1.3 Molecular Level Presentation

Size: px

Start display at page:

Download "1.3 Molecular Level Presentation"

Lorena Newman
5 years ago
Views:

1 1.3.1 Introduction A molecule is the smallest chemical unit of a substance that is capable of stable, independent existence. Not all substances are composed of molecules. Some substances are composed of electrically-charged particles known as ions. To get an idea of the extremely small size of molecules, A molecule of water is about m (3 Angstroms, Å) in diameter. Molecules of more complex substances may have sizes of more than 200 Å. 1

2 1.3.1 Introduction If a molecule is analyzed further, it is found to consist of particles of simpler kinds of matter, called elements. For sugar, three kinds of matter are carbon, hydrogen and oxygen. An atom is the smallest unit of an element that can exist either alone or in combination with other atoms of the same or different elements. The smallest atom, an atom of hydrogen, has a diameter of 0.6 Å. The largest atoms are slightly larger than 6 Å in size. 2

3 1.3.1 Introduction Under normal conditions at the macroscopic level, there are three phases (or states) of matter: solid phase liquid phase gaseous phase * Plasma is sometimes called the fourth phase of matter The phase of matter indicates how particles group together to form a substance. The structure of a substance can vary from compactly-arranged particles to highly-dispersed ones. In a solid, the particles are close together in a fixed pattern. In a liquid the particles are almost as close together as in a solid but are not held in any fixed pattern. In a gas, the particles are also not held in any fixed pattern, but the average distance between particles is large. 3

4 1.3.2 Kinetic Theory Elementary kinetic theory of matter The molecules of a substance are in constant motion. Motion depends on the average kinetic energy of the molecules. Average kinetic energy depends on the temperature of the substance. Collisions between molecules are perfectly elastic. Except when chemical changes or molecular excitations occur. 4

5 1.3.2 Kinetic Theory 10 The size of molecules is on the order of 10 m. The distance between the molecules of gas is on the 9 order of 10 m. Large distance between gas molecules Intermolecular forces are very weak, except when molecules collide with each other. The distance that a molecule travels between two 7 collisions is on the order of 10 m. Average velocity of molecules is about 500m/s. 10 The molecules collide with each other every 10 s, or at the rate of 10 billion collisions per second. 13 The duration of each collision is approximately 10 s. 5

6 1.3.2 Kinetic Theory Assumptions about the structure of the gases The size of the gas molecules is negligible compared with the distance between gas molecules. The molecules collide infrequently because the collision time is much shorter than the free motion time. The effects of gravity and any other field force are negligible, thus the molecules move along straight lines between collisions. The motion of gas molecules obeys Newton s second law. The collision of gas molecules is elastic. The kinetic energy before and after a collision is the same. 6

7 1.3.2 Kinetic Theory The average magnitude of the molecular velocity is given by kinetic theory k b c 8k b T π m where is Boltzmann constant, and m is the mass of the molecule. = (1.15) For any stationary surface exposed to the gas, the frequency of the gas molecular bombardment per unit area on one side is given by 1 f = N c 4 (1.16) where N is the number density of the molecules, defined as number of molecules per unit volume ( N = N / V ). 7

8 1.3.2 Kinetic Theory The mean free path, defined as average distance traveled by a molecule between collisions, is 1 λ = 2 2π σ N (1.17) where σ is the molecular diameter. The relaxation time,, which is the average time between two subsequent collisions, is The collision rate, is the average number of collisions an individual particle undergoes per unit time. τ τ 1, τ = (1.18) After the last collision with other molecules, the molecule travels an average distance of 2 λ / 3 before it collides with the plane. λ c 8

9 1.3.2 Kinetic Theory Table 1.3 Kinetic properties of gases at 25 C and atmospheric pressure (Lide, 2004)* Gas σ (m) 8 λ 10 (m) c m(m/s) τ (ps) Air Ar CO H He Kr N NH Ne O Xe *Reproduced by permission of Routledge/Taylor & Francis Group, LLC. 9

