Molecular interactions in beas notable advanceent in the experiental study of interolecular forces has coe fro the developent of olecular beas, which consist of a narrow bea of particles, all having the sae velocity, traveling through an evacuated vessel. The bea is directed towards other olecules, and the scattering of olecules that occurs on ipact is used to study interolecular interactions. The basic arrangeent for a olecular bea experient is shown in the figure. The slotted disks ake up the velocity selector. They rotate in the path of the bea and allow only those olecules having a certain velocity to pass through. ny desired velocity can be chosen by altering the speed of rotation. The target gas ay be enclosed in a container, or ay be in the for of another olecular bea in a perpendicular direction. The latter is called crossed bea technique and gives a lot of useful inforation because the velocities of both target and projectile olecules can be controlled. The intensity of the incident bea is easured by the incident bea flux, I, which is the nuber of particles per unit area per unit tie. The detectors ay consist of a chaber fitted with a sensitive pressure gauge or an ioniation detector, in which the incoing olecule is first ionied and then detected electronically. The state of the scattered olecules ay also be deterined spectroscopically, and is of interest when the collisions change their rotational and vibrational states. The priary experiental inforation fro a olecular bea experient is the fraction of olecules in the incident bea that are scattered in a particular direction (angle ). The fraction is norally expressed in ters of di, the nuber of olecules that are scattered per unit tie into a cone that represents the area covered by the eye of the detector. This nuber is reported as the differential scattering cross section σ, and is the constant of proportionality between the value of di and the intensity of the incident bea I, the nuber density of target olecules N and the infinitesial path length dx through the saple. di = σ I N dx The differential scattering cross section σ has the diensions of area. Its value depends upon the ipact paraeter b, which is the initial perpendicular separation of the paths of the colliding olecules, and the details of the interolecular potential. The role of the ipact paraeter is ost easily seen by considering
the ipact of two hard spheres. If b = 0, the lighter projectile is on a trajectory that leads to head-on collision, so that the only scattering intensity is detected when the detector is at = 80. If the ipact paraeter is so large (b > R +R B, where R and R B are the radii of the two spheres respectively) that the spheres do not ake contact, there is no scattering and the scattering cross section is ero at all angles except = 0. Glancing blows with 0 < b < R +R B lead to scattering intensity in cones around the initial line of flight direction. Scattering effects: The scattering pattern of real olecules, which are not hard spheres, depends on the details of the interolecular potential, including the anisotropy that is present when the olecules are non-spherical. The scattering also depends on the relative speed of approach of the two particles: a very fast particle ight pass through the interaction region without uch deflection, whereas a slower one on the sae path ight be teporarily captured and undergo considerable deflection. The variation in the scattering cross-section with the relative speed of approach should therefore give inforation about the strength and range of the interolecular potential. further point is that the outcoe of collisions is deterined by quantu, not classical, echanics. The wave nature of the particles can be taken into account, at least to soe extent, by drawing all classical trajectories that take the projectile particle fro source to detector, and then considering the effects of interference between the. The second effect is a strongly enhanced scattering in a nonforward direction. This effect is called rainbow scattering because the sae echanis accounts for the appearance of an optical rainbow. s the ipact paraeter decreases, there coes a stage when the scattering angle passes through a axiu and the interference between the paths results in a strongly scattered bea. The rainbow angle is the angle for which dθ/db = 0 and the scattering is strong. The detailed analysis of scattering data can be very coplicated, but the ain outcoe is clear: the intensity distribution of the scattered particles can be related to the interolecular potential, and a detailed picture can be built up of its radial and angular variation. Testing of the effects of van der Waals interaction and Lennard-Jones potential becoes possible. Two quantu echanical effects are of great iportance. particle with a certain ipact paraeter ight approach the attractive region of the potential in such a way that it is deflected towards the repulsive core, which then repels it out through the attractive region to continue its flight in the forward direction. Soe olecules, however, also travel in the forward direction because they have ipact paraeters so large that they are undeflected. The wave functions of the particles that take the two types of path interfere, and the intensity in the forward direction is odified. This effect is called glory scattering. [To see this effect with light rays, look at a bright oon in a sky with thinly dispersed clouds. bright halo is seen around the oon.] MOLECULR COLLISION THEORY & GS-PHSE RECTION DYNMICS ssuptions of the kinetic theory of gases and rates of gas-phase reactions (cheical kinetics):. Gases consist of particles oving in all directions with velocities ranging fro ero to very high values.. The velocities of the particles at any given teperature follow a Gaussian (or Maxwell-Boltann) distribution. 3. The ean velocity of the particles is directly proportional to the teperature. 4. The particles collide with each other and with the walls of the vessel during their rando otion. 5. These collisions are perfectly elastic (the total oentu is conserved). 6. Reactions take place when olecules collide with each other. Therefore rate of a reaction ust be proportional to the collision frequency. 7. The rates of reactions increase with teperature since the velocity of the olecules and therefore the collision frequency increases.
