ − − impart velocity v to a mass m in time e E kT ≈ e 80 ~ 10 35 . δt ≈ r ⁄ v. The inertial force it applies to the liquid However, liquid water has a pH of 7—meaning is thus − that the fraction of dissociated molecules is 10 7. v2 Hence we conclude the binding energy of the F ≈ Ω ρ . in r water molecule is very much less in the liquid environment than in the gas phase. Ergo, water On the other hand, the viscous force is dissolves water. = Γ η Fvisc v ; 4. Viscosity The ratio of inertial to viscous force (stripped of geometric factors) is called Reynolds’s number, The viscosity of a liquid is defined as follows: df 2 consider two parallel plates of area A with a film of Fin Ω ρ v ρ r v R = = → . 2 η the liquid between them, as shown below: Fvisc Γ r v η PHYS 304 Physics of the Human Body 75 Chapter 8 The properties of water

The viscosity of water in cgs units is about 0.01, As we have seen, the water molecule is triatomic. hence for a barracuda of length 100 cm swimming Its center of mass has 3 translational modes and at—say—10 m/sec (about 20 mi/hr) the Reynolds since it has 3 large principal moments of inertia, number is 107. Inertia dominates viscosity by an there are 3 rotational modes. Additionally water enormous factor. has 3 (normal) modes of internal vibration. Two of them will have the same frequency (because of the But for an E. coli bacterium of dimension 10-4 cm, symmetry) and the third will be much lower in swimming at 3×10-3 cm/sec, the Reynolds number frequency (because it involves a scissors motion of is 3×10-5. Here viscosity dominates inertia by a the hydrogen atoms, rather than the stretching of large factor. We saw that the coasting distance for the strong oxygen-hydrogen bonds). Quantum a bacterium that stops turning its propeller is mechanics says that vibrational modes cannot be 2 2 ρ r v excited until the temperature is high enough: x = 0 . stop 9 η − kT ≈ hω . The stopping distance is 0.07 Å—about 0.07 of an Moreover, from the perfect gas law we can deduce atomic radius! For practical purposes, when a bac- that the molar specific heat at constant pressure is terium stops swimming it stops dead in the water. the specific heat from the internal energy plus R. 5. Specific heat Thus a gas of water molecules (water vapor or The specific heat of water is defined by the heat steam) should have molar specific heat at constant required to raise the temperature of one gram by pressure increasing with temperature from 4R to 7R. In fact, from tables we find for water vapor in one degree Celsius. The heat required is the calorie, o whose mechanical equivalent is about 4.2 Joule. the temperature range 100–500 C ≈ Thus the molar specific heat of liquid water is cp 4.5R , 75.3 J, almost exactly 9R (R is the perfect gas o so we are on the right track: the internal energy constant, about 8.32 J/gm-mol/ C). involves a bit more than 6 degrees of freedom (that Is there any easy way to see why this is so? The Law is, a vibrational mode is partially excited). Below of Equipartition in thermodynamics says that the we see a plot of the specific heat of water vapor in average thermal energy of a particle (say, an atom) the temperature range 400–6000 oK: is

〈E〉 = 1 k T 2 B per “degree of freedom”, where Boltzmann’s con- df = ⁄ stant is kB R NA. Now we count degrees of freedom as follows: each translational motion (and there are three, in 3-dimensional space) counts as one; each rotational mode counts as one; and each vibrational mode counts as two (because the aver- age potential energy in a harmonic oscillator is the same as the average kinetic energy). Thus for a At the lowest temperature in the Figure the spe- monoatomic gas the average energy per molecule cific heat is 4.2R, rising asymptotically to 7.5R at the upper end of the range. Since is 3 kT and the molar specific heat is 3R . 2 2 76 Physics of the Human Body Heat of vaporization