10 1.3.2 Kinetic Theory The mass flux of molecules in one direction at a point in a gas is given as 1/ 2 1/ 2 N cm kbt kbtm m& molecules = = N m 4 = 2π m N 2π (1.19) The pressure in a container is related to the number and average velocity of the molecules by p = 1 c Nm 3 V (1.20) where N is the number of molecules in the container, m is the mass 2 of each molecule, V is the volume of the container, and c is the average of the square of the molecular velocity. 2 c 2 1 N 2 = cn N n = 1 (1.21) 10

11 1.3.2 Kinetic Theory The average of the square of the molecules velocity is related to its three components by u = v = w = c 3 (1.22) The average kinetic energy of a molecule is defined as Substituting eq. (1.23) into eq. (1.20) yields E pv = = mc 2 NE (1.23) (1.24) 11

12 1.3.2 Kinetic Theory The monatomic ideal gas also satisfies the ideal gas law, i.e., pv nr T (1.25) where R u = kj/kmol-k is the universal gas constant, which is the same for all gases. Combining eqs. (1.24) and (1.25 )yields 3 Ru E = T 2 N N = N / n where A is the number of molecules per mole, which is a constant that equals is referred to as Avogadro s number. = A u 23 (1.26) and 12

13 1.3.2 Kinetic Theory Equation (1.26) can also be rewritten as where the Boltzmann constant is k b E 3 k T 2 b Ru = = = 23 N A The specific heat at constant volume, c v, is given by kinetic theory. c v = = J/K From eq. (1.26) it is evident that the average kinetic energy of molecules increases with increasing temperature. 3 2 k b m (1.27) (1.28) (1.29) 13

14 1.3.2 Kinetic Theory Internal energy (E) can be illustrated using kinetic theory. E = the sum total of all the energy of all the molecules in an object. The internal energy of the ideal gas equals the sum of all the kinetic energies of all the atoms. This sum can be expressed as the total number of molecules, N, times the average kinetic energy per atom, i.e., 3 E = N E = NkbT 2 (1.30) which shows that the internal energy of an ideal gas is only a function of mole number and temperature. 14

15 1.3.2 Kinetic Theory Similarly, viscosity, μ, thermal conductivity, k, and mass selfdiffusion coefficient can be obtained using simple kinetic theory. Following are the results: µ = 2 3π mk T b 3/ 2 2 σ (1.31) k = π 3 1 b 3/ 2 2 σ k T m (1.32) D = 2 σ 11 3/ 2 2 3π P k T 3 3 b m (1.33) 15

16 1.3.2 Kinetic Theory Equations (1.31) (1.33) can be used for binary systems with components 1 and 2, if σ and m are replaced by ( σ 1 + σ 2) / 2 and m m /( m + m ), respectively The significance of the above results should not be overlooked even though some simplified assumptions were used in their developments. Equations (1.31) and (1.32) for µ and k are independent of pressure for a gas for pressure up to 10 atmospheric pressures. Viscosity and thermal conductivity are proportional to 1/2 power of absolute temperature Diffusion coefficient is proportional to 3/2 power of absolute temperature. 16

17 1.3.2 Kinetic Theory To better model the temperature effects, one needs to replace the rigid sphere model and the mean free path concepts and use the Boltzmann equation to describe the nonequilibrium phenomena accordingly. Note that the equations described above are valid only for an ideal monatomic gas. For ideal gas molecules containing more than one atom, the molecules can rotate and the different atoms in the molecule can vibrate around their equilibrium position. The kinetic energy of molecules with more than one atom must include both rotational and vibrational energy. Their internal energy at a given temperature will be greater than that of a monatomic gas at the same temperature. 17

18 1.3.2 Kinetic Theory The internal energy of an ideal gas with a molecule containing more than one atom still depends solely on mole number and temperature. Figure 1.4 Modes of molecular kinetic energy: (a) rotational energy, (b) vibrational energy. 18

19 1.3.2 Kinetic Theory As noted above the simple kinetic theory of ideal gas was based on the mean free path concept. It provides the first order of magnitude approximation for several key transport phenomena properties. The simple kinetic theory is limited to local equilibrium. Therefore it is for time duration much larger than relaxation time. The advanced kinetic theory is based on the Boltzmann transport equation, which is presented in the next section. 19

20 1.3.3 Intermolecular Forces and Boltzmann Transport Equation A keen understanding of intermolecular forces is imperative for discussing the different phases of matter. The intermolecular forces of a solid are greater than those of a liquid. Most solids and liquids are deemed incompressible. The molecules repel each other when they are forced closer than their normal spacing. The closer they become, the greater the repelling force (Tien and Lienhard, 1979). A gas differs from both a solid and a liquid. Its kinetic energy is great enough to overcome the intermolecular forces, causing the molecules to separate without restraint. The intermolecular forces in a gas decrease as the distance between the molecules increases. 20

21 1.3.3 Intermolecular Forces and Boltzmann Transport Equation Potential function - the energy required bringing two molecules, which are initially separated by an infinite distance, to a finite separation distance r. The form of the function always depends on the nature of the forces between molecules, which can be either repulsive or attractive depending on intermolecular spacing. An accurate representation of the potential function should Account for all of the forces discussed above. Be able to reflect repulsive forces for small spacing and attractive forces in the intermediate distance. 21

22 1.3.3 Intermolecular Forces and Boltzmann Transport Equation When the molecules are close together, a repulsive electrical force is dominant. The repulsive force is due to interference of the electron orbits between two molecules, and it increases rapidly as the distance between two molecules decreases. When the molecules are not very close to each other, the forces acting between molecules are attractive in nature. Electrostatic forces occur between molecules that have a finite dipole moment, such as water or alcohol. Induction forces occur when a permanently-charged particle or dipole induces a dipole in a nearby neutral molecule. Dispersion forces are caused by transient dipoles in nominallyneutral molecules or atoms. 22

23 1.3.3 Intermolecular Forces and Boltzmann Transport Equation When the distance between molecules is very large, there should be no intermolecular forces. While the exact form of φ ( r) is not known, the following Lennard-Jones 6-12 potential provides a satisfactory empirical expression for nonpolar molecules: 12 6 r0 r0 φ ( r) 4ε = ε 0 r r r where is a constant and is a characteristic length. Both of them depend on the type of the molecules. (1.34) 23

24 1.3.3 Intermolecular Forces and Boltzmann Transport Equation Figure 1.5 shows the Lennard- Jones 6-12 potential as a function of distance between two molecules. When the distance between molecules is small, the Lennard- Jones potential decreases with increasing distance between molecules and the repulsive force dominates. It is necessary to add energy to the system in order to bring the molecules any closer. Figure 1.5 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules. 24

25 1.3.3 Intermolecular Forces and Boltzmann Transport Equation r min, As the molecules separate, there is a distance, at which the Lennard-Jones potential becomes minimum. As the molecules move further apart, the Lennard-Jones potential increases with increasing distance between molecules, the attractive force dominates. The Lennard-Jones potential approaches zero when the molecular distance becomes very large. 25

26 1.3.3 Intermolecular Forces and Boltzmann Transport Equation When the Lennard-Jones potential is minimal, the following condition is satisfied: dφ ( r ) dr min = 0 (1.35) Substituting eq. (1.34) into eq. (1.35), one obtains r = 2 r 1.12r 1/ 6 min 0 0 (1.36) Figure 1.5 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules. 26

27 The Lennard-Jones potential at this point is φ ( r ) = ε (1.37) One example that requires molecular dynamics simulation is heat transfer and phase change during ultrashort pulsed laser materials processing. The motion of each molecule in the system is described by Newton s second law, i.e., N j = 1( j i) F (1.38) The force between the i th and j th molecules can be obtained from r ij F ij min (1.40) where is the distance between the i th and j th molecules. The Lennard-Jones potential between the i th and j th molecules is 12 6 obtained by r 0 r 0 φ ij = 4ε r (1.39) ij r ij ij = d r 2 i mi dt 2 = φ ij = rο r r r ε r ο r r ο ij ij ij ij 27

28 For a pure gas, the self-diffusivity, D, viscosity,, and thermal conductivity, k, are. 3 π mkbt 1 D = 2 8 π σ Ω ρ (1.41) σ µ k = = c v (1.42) (1.43) where is collision diameter, is the molar specific heat under constant volume, and Ω µ = Ω k 1.1 Ω D are the dimensionless collision integrals, which are slow varying functions of k b T/ε (ε is a characteristic energy of molecular interaction). 2 D π mk T π σ 2 Ω b b k µ π mk T π σ Ω c v µ 28

29 1.3.3 Intermolecular Forces and Boltzmann Transport Equation For a nonequilibrium system, the mean free path theory is no longer valid The Boltzmann equation should be used to describe the molecular velocity distribution in the system. For low-density nonreacting monatomic gas mixtures, the random molecular movement can be described by the molecular velocity distribution function fi ( c, x, t), where c is the particle velocity and x is the position vector in the mixture. At time t, the probable number of molecules of the ith species that are located in the volume element dx at position x and have velocity within the range dc about c is f ( c, x, t) dcdx. i 29

30 1.3.3 Intermolecular Forces and Boltzmann Transport Equation The evolution of the velocity distribution function with time can be described using the Boltzmann equation Df i Dt fi = + c x fi + a c f f = Ω i ( f ) t (1.44) x c where and are operator with respect to x and c, respectively (see Appendix C), a is the particle acceleration (m/s 2 ), and Ω i is a five-fold integral term that accounts for the effect of molecular collision on the change of velocity distribution function f i. 30

31 1.3.3 Intermolecular Forces and Boltzmann Transport Equation The Boltzmann equation can also be considered as a continuity equation in six dimensional position-velocity space ( x and c ). The velocity distribution function is related to the number density (number of particles per unit volume) by fi ( c, x, t) dc = N i ( x, t) (1.45) Note that the density is ρ ( x, t) = mn ( x, t) where m is the mass of particle. The total number of particles N inside the volume V as a function of time is N ( t ) f (,, t ) d d = c x c x V c (1.46) 31

32 1.3.3 Intermolecular Forces and Boltzmann Transport Equation In thermodynamic equilibrium, f is independent of time and space, i.e., f ( c, x, t) = f ( c). It can be demonstrated that the stress tensor, eq. (1.60), heat flux, eq. (1.71), and diffusive mass flux, eq. (1.118) can be obtained from the solution of velocity distribution function f i. 32

33 1.3.3 Intermolecular Forces and Boltzmann Transport Equation More detailed information about Boltzmann equation and its applications related to transport phenomena in multiphase systems can be found in Chapter 2. Most macroscopic transport equation such as Fourier s law of conduction, Navier-Stokes equation for viscous flow, or the equation of radiation transfer for photons and phonons can be developed from Boltzmann transport equation using local equilibrium assumptions. 33

34 1.3.4 Cohesion and Adhesion Cohesion - the intermolecular attractive force between molecules of the same kind or phase. For a solid, cohesion is significant only when the molecules are extremely close together. Viscosity - the resistance of a liquid or a gas to shear forces. The fundamental basis for viscosity observed in fluids is cohesion within the fluids, measured as a ratio of shear stress to shear strain. Dependant on temperature: as temperature increases the viscosity of a gas increases, while that of a liquid decreases. Adhesion - the intermolecular attractive force between molecules of different kinds or phase. 34

35 1.3.5 Enthalpy and Energy Phase change phenomena can be viewed as the destruction or formation of intermolecular bonds as the result of changes in intermolecular forces. Solid Liquid Gas Intermolecular forces and bonds strongest Smallest distance between molecules Intermolecular forces and bonds weaker Larger distance between molecules Intermolecular forces weakest, no intermolecular bonds Largest distance between molecules 35