8. ll collisions do not lead to reaction. The colliding olecules ust have a certain iniu energy called activation energy for successful reaction. Dependence of the reaction rate on activation energy and teperature are given by the epirical rrhenius equation: d ln k dt or ln k ln or k e The rates of reactions calculated on the basis of the aforesaid assuptions give very high values copared to the experientally obtained values. Therefore odifications are necessary taking into account the following factors also:. Collisions between real olecules are not perfectly elastic.. ll collisions with the right activation energy cannot be fruitful. The orientations of the olecules at the tie of collision are also iportant. Till now, exaination of kinetics at a olecular level was not possible, and all the above were only intelligent assuptions about the nature of cheical reactions. However, with the developent the olecular bea apparatus in the 950s by Kusch and coworkers at the Colubia University, narrow beas of olecules having a fixed velocity could be obtained (for description and diagra of the apparatus, see earlier discussion on olecular interaction in beas). Using this, the actual nuber distribution of olecules having different velocities could be obtained and the Maxwell-Boltann distribution verified. The other assuptions of the kinetc theory and rates of reactions could also be studied and verified at a olecular level. The ensuing discussion is an introduction to such studies. The relationship of ean free path and collision frequency to collision cross section: Iagine the olecules to be hard spheres (so that collisions are elastic) of diaeter d. For the tie being, let us also iagine that only the olecule under consideration is oving while all the others are stationary. Then, when our olecule oves along a straight line, it will collide with all other olecules that lie within a cylinder of diaeter d surrounding its path. The area of cross section of this collision cylinder is the collision cross section, σ. The area σ = πd. If we are considering the otion of our olecule with a velocity u over an infinitesial tie interval dt, then the length of the collision cylinder is u.dt. The volue of the cylinder is σ.u.dt. If the nuber density of olecules (ie. nuber of olecule per unit volue) is ρ, then the nuber of collisions dn coll in the tie interval dt will be dn coll = ρ.σ.u.dt Therefore the collision frequency (ie. nuber of collisions per second) is given by dn coll 8k B T u dt where k B is the Boltann constant, T the teperature and the ass of the particles. If all collisions lead to reactions, then the collision frequency should be equal to the rate of reaction. In the above treatent, we have considered only one particle as oving while all others were fixed. If two different particles of ass and are oving relative to each other, then the velocity of one relative to the other can be obtained by considering the as one particle with a reduced ass µ = /( + ) oving with respect to the other being fixed. If both particles are of the sae ass, then µ = / and it can be shown that the relative velocity u r = / u. Therefore: 3 /
/ u u () r Since the velocity (distance traveled per second) is u and the nuber of collisions per second is, the ean free path l (distance between collisions) is given by: l u / u u / Calculation of the rate of a biolecular gas-phase reaction using hard-sphere collision theory and an energy-dependent reaction cross section Biolecular gas-phase reactions are aong the siplest eleentary kinetic processes that occur in nature. The siplest gas-phase reaction is the hydrogen exchange reaction: nother reaction ore convenient to study is: H + H B -H C H -H B + H C F(g) + D (g) DF(g) + D(g) In the following discussions, we will exaine soe of the current odels used in the study of such reactions. The siple general reaction we shall consider is: (g) + B(g) products d[ ] The rate of this reaction is given by: k[ ][ B] () dt ccording to the earlier discussion of hard sphere collisions, this should be equal to the collision frequency. Therefore using equation (), we get v = Z B = σ B.u r.ρ.ρ B (3) where Z B is the frequency of collision between and B olecules, σ B is the collision cross section and ρ, ρ B are the nuber densities of and B olecules. The collision cross section is given by: σ B = (d B ) where d B is the su of the radii of the two colliding spheres. The collision frequency Z B has units of collisions. -3.s -, where the unit collisions is usually oitted. Z B also gives the nuber of product olecules fored per unit volue per unit tie, which is the rate constant. Coparing equations () and (3), we get: k = σ B.u r (since [] = ρ and [B] = ρ B ) (4) The rate constant k has units of Z B /ρ.ρ B or olecules. -3.s - /(olecules. -3 ) or olecule -. 3.s -. To convert into the failiar units of d 3.ol.s - ( ole = N olecules where N is the vogadro nuber and 3 = 000 litres), k has to be divided by 000 N. Therefore: k = 000.N.σ B.u r d 3.ol.s -. The rate constants calculated by the above hard-sphere collision theory equation are found to be about 30 orders of agnitude larger than experientally observed rate constants! Further, since u r T /, this theory predicts that k should show a teperature dependence of T /, whereas rrhenius equation and experiental easureents generally show that k is exponentially dependent on / T. Therefore the above theory requires odification. 4
Effect of collision energy We assued that each pair of reactant olecule approach with a relative speed of u r, but actually they approach each other with a distribution of speeds. s two olecules collide, the valence electrons of the two olecules repel each other so that no reaction can occur unless the relative speed is sufficient to overcoe the repulsive force. The first step in the odification to take the speed dependence into account is to introduce a reaction cross section σ r (u r ) which depends on the speed of reactant olecules in place of the earlier collision cross section σ B. The rate constant for all olecules colliding with a relative speed of u r is given by an equation siilar to equation (4). k(u r ) = u r.σ r (u r ) (5) To obtain the calculated rate constant k, we ust average the right hand side of the above equation over all possible collision speeds ie. integrate the equation within liits 0 to. Further, to copare the result with the traditional rrhenius for of k, we have to change the dependent variable fro u r to E r, the relative kinetic energy. E r = ½ µ.u r u r = (E r /µ) / Then the energy dependence of the reaction cross section is represented by σ r (E r ) such that σ r (E r ) = 0 when E r < E 0 and σ r (E r ) = π(d B ) when E r > E 0, where E 0 is the threshold energy to overcoe repulsion. Introducing these changes and avoiding the involved atheatics, we get an expression for k of the for: k E E 0 / kbt u e 0 r. B. (6) k BT where σ B is the hard-sphere collision cross section. The rate constant calculated using this expression gives a value a few orders of agnitude higher than the experiental value. Therefore the theory has to be further odified. Different odels for σ r (E r ) ust now be tested which will give different expressions for k. The validity of any odel has to be tested experientally. Effect of collision geoetry The siple energy-dependent reaction cross section used to derive equation (6) is not realistic. To see why, let us consider the following reaction geoetries. The line joining the centres of the two colliding spheres is called the line of centres (loc). In the first case, the olecules approach along the line of centeres and will coe to a stop after a head-on collision. The full energy of the ipact will be used for reaction. But in the second case, the olecules are not approaching along the line of centres, and will undergo only a graing collision. Most of the kinetic energy of the olecules still reains with the as they fly apart. Only that coponent of the kinetic energy which lies along the line of centres will be useful for reaction. Thus a reasonable odel for the energy-dependent reaction cross section σ r (E r ) ust depend on the coponent of the kinetic energy which lies along the line of centres. This is called the line-of-centres odel for σ r (E r ). If we denote the relative kinetic energy along the line of centeres by E loc, then we are assuing that the reaction occurs when E loc > E 0. The reaction cross section ust depend on the ipact paraeter b, which is the perpendicular distance between the lines of otion of the two spheres. If b = 0, the olecules hit head-on, and if b > d B (d B = R + R B ), the olecules will pass on without colliding and the collision cross section ust be equal to ero. The derivation is a bit geoetrically involved and is not discussed here. But the result is such that σ r (E r ) = 0 when E r < E 0 and E σ r (E r ) = π(d B ) 0 when Er > E 0. Then the rate equation for the Er biolecular gas-phase reaction becoes: k E / kbt ur. B. e 0 (7) 5
E which differs fro equation (6) by a factor of 0. But the results calculated using equation (7) are still kbt higher than the experiental values by several orders of agnitude. Therefore the theory requires further odifications. Effect of the orientation of colliding olecules Results show that the hard-sphere collision theory does not accurately account for the agnitude of the rrhenius factor. One of the fundaental flaws of this odel is the assuption that every collision of sufficient energy is reactive. In addition to an energy requireent, the reacting olecules ay need to collide with a specific orientation for the cheical reaction to occur. For exaple, consider the reaction: Rb(g) + CH 3 I(g) RbI(g) + CH 3 (g) Experiental studies reveal that this reaction occurs only when the rubidiu ato collides with the iodoethane olecule in the vicinity of the iodine ato. Collisions between rubidiu and the ethyl end of the olecule do not lead to reaction. This is indicated by the cone of nonreactivity in the figure. Since the earlier odel does not take the effect of olecular orientation into account, it ust overestiate the rate constants for reactions that are orientation dependent. However, this steric factor alone cannot account for the significant differences observed between the experiental and calculated rrhenius factors. Therefore the theory requires still further odification. Effect of internal energy of the reactants Reaction cross sections for hydrogen olecular ion with atoic heliu are plotted in the figure. H + (g) + He(g) HeH + (g) + H(g) ch curve corresponds to the reactant H + in a specific vibrational state. For vibrational states v = 0 to v = 3, there is a threshold energy of about 70 kj.ol -. This is because for H + olecules with vibrational quantu nuber v = 0 to v = 3, the total vibratonal energy is less than E 0. dditional translational energy is required to initiate reaction in these states. Molecules having v > 3 do not require additional translational energy sice their vibrational energy is greater than E 0. The internal energy of a olecule is distributed aong the discrete rotational, vibrational and electronic states. Data such as in the figure tell us that cheical energy depends not only on the total energy of the reacting olecules but also on how that energy is distributed aong these internal levels. Siple hard-sphere collision theory considers only the translational energy of the olecules. Energy can also be exchanged between the different degrees of freedo during the reactive collisions. For exaple, vibrational energy can be converted to translational energy and vice versa. 6
reactive collision can be described by a centre-of-ass coordinate syste Consider the reaction (g) + B(g) C(g) + D(g). For siplicity we will assue there are no interolecular forces. Before collision the olecules and B are traveling with velocities u and u B respectively. The collision generates products C and D oving away with velocities u C and u D respectively. We will describe the collision process in the centre-of-ass coordinate syste. The idea is to view the collision fro the centre of ass of the two colliding olecules. The centre of ass lies along the vector r = r r B that connects the centres of the two colliding olecules. The location of R, the centre of ass, on this vector depends on the asses of the two olecules and is defined by: R r Br B B and u CM u Bu B B where u CM is the velocity of the centre of ass. Then it can be shown that the total kinetic energy of the reactants consist of two coponents, one depending on the velocity of the centre of ass, and the other on the relative velocity of the olecules. KE react MuCM u r where M is the total ass ( + B ) and µ is the reduced ass B /( + B ). lternative treatent of gas-phase reaction dynaics: The sybols used in the equations and the general approach given in tkins, Physical