The properties of water“From water does all life begin.”—OC Bible, 457 Kalima.1. Static propertiesThe chemical formula of water is H2O. Its molecu-lar structure is roughly as shown below:The density of water, at standard temperature andpressure is, by definitionρ = 1 gm ⁄ cm3 = 103 kg ⁄ m3 .Since the molecular weight of water (ignoringisotopes like 18O and 2H) is 18, the number densityof molecules in liquid water isn = NA18 ≈ 3×1022 cm−3 .The spacing between the hydrogen and oxygenatoms in the molecule is aboutr ≈ 1 Å =df 10−8 cm .We may estimate the collision cross-section of awater molecule (with a point-like particle) as itsgeometric cross section:σ ≈ π r2 ;the cross section for a collision between two suchmolecules is therefore about four times as large,σH2O−H2O ≈ 10−15 cm2 .The mean free path of a molecule (that is, theaverage distance it can travel between collisions)is thereforeλ = 1n σ ≈ 2.5 Å .We can estimate the average inter-molecular spac-ing in liquid water by computing the volume permolecule (often called the specific volume):ω =df 1n = 4π3 r03giving r0 ≈ 2 Å , which means the inter-molecu-lar spacing is 2r0 = 4 Å . Thus the molecules inwater are not actually touching each other (whichis why it is a liquid rather than a solid) but they arevery close together. The mean free path is smallerthan the average inter-molecular spacing.2. Solvent propertiesWater has a very large dielectric constant, about80 (times the dielectric constant of vacuum). Thusthe Coulomb potential between—say—the atomsof the NaCl molecule (salt) is modified toVH2O = −e2ε rThe interatomic potential looks like an attractiveCoulomb potential at long distances, but has ashort-ranged repulsive part, so the total lookssomething like the figure below:PHYS 304 Physics of the Human Body 73Chapter 8 The properties of waterThe actual binding energy of the NaCl molecule isabout 4.3 eV, but when the molecule is immersedin water and the Coulomb potential is reduced bya factor 80, the minimum of the potential is nowof the order of the thermal energy at room tem-perature (T ≈ 300 oK). That is, the molecule isessentially no longer bound. 3. Solubility of H2O in H2OThe title of this section sounds almost paradoxi-cal—how can something dissolve itself? The inter-esting thing is that water does just that. The energyrequired to remove a hydrogen ion from a freewater molecule is of the order of several eV. If thatremoval energy were the same for a water moleculein the liquid phase as for a gas molecule, theBoltzmann distribution would then predict thatthe fraction of H+ ions in liquid water would bee−∆E ⁄ kT ≈ e−80 ~ 10−35 .However, liquid water has a pH of 7—meaningthat the fraction of dissociated molecules is 10−7.Hence we conclude the binding energy of thewater molecule is very much less in the liquidenvironment than in the gas phase. Ergo, waterdissolves water.4. ViscosityThe viscosity of a liquid is defined as follows:consider two parallel plates of area A with a film ofthe liquid between them, as shown below:The lower plate is fixed, the upper one is draggedat constant speed vx in the x-direction. The forcerequired to drag it is proportional to the area of theplates, the viscosity η of the liquid, and to thegradient of the relative velocity in the directionnormal to the plates.Objects moving at small speeds through liquidexperience a viscous drag force proportional to andopposing their velocity. The force has the formF→ = −Γ a v→ ,where Γ is a geometric factor and a is the lineardimension. Thus for example, a sphere of radius rexperiences the drag force (Stokes’s Law)F = −6πrηv .We previously introduced the idea of Reynolds’snumber: an object of volume Ω and linear dimen-sion r, moving through liquid of density ρ, has toimpart velocity v to a mass m = Ω ρ in timeδt ≈ r ⁄ v. The inertial force it applies to the liquidis thusFin ≈ Ω ρ v2r .On the other hand, the viscous force isFvisc = Γ η v ;The ratio of inertial to viscous force (stripped ofgeometric factors) is called Reynolds’s number,R =df FinFvisc = Ω ρ v2Γ r2 v η → ρ r vη .vx →Fx = ηA ∂vx∂y → y^74 Physics of the Human BodySolubility of H2O in H2OThe viscosity of water in cgs units is about 0.01,hence for a barracuda of length 100 cm swimmingat—say—10 m/sec (about 20 mi/hr) the Reynoldsnumber is 107. Inertia dominates viscosity by anenormous factor.But for an E. coli bacterium of dimension 10-4 cm,swimming at 3×10-3 cm/sec, the Reynolds numberis 3×10-5. Here viscosity dominates inertia by alarge factor. We saw that the coasting distance fora bacterium that stops turning its propeller isxstop = 29 ρ r2 v0η .The stopping distance is 0.07 Å—about 0.07 of anatomic radius! For practical purposes, when a bac-terium stops swimming it stops dead in the water.5. Specific heatThe specific heat of water is defined by the heatrequired to raise the temperature of one gram byone degree Celsius. The heat required is the calorie,whose mechanical equivalent is about 4.2 Joule.Thus the molar specific heat of liquid water is75.3 J, almost exactly 9R (R is the perfect gasconstant, about 8.32 J/gm-mol/oC).Is there any easy way to see why this is so? The Lawof Equipartition in thermodynamics says that theaverage thermal energy of a particle (say, an atom)is〈E〉 = 12 kBTper “degree of freedom”, where Boltzmann’s con-stant is kB =df R ⁄ NA. Now we count degrees offreedom as follows: each translational motion (andthere are three, in 3-dimensional space) counts asone; each rotational mode counts as one; and eachvibrational mode counts as two (because the aver-age potential energy in a harmonic oscillator is thesame as the average kinetic energy). Thus for amonoatomic gas the average energy per moleculeis 32 kT and the molar specific heat is 32R .As we have seen, the water molecule is triatomic.Its center of mass has 3 translational modes andsince it has 3 large principal moments of inertia,there are 3 rotational modes. Additionally waterhas 3 (normal) modes of internal vibration. Two ofthem will have the same frequency (because of thesymmetry) and the third will be much lower infrequency (because it involves a scissors motion ofthe hydrogen atoms, rather than the stretching ofthe strong oxygen-hydrogen bonds). Quantummechanics says that vibrational modes cannot beexcited until the temperature is high enough:kT ≈ h−ω