36 1.3.5 Enthalpy and Energy Sublimation occurs when the intermolecular bonds between the molecules in a solid are completely broken. The internal energy of a substance with molecules containing more than one atom (such as H 2 O) is the sum of the kinetic, rotational, and vibrational energies. Since the molecules in a solid are held in a fixed pattern and are not free to move or rotate, the lattice vibrational energy is the primary contributor to the internal energy of the ice. The enthalpy of melting and vaporization are much larger than the lattice vibrational energy, which explains why the latent heat is usually much greater than the sensible heat. 36

37 1.3.5 Enthalpy and Energy T a b l e 1. 4 Energies of the H 2 O molecule in the vicinity of 273 K Types of energy Approximate magnitude per molecule (ev) Lattice vibration Intermolecular hydrogen bond breaking 0.58 Enthalpy of melting 0.06 Enthalpy of vaporization 0.39 Enthalpy of sublimation

38 1.3.5 Enthalpy and Energy Different phases are characterized by their bond energy and their molecular configurations. From a microscopic point of view, the entropy of a system, S, is related to the total number of possible microscopic states of that system, known as the thermodynamic probability, P, by the Boltzmann relation S = k P b ln ( ) (1.47) 23 where k b is the Boltzmann constant, J/K. Therefore, the entropy of a system increases when the randomness or thermodynamic probability of a system increases. 38

39 1.3.5 Enthalpy and Energy Sublimation, melting, and vaporization are all processes that increase the randomness of the system and therefore produce increases of entropy. Since phase changes occur at constant temperature, the increases of entropy in these processes follows h s = = constant T (1.48) where h is the change of enthalpy during phase change, i.e., the latent heat, and T is the phase change temperature. The constant in eq. (1.48) depends on the particular phase change process but is independent of the substance. 39

40 1.3.5 Enthalpy and Energy For vaporization and condensation, Trouton s rule is applicable s v hl s = T v l ; sat 83.7J/(mol-K) (1.49) while Richards rule is valid for melting and solidification s h sl l ss = ; Tm 8.37J/(mol-K) (1.50) 40

41 1.3.5 Enthalpy and Energy Example 1.1: Calculate the time between subsequent collisions (relaxation time) τ, mean free path λ, and the number of collisions each molecule experiences per second for air at 25 C and 1 atm. Using the speed of sound show why we can smell odor very frequently even when we may be far from the source. Solution: The number density of air molecule is 5 p Pa N = = = molecules/m 23 k T ( J/K) (298.15K) b The mean distance between molecules is L -1/ 3 9 = N = = m 3.4nm

42 1.3.5 Enthalpy and Energy The diameter of molecules, according to Table 1.3, is 10 σ = 0.366nm = m The mean free path is can be obtained form eq. (1.17), i.e. 1 1 λ = = = = π σ N 2 π ( ) m 66nm The average magnitude of the molecular velocity can be obtained from eq. (1.15), i.e., c 23 8kbT = = = π m π m/s where molecular mass of air is atomic mass units (each -27 atomic mass unit is kg). 42

43 1.3.5 Enthalpy and Energy The relaxation time is The speed of sound in air is sound λ τ = = c 0.14ns which is less than the molecular velocity of 465m/s. The number of collisions for each molecule per second is 1 τ = 7 bilions/s Since the average speed of molecules is higher than the speed of sound and the number of collisions that each molecules experiences is high, it does not take very long for the nose to detect molecules. V = γ R T = g 345m/s 43

CHAPTER 13. States of Matter. Kinetic = motion. Polar vs. Nonpolar. Gases. Hon Chem 13.notebook

CHAPTER 13. States of Matter. Kinetic = motion. Polar vs. Nonpolar. Gases. Hon Chem 13.notebook CHAPTER 13 States of Matter States that the tiny particles in all forms of matter are in constant motion. Kinetic = motion A gas is composed of particles, usually molecules or atoms, with negligible volume