at much higher temperatures, the internal vibra- ∂U c = + R tional modes are not excited. So how can they be P ∂T P excited in liquid water? The answer, once again, is we see that over this range the internal energy of the high dielectric constant of liquid water. As we a gas of water molecules can be written saw in our discussion of pH, the effective spring constants between the atoms in a given molecule U = 3 R + 3 R T + U (T) + U (T) 2 2 vib e are weakened 80-fold, leading to an almost 9-fold decrease in the temperature at which they can be ( ) where Uvib T runs from slightly more than 0, to excited. That is, rather than lying at several thou- 3R (when the temperature is well above the vibra- sand oK, the excitation temperature is depressed tional excitation energy). At the highest tempera- well below room temperature. ture, we clearly begin to excite internal electronic states of the molecule. The lowest such excitation 6. Heat of vaporization energy must be of the order of electron volts; since The molecules of liquid water interact fairly 1 eV/k = 12,000 oK, it should not surprise us that B strongly. Thus to break a molecule loose requires at 6000 oK such states begin to play a role in the considerable energy. For water this so-called heat thermodynamics1 of water vapor. of vaporization is about3 540 cal/gm, or about But what about liquid water? Here the volume 0.4 eV/molecule. remains (more-or-less) constant, and the specific We can use the heat of vaporization to estimate the heat per molecule is 9k , implying 18 degrees of B size of molecules. A molecule in a liquid interacts freedom. How can this be? only with its nearest neighbors—say 6 of them, so Suppose water were really a solid. Then according its average potential energy will be roughly to the law of Dulong and Petit we would expect— 〈V〉 ≈ −6B at sufficiently high temperature2—to find a spe- where B is the energy per bond. The energy of N cific heat of 3kB per atom, or 9kB per (triatomic) molecule. That is, this law would predict the molar molecules is therefore specific heat of 9R, which would imply that both N E ≈ −6NB = −6B Ω ≡ −6Bn Ω , the rotational and translational degrees of freedom vol Ω of a given molecule are constrained by springs— where n is the number-density (assumed constant) intermolecular chemical bonds—and additionally, and Ω is the volume. That is, the dominant con- that the 3 internal vibrational states were being tribution to the energy is proportional to the vol- excited by thermal motions. ume. However, this overcounts because the On the other hand, if the internal vibrational molecules at the surface have fewer neighbors than modes were not excited, the specific heat would be those in the interior, hence have fewer bonds. The only 6R. Now we have seen that for water vapor, number of surface molecules is

1. If the first elctron excitation has energy 2 eV, about 2% of the molecules will be excited at 6000 oK, raising the specific heat to 6.6R. 2. For many solids, “sufficiently high” is room temperature; however, for diamond, whose “springs” are stiffer than those of other materials, the specific heat at room temperature is substantially lower than that of other materials. 3. The heat of vaporization is temperture dependent. At 0 oC it is closer to 600 cal/gm. The average is 573 cal/gm. PHYS 304 Physics of the Human Body 77 Chapter 8 The properties of water

= τ But if the force exerted falls from fmax to 0 over the Nsurf nA distance τ, then we have where τ is the surface thickness and A the surface ∆E = 2aS ≈ 1 f τ . area. If we suppose the surface molecules have one 2 max fewer bond apiece, the correction is Now we may re-express this in terms of the heat df ≈ = τ = of vaporization and the volume per molecule: Esurf BNsurf Bn A S A f 4S |E | where we have defined the surface tension S as the Y = max = ≈ 2 vap max a τ 3 v correction to the total energy from the surface of the body of water. where v is the molar volume. For water this number is then (in dynes/cm2) Now, suppose we could break up the entire volume ≈ × × × 7 ⁄ 2 of water into individual molecules. Wvidently this Ymax 0.67 590 4.2 10 dyne cm would require energy proportional to the total 10 2 area: if the molecules are supposed little spheres of = 1.7×10 dyne ⁄ cm . radius r, the total surface area is A = N 4πr2 and the energy required is4 7. Vapor pressure When water and water vapor coexist in thermal ∆ = π 2 E SN 4 r . equilibrium at the same temperature, the number = If we let N NA and set the energy equal to the of molecules in the liquid, that in a given time vaporization energy per mole, we find randomly gain enough energy to escape, must be balanced by an equal number (on average) from Evap ≈ S 4πr2 . the gas that inelastically collide with the surface NA and are captured. Thus thermodynamics leads to The surface tension of water at 20 oC is about a relation between the partial pressure pw of water 73 erg/cm2 so we find a radius of about 2.8 Å. If we vapor in saturated air above the surface of a body had assumed cubic rather than spherical molecules of water, and the molar heat of vaporization, Evap : the cubes would have been about 4 Å on a side. − ⁄ = Evap RT pw p0 e . From the above considerations we can ascribe a 5 The constant p depends on such things as the tensile strength to a column of water . Imagine the 0 water in a long thin column, and now break the capture probability, which in turn depend on de- column. The total area increases by twice the tails of the interactions. That is, although p0 could cross-sectional area, a, of the column, requiring an in principle be computed from the laws of quantum energy input mechanics, in our present state of knowledge it remains an empirical constant. The correctness of ∆E = 2aS .

4. This area so greatly exceeds the area of a macroscopic contiguous body of fluid that we can ignore the energy associated with the latter. 5. In order for this to be either meaningful or measurable we must put the water in a capillary tube to sta- bilize it so the sides do not neck in (as they would for a free strand of water) causing the strand to break. 78 Physics of the Human Body Vapor pressure this law is seen from the semi-logarithmic plot below: