Internal Energy, Temperature, Collision Number and Mean Free Path

3. Internal Energy, Temperature, Collision Number and Mean Free Path

a. Internal Energy and Temperature At low pressures and high temperatures the intermolecular spacing in gases is usually large and the molecules move incessantly over a wide range of velocities. The molecules also vibrate and rotate. The total energy possessed by them is due to these translational, rotational, and vibrational modes ( Figure 19 ).

For the sake of illustration, consider H 2 O vapor–phase molecules at a pressure of 1 bar and a temperature of 200ºC. Typically, these molecules move with an average velocity of 350 m s –1 at temperatures around 300 K. Since l » 3l imit , attractive forces can be ignored. As the water vapor is compressed, the intermolecular distance decreases and attractive forces become significant as the gas reaches a certain volume (or pressure). Upon further compression, the attractive forces become so strong that the vapor changes phase to become liquid. According to liquid cell theory, each molecule is confined to a small cell of volume v´ (which is the total volume divided by the number of molecules contained in it). If the molecular diameter is small compared to the cell volume, a molecule is free to move within its cell without interacting with its nearest neighbors. Therefore, the translational energy of that molecule decreases, although it possesses the same rotational and vibrational energies. As the liquid is further compressed it becomes a solid. The interactions of a molecule with its neighbors are strongest when motion

is restricted to conditions corresponding to the minimum potential energy, i.e., when l = l min . At this state the molecules possess most of their energy in the vibrational mode. The relative

position of molecules (or their configuration) is fixed in solids. Gases correspond to the other extreme and contain a chaotic molecular distribution and motion. Liquids fall in a regime in- termediate between gases and solids, since their molecular kinetic energies are comparable to the maximum potential energies. Therefore, the molecular energy changes significantly with compression and phase change.

The position of an atom within a molecule can be fixed by three spatial coordinates (say, x,y and z). A polyatomic molecule containing δ atoms requires 3δ coordinate values in

order to fix the atomic positions, and, consequently, has 3 δ degrees of freedom. Molecules can have three translational energy modes. A monatomic gas ( δ =1) has three translational energy

modes, and a linear molecule such as CO 2 , which has all of its atoms arranged in a straight line, possesses two rotational degrees of freedom (since rotation about its own axis is negligi- ble) while H 2 O, which is a nonlinear molecule, possesses three rotational degrees of freedom. Therefore, the number of vibrational energy modes for a nonlinear molecule must be equal to the difference between the total degrees of freedom and the sum of the translational and rota- tional energy modes, i.e., (3 δ–6). Since a linear molecule possesses three translational and two

rotational modes, its vibrational energy modes must number (3 δ–5). The total energy associ- ated with a molecule u´ = e´ T + e´ R + e´ V is known as the molecular internal energy, where e´ T ,

e´ R , and e´ V , respectively, represent the total translational, rotational, and vibrational energies of that molecule.

b. Collision Number and Mean Free Path Molecules contained in matter travel a distance l mean before colliding with another molecule. Consider a molecule A that first collides with another molecule after traveling a distance l mean , then undergoes another collision after moving a distance of 2l mean , and so on, until colliding for the Nth time with another molecule after having moved along a distance N ×l mean . If these N collisions occur in one second, the molecule A is said to undergo N colli- sions per unit time (also known as the collision number).

If the molecular diameter of a molecule is σ (also called the collisional diameter), this is the closest distance at which another molecule can approach it. At this distance the repulsive

force between the two molecules is infinitely large as shown in Figure 17 . Assume that the average molecular velocity V avg is the distