Chapter 2, page 1 2 Size, Mass and Kinetics of Molecules 2.1 The Methods Determining the Size 2.1.1 Gas Laws and Density The ideal gas law is hardly helpful in determining the size of molecules: p Vm = R T (2.01) The volume Vm is the molar volume. For n moles, we have Vm = V/n. The gas constant has the value R = 8,314510(70) J K−1mol−1. If we put into equ. (1) the pressure 0,1 MPa, the temperature 273,15 K and R, we get the molar volume of an ideal gas, which is considered a fundamental physical constant, and is 22,71108(19) L mol−1 (10−3m3 mol−1). The values 0,1 MPa and 273,15 K should not be fused with the corresponding values for the standard condition in the physical chemistry: Ä The standard condition convention of a substance in physical chemistry is the state of a pure substance at the normal pressure of 1 atm (101,33 kPa) and temperature of 25 °C (298,15 K), where gases should behave ideally. 23 If we divide the molar volume of an ideal gas by Avogadro’s number (NA = 6,0221367 × 10 mol−1), we get the volume occupied by a molecule, which is approximately 3,77 × 10−26 m3. The length of a side of this volume is about 3,35 nm for an ideal gas at 0 °C and 1 bar. Now we use the volume of a mol of liquid water instead (about 18 cm3), divide this by Avogadro’s number and calculate the third root, we get a "diameter" of about 3,1 Å, which is more realistic. Here we use the unit Å parallel to the unit nm (10 Å = 1 nm). Much more accurate results are obtained, if we calculate the molecular size from the 2 covolume b in the real gas equation (a/Vm is the internal pressure) ⎛ a ⎞ ⎜ p + ⎟ =− TRbV with b = 4 N V = 4V . (2.02) ⎜ 2 ⎟ ()m A particle mole particles ⎝ Vm ⎠ The right hand part of equ. (2.02) defines that for spherical molecules (atoms), the covolume b is equal to four times the internal volume of NA particles. By measuring the pV-diagram, and determining the parameter b by fitting the real gas equation, we get for water b = 30,49 × 10−6 m3 mol−1 or 12,66 × 10−30 m3 per molecule. That gives us a diameter of 2,89 Å. Some further examples: Molecule Covolume b/L mol−1 Covol.-diameter d/Å H2 0,0266 2,76 O2 0,0318 2,93 CO2 0,0427 3,24 C6H6 0,155 4,50 Molecular Physics © D. Freude Chapter Size, version October 2004 Chapter 2, page 2 2.1.2 Transport Phenomena (Diffusion, Heat Conduction, Viscosity) The three named phenomena can be mathematically described. We have: j = −C gradϕ and ϕ& = C Δϕ , (2.03) (Do you know the meaning of the Nabla-operator (grad) ∇ and Laplace-operator Δ?) Diffusion Heat Conduction Viscosity j particle flow heat flow shearing stress C diffusion coefficient D temperature conductivity λ/cvρ viscosity η/ρ ϕ concentration n temperature T flow velocity Equations (2.03) are Fick’s first and second law of diffusion. We can make statements about the mean free path using diffusion, heat conduction and viscosity: Consider the flow of one type of molecule through a surface. This type is flow is due primarily to particles which are within one mean free path length l from both sides of the surface. Molecules that reach the center surface per unit time 2 x0−l x0 x0+l (and per m ) come from both the right and left side. That is 1 1 ⎛ d n ⎞ from the right side: nx+ l v≈⎜ nx + l ⎟ v , (2.04) 6 0 6 ⎝ 0 d x ⎠ 1 1 ⎛ d n ⎞ and from the left side: nx− l v≈⎜ nx − l ⎟ v , (2.05) 6 0 6 ⎝ 0 d x ⎠ where 〈v〉 is the mean of the absolute value of the velocity of the particles, which unlike the concentration should not depend on x. The difference of these two currents is 1 d n l v . (2.06) 3 d x A comparison of equ. (2.06) with equ. (2.03) (replace C with D and ϕ with n) gives D = 〈v〉 l /3. (2.07) We can measure D, and 〈v〉 can be estimated with the aid of kinetic gas theory. Maxwell’s distribution function for the mean square of the velocity is a function of the density ρ and pressure p: 2 3p v= v2 = . (2.08) ρ Molecular Physics © D. Freude Chapter Size, version October 2004 Chapter 2, page 3 Similar considerations to mass transfer lead for the heat conductivity λ to 1 CV 2 λ = N lv , (2.09) 3 N A 3 where N is the number of molecules per m , NA is Avogadro’s number, and CV is the specific heat at constant volume. The inner viscosity η of the molecules is pρ η = l . (2.10) 3 We can thus make statements about the mean free path from the measured parameters of transport phenomena. If we set this length equal to the size of the molecule, we run into similar problems as with the use of the ideal gas law. We will derive a satisfactory correlation in the following chapter. 2.1.3 Scattering Cross Section Experimental setup gas 2 x gas 1 heater diaphragm detector pump For the sake of simplicity, we will consider the identical particles (atoms or molecules) in the oven and scattering chamber, and assume that the concentration in the scattering chamber is so small, that no particles in the scattering direction are aligned. The probability W that a collision occurs is thus W = sum of all scattering cross sections in the flight path . total area A The scattering cross section σ in a collision of particle type 1 with type 2 would be 2 2 (r1 + r2) π, for identical particles it is σ = 4r π. The sum of all scattering cross sections in the flight path is given by the product of the scattering cross section σ times the particle density per unit volume n time the surface area A time the length of the scattering chamber Δx. Molecular Physics © D. Freude Chapter Size, version October 2004 Chapter 2, page 4 If N particles pass the point x in the scattering chamber, WN particles are scattered out of the beam. We have σ n AΔ x ΔN = − N . (2.11) A Instead of a scattering chamber of length Δx let us consider many scattering chambers in series, each of the infinitesimal length dx, whose total length is L. From equ. (2.11), we get d N = −nσ d x , (2.12) N and integration leads us to ln N = −n σ x + ln N0, (2.13) where N0 is the number of particles entering the chamber at x = 0. The measurement of the particle concentration at the detector gives: −nσL N = N0 e . (2.14) nσ ≡ α is called the macroscopic scattering coefficient and σ the microscopic total scattering cross section. The direct determination of the scattering cross section in gases is often replaced by the measurement of the mean free path l, which we can define with equ. (2.14) as the path L, in which N has been reduced from N0 to N0/e. From equ. (2.14) we can also see that l is reciprocal of the macroscopic scattering coefficient: 1 1 1 l = = = . (2.15) σ n4π r2 nπ d2 n Without proof we state that the consideration of the motion of the scattered molecules gives an extra factor of √2 in the denominator of equ. (2.15). Now we can determine l by the method mentioned in the previous chapter 2.1.2. We know the value n and obtain the size d of the molecule by means of equ. (2.15). Gas kinetic diameters obtained from the mean free path l are given below and compared with the values obtained from the covolume, cf. chapter 2.1.1: Molecule Kinetic diameter d/Å Covol.-diameter d/Å H2 2,3 2,76 O2 3,0 2,93 CO2 3,4 3,24 Molecular Physics © D. Freude Chapter Size, version October 2004 Chapter 2, page 5 2.1.4 X-ray, Electron, and Neutron Diffraction The distances between atoms or molecules in solids can be measured using diffraction (interference) of radiation. Single crystals are preferable, but the methods also work with powdered samples of crystalline substances. The wave length of the diffracted light should be of the same order of magnitude as the distance between the atoms, about 1 Å. That is true for X-rays, slow neutrons (4 km s−1) and electrons accelerated with 50 kV. Examinations of molecular crystals help us understand biological structures. In "X-ray amorphous" solids, for example glass, or in liquids the information we can obtain is limited to the short range order. Structural examinations using X-ray beams and slow neutrons have found a wide range of applications, but are not the subject of this chapter. The size of the molecules can be obtained from the distances between the atoms of a molecule by analyzing the distribution of the scattering intensities in the electron diffraction diagram, cf. figure on the left taken from Haken and Wolf. A complete structural analysis including the X-H-distances (X = C, N, O, P, etc.) can be obtained from the neutron diffraction.