Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Transition state theory wikipedia , lookup
Bose–Einstein condensate wikipedia , lookup
Heat transfer physics wikipedia , lookup
Particle-size distribution wikipedia , lookup
Rotational spectroscopy wikipedia , lookup
State of matter wikipedia , lookup
Physical organic chemistry wikipedia , lookup
Gibbs paradox wikipedia , lookup
Lecture 2/3. Kinetic Molecular Theory of Ideal Gases Last Lecture …. IGL is a purely empirical law - solely the consequence of experimental observations Explains the behavior of gases over a limited range of conditions. IGL provides a macroscopic explanation. Says nothing about the microscopic behavior of the atoms or molecules that make up the gas. 1 Today ….the Kinetic Molecular Theory (KMT) of gases. KMTG starts with a set of assumptions about the microscopic behavior of matter at the atomic level. KMTG Supposes that the constituent particles (atoms) of the gas obey the laws of classical physics. Accounts for the random behavior of the particles with statistics, thereby establishing a new branch of physics statistical mechanics. Offers an explanation of the macroscopic behavior of gases. Predicts experimental phenomena that suggest new experimental work (MaxwellBoltzmann Speed Distribution). Kotz, Section 11.6, pp.532-537 Chemistry3, 1st edition: Section 7.4, pp.316-319; Section 7.5, pp.319-323. 2nd edition: Section 8.4, pp.354-357, section 8.5, pp. 358-368. Kinetic Molecular Theory (KMT) of Ideal Gas • • • • • • • Gas sample composed of a large number of molecules (> 1023) in continuous random motion. Distance between molecules large compared with molecular size, i.e. gas is dilute. Gas molecules represented as point masses: hence are of very small volume so volume of an individual gas molecule can be neglected. Intermolecular forces (both attractive and repulsive) are neglected. Molecules do not influence one another except during collisions. Hence the potential energy of the gas molecules is neglected and we only consider the kinetic energy (that arising from molecular motion) of the molecules. Intermolecular collisions and collisions with the container walls are assumed to be elastic. The dynamic behaviour of gas molecules may be described in terms of classical Newtonian mechanics. The average kinetic energy of the molecules is proportional to the absolute temperature of the gas. This statement in fact serves as a definition of temperature. At any given temperature the molecules of all gases have the same average kinetic energy. Air at normal conditions: ~ 2.7x1019 molecules in 1 cm3 of air Size of the molecules ~ (2-3)x10-10 m, Distance between the molecules ~ 3x10-9 m The average speed - 500 m/s The mean free path - 10-7 m (0.1 micron) The number of collisions in 1 second - 5x109 2 Random trajectory of individual gas molecule. Assembly of ca. 1023 gas molecules Exhibit distribution of speeds. Gas pressure derived from KMT analysis. Pressure The pressure of a gas can be explained by KMT as arising from the force exerted by gas molecules impacting on the walls of a container (assumed to be a cube of side length L and hence of Volume L3). We consider a gas of N molecules each of mass m contained in cube of volume V = L3. When gas molecule collides (with speed vx) with wall of the container perpendicular to x co-ordinate axis and bounces off in the opposite direction with the same speed (an elastic collision) then the momentum lost by the particle and gained by the wall is px. p x mvx ( mvx ) 2mvx t 2L vx y The force F due to the particle can then be computed as the rate of change of momentum wrt time (Newtons Second Law). F L z The particle impacts the wall once every 2L/vx time units. Derivation: Box 8.2 pp.356-357. x vx -vx p 2mvx mv x2 t 2 L v x L 3 Force acting on the wall from all N molecules can be computed by summing forces arising from each individual molecule j. F m N 2 vx, j L j 1 The magnitude of the velocity v of any particle j can also be calculated from the relevant velocity components vx, vy, and vz. v 2 v x2 v 2y v z2 The total force F acting on all six walls can therefore be computed by adding the contributions from each direction. F 2 N N m N 2 m N 2 m N 2 2 2 2 2 v x , j v y , j v z , j 2 v x , j v y , j v z , j 2 v j L j 1 L j 1 L j 1 j 1 j 1 Assuming that a large number of particles are moving randomly then the force on each of the walls will be approximately the same. F 1 m N 2 1 m N 2 2 v j vj 6 L j 1 3 L j 1 2 v 2 vrms The force can also be expressed in terms of the average velocity v2rms where vrms denotes the root mean square velocity of the collection of particles. F 1 N N v j 1 2 j 2 Nmvrms 3L The pressure can be readily determined once the force is known using the definition P = F/A where A denotes the area of the wall over which the force is exerted. P 2 2 F Nmvrms Nmvrms A 3 AL 3V AL V The fundamental KMT result for the gas pressure P can then be stated in a number of equivalent ways involving the gas density , the amount n and the molar mass M. PV 1 2 v 2 1 M 2 1 N 2 1 2 2 Nmvrms Mvrms nMvrms rms N vrms 3 3 2 3 NA 3 NA 3 m Avogadro Number = 6 x 1023 mol-1 Nm V M NA Using the KMT result and the IGEOS we can derive a Fundamental expression for the root mean square Velocity vrms of a gas molecule. PV nRT IGEOS KMT result PV 1 1 2 2 Nmvrms nMvrms 3 3 1 2 nMvrms nRT 3 2 Mvrms 3RT vrms 3RT M 4 Gas 103 M/kg mol-1 Vrms/ms-1 H2 2.0158 1930 H2O 18.0158 640 N2 28.02 515 O2 32.00 480 CO2 44.01 410 Internal energy of an ideal gas We now derive two important results. The first is that the gas pressure P is proportional to the average kinetic energy of the gas molecules. The second is that the internal energy U of the gas, i.e. the mean kinetic energy of translation (motion) of the molecules is directly proportional to the temperature T of the gas. This serves as the molecular definition of temperature. Average kinetic Energy of gas molecule E P Nmv 3V 2 rms 2N 1 2 2N E mvrms 3V 2 3V Boltzmann Constant n P nRT NRT Nk BT V N AV V 1 2 mvrms 2 N NA kB R 8.314 J mol1 K 1 1.38x1023J K 1 6.02x1023 mol1 NA nR Nk B Nk BT 2N E V 3V 3 E k BT 2 3 3 U N E Nk BT nRT 2 2 5 Maxwell-Boltzmann velocity distribution • • • • In a real gas sample at a given temperature T, all molecules do not travel at the same speed. Some move more rapidly than others. We can ask : what is the distribution (spread) of molecular velocities in a gas sample ? In a real gas the speeds of individual molecules span wide ranges with constant collisions continually changing the molecular speeds. Maxwell and independently Boltzmann analysed the molecular speed distribution (and hence energy distribution) in an ideal gas, and derived a mathematical expression for the speed (or energy) distribution f(v) and f(E). This formula enables one to calculate various statistically relevant quantities such as the average velocity (and hence energy) of a gas sample, the rms velocity, and the most probable velocity of a molecule in a gas sample at a given temperature T. James Maxwell 1831-1879 3/ 2 m F (v) 4 v 2 2 k BT F (E) 2 E k BT 3 mv 2 exp 2k BT E exp k BT http://en.wikipedia.org/wiki/Maxwell_speed_distribution http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution Maxwell-Boltzmann velocity Distribution function m F (v) 4 v 2 k BT 3/ 2 2 m = particle mass (kg) kB = Boltzmann constant = 1.38 x 10-23 J K-1 Gas molecules exhibit a spread or distribution of speeds. F (v) mv 2 exp 2 k BT Ludwig Boltzmann 1844-1906 • The velocity distribution curve has a very characteristic shape. • A small fraction of molecules move with very low speeds, a small fraction move with very high speeds, and the vast majority of molecules move at intermediate speeds. • The bell shaped curve is called a Gaussian curve and the molecular speeds in an ideal gas sample are Gaussian distributed. • The shape of the Gaussian distribution curve changes as the temperature is raised. • The maximum of the curve shifts to higher speeds with increasing temperature, and the curve becomes broader as the temperature increases. • A greater proportion of the gas molecules have high speeds at high temperature than at low temperature. v 6 Properties of the Maxwell-Boltzmann Speed Distribution. Maxwell Boltzmann (MB) Velocity Distribution vmax = vP 0.0025 T = 300 K T = 400 K T = 500 K T = 600 K T = 700 K T = 800 K T = 900 K T = 1000K 0.0020 Ar M = 39.95 kg mol-1 F(v) 0.0015 0.0010 vP (300K) = 353.36 ms-1 vrms (300K) = 432.78 ms-1 ‹v› (300K) = 398.74 ms-1 0.0005 0.0000 0 200 400 600 800 1000 v / ms-1 Features to note: The most probable speed is at the peak of the curve. The most probable speed increases as the temperature increases. The distribution broadens as the temperature increases. 1200 1400 1600 1800 Relative mean speed (speed at which one molecule approaches another. vrel 2 v 7 MB Velocity Distribution Curves : Effect of Molar Mass T = 300 K 0.005 He Ne Ar Xe 0.004 M = 4.0 kg/mol M = 20.18 kg/mol M = 39.95 kg/mol M = 131.29 kg/mol F(v) 0.003 0.002 0.001 0.000 0 500 1000 1500 2000 v / ms-1 Determining useful statistical quantities from MB Distribution function. Average velocity of a gas molecule v v F v dv MB distribution of velocities enables us to statistically estimate the spread of molecular velocities in a gas 0 m F (v) 4 v 2 2 k BT 3/ 2 mv 2 exp 2 k BT Maxwell-Boltzmann velocity Distribution function Some maths ! v Most probable speed, vmax or vP derived from differentiating the MB distribution function and setting the result equal to zero, i.e. v = vmax when dF(v)/dv = 0. vmax vP Derivation of these formulae Requires knowledge of Gaussian Integrals. 8k BT 8 RT m M Mass of molecule Molar mass vP v vrms 2 k BT 2 RT m M Root mean square speed Yet more maths! 1/ 2 vrms v 2 F (v)dv 0 3k BT 3RT m M 8 vP v vrms Gas 103 M/ kg mol-1 vrms/ms-1 ‹v›/ms-1 Vrel/ms-1 vP/ms-1 H2 2.0158 1930 1775 2510 1570 H2 O 18.0158 640 594 840 526 N2 28.02 515 476 673 421 O2 32.00 480 446 630 389 CO2 44.01 410 380 537 332 2200 Typical molecular velocities Extracted from MB distribution At 300 K for common gases. 2000 H2 1800 1400 v 1200 1000 8k BT 8 RT m M H2O 800 N2 O2 600 vmax vP CO2 400 2 k BT 2 RT m M 200 0 10 20 30 3 10 M/kg mol 40 50 -1 vrms vrel 2 v 3k BT 3RT m M Maxwell Boltzmann Energy Distribution 0.00025 T = 300 K T = 400 K T = 500 K T = 600 K T = 700 K T = 800 K T = 900 K T = 1000 K Ar 0.00020 0.00015 F(E) vrms/ms -1 1600 0.00010 0.00005 0.00000 0 2000 4000 6000 E /J F (E) 2 E E F ( E ) dE 0 E k BT k BT 3 / 2 E1/ 2 exp 2 E 3 dE k BT k 2 BT k BT 3 / 2 E 3 / 2 exp 0 9 T v L 2L v Apparatus used to measure gas molecular speed distribution. Rotating sector method. Chemistry3 1st edition: Box 7.3, pp.322-323. 2nd edition: Box 8.3, pp.360-361. 10 Effusion and Diffusion of Gases. Effusion is the process by which gas molecules pass through a small hole such as a pore in a membrane. Diffusion occurs when two or more gases come together and mix. Diffusion arises due to the presence of a concentration gradient. Grahame’s Law of effusion: At a given temperature and Pressure the rate of effusion (number of molecules Passing through hole per second) is inversely proportional to square root of molar mass M of gas. Reffusion K M K = constant For a mixture of two gases A and B With molar masses MA and MB the Relative effusion rate is given by: RA RB MB MA We conclude that gases with different molar masses Will effuse at different rates. A gas with a low molar mass (e.g. He) will effuse Faster than a gas with a higher molar mass (e.g. N2). Nitrogen passes through membrane 2.6 times more Slowly than helium. 1 MN2 RHe 28gmol 7 2.6 RN2 MHe 4gmol1 Gas mixing takes time. Rate of diffusion given by Diffusion coefficient D (units: cm2s-1). Mean distance L travelled by a diffusing molecule is: L 6 Dt Molecular collisions in gases. Chemical reactions occur when molecules collide With each other. We can use KTG to estimate collision frequency in gases. Molecular size will affect probability of collision. 2 molecules A,B will collide if the center of one (molecule B) Comes within a distance of two radii – one diameter- of Molecule A. This area is the target presented by A. The area of the target for molecule B to hit Molecule A is a circle of area s called the Collision Cross Section . The collision cross section is related to Molecular size. d2 11 Collision frequency & mean free path. Collision frequency Z = mean number of collisions that a molecule undergoes per second. Number of collisions per second depends on distance travelled, and number of molecules per unit volume (concentration). This is the pressure. In a given time period, a molecule will collide with any other molecule whose centre lies within a cylinder with cross sectional area . Using Kinetic theory we can show that the collision frequency is: Z 2 N v P RT Knowing average speed that molecules move and number of collisions they undergo, the distance travelled between collisions can be determined. This is called the mean free path . v Z v 2 N v P RT RT 2 N P Example: N2 gas at SATP. Collision Cross Section = 0.43 nm2. Mean speed <v> = 475 ms-1. Collision Frequency Z: Z = 21/2 x(6.02x1023 mol-1)x(475 ms-1)x(0.43 x 10-18 m2)x{(1x105Pa)/(8.314JK-1mol-1)x(298K)} Hence collision frequency Z = 7.0 x 109 s-1. Mean free path: = RT/21/2 NP, Hence ={ (8.314 Jmol-1K-1)x (298K)}/{21/2x(6.02x1023mol-1)x(0.43x10-18m2x(1x105Jm-3)} So mean free path = 6.7x10-8 m = 67 nm. Note significant orders of magnitude for N2: molecular size typically 0.3 nm. Mean free path typically 67 nm. At atmospheric pressure gas molecules collide every 0.1 ns. 12