Cheistry 6 th Edn., is slightly different, but gives the sae overall results. This treatent is discussed below: In this treatent, the sybol c rel is used to represent relative velocity of the colliding olecules instead of u r in the earlier discussion. Nuber density (nuber of olecules per litre) of olecules of is given by N [], where N is the vogadro nuber and [] the nuber of oles of per litre. The collision frequency, (which is the nuber of collisions encountered by a single olecule in unit tie) is then given by: c N [] (8) rel which is siilar to equation () in the earlier discussion. But 8kT c ; c rel c and = ½ for a single type of olecule. π Substituting in equation (8) we get collision frequency for one type of olecules as: 7
8kT c rel N [ ] N [ ] (9) Then the total collision density Z (which is the total nuber of collisions encountered by all the olecules together) is obtained by ultiplying the collision frequency with the nuber density of olecules: Z 8kT N [ ] N [ ] N [ ] The factor ½ is introduced to account for the fact that a single collision of olecule with another ay be counted as two different collisions (first with second, and also as second with first ). Thus: Z 8kT crel N [ ] N [ ] (0) or if the ass of is directly used, Z 4kT N [ ] () If collisions are between two different types of olecules and B, then the factor ½ is not necessary, and the collision density is given by: 8kT Z B crel N [ ][ B] N [ ][ B] () Collision densities ay be very large. For exaple, in nitrogen at roo teperature and pressure, with d = 80 p, Z = 50 34 3 s. Using the sae arguents for the influence of activation energy E a, and coparing with equation (7), k 8kT c rel N [ ] e N [ ] e (3) Correction for all other factors influencing the reaction, such as orientation, internal energy of olecules etc. are introduced through a steric factor P, so that the reactive cross section * = P. Thus, k 8kT P c rel N [ ] e P N [ ] e (4) Coparing with the epirical rrhenius equation k e where is called the pre-exponential factor, the preexponential factor as given by olecular dynaical calculations is 8kT P c rel N [ ] or P N [ ] (5) 8
Collision frequency when pressure of the gas is given: For equation (8), c rel N [], or c rel N for ole. Since nuber density of olecules (nuber of olecules per unit volue) is N, and V V, P nuber density = PN P kt. Therefore, P c rel (6) kt In dierisation of ethyl radicals at 5C, the experiental pre-exponential factor is.4 0 0 L ol - s -. What are (a) the reactive cross section and (b) the p-factor for the reaction, if C H bond length is 54 p. {pril 005} [Ref: () Physical Cheistry, tkins, 6 th edn, page 846, proble 7.; () valuation schee for this exa.] d = 54 p = 308 p = 308 0. (a) Collision cross section = d = 3.4 (308 0 ) = 9787.96 0 4 =.98 0 9 -E (b) ccording to rrhenius theory, rate constant a / k e, where is the collision frequency factor, or the pre-exponential factor. ccording to reaction dynaics predictions, σ c rel N 8kT d N, where is the reduced ass; = 5 3-6 Mass of one ethyl radical = 0 kg 3 6.030-6 -6.490 kg.490 kg µ.450-6 -6.490 kg.490 kg -6 kg 3.4 3080 = - -4 6.030 3.490-3 8.380 JK 3.4.450 3-6 kg 98K kg 3.4 948640 6.030 97.39 s = 6.458 0 7 3 ol s. But given that the experiental value of =.4 0 0 L ol - s - =.4 0 7 3 ol - s - ;. P = steric factor = = 0.458. experiental value of calculated value of.40 7 6.4580 7 3 ol 3 - ol s - - s - Then in (a), Reaction cross section * = P =.98 0 9 0.458 = 0.4345 0 9. 9
Calculate the collision frequency and the collision density Z in CO. R = 80 p at 5C and 00 kpa. What is the percentage increase when the teperature is raised by 0 K at constant volue? {pril 005} [Ref: () Physical Cheistry, tkins, 6 th edn, page 844, Exercise 7.(b); () valuation schee for this exa.] Collission frequency = = Therefore = = σ c rel P, where P is the pressure and kt for a single type of olecule; c rel 8kT 8 3 0 kg =.344 0 6 kg. and = d. 3 6.030 = = σ c rel P kt =.3440 8 ( kt ) P 8 d = d P kt kt 8 3.4 5 (800 ) 0 kg s -6 3 kg.380 kg s K 98K = 6.6433 0 9 s. N N P Collision density Z = ; But ; V V kt Therefore: 5 9 P 0 kg s 6.6433 0 s Z = = 3 kt.38 0 kg s K 98K = 8.077 0 34 3 s. d d dc Percent increase in collision frequency = 00 ; But dt c dt T d dt 0K Hence 0.067 ; Therefore percentage increase =.67% T 98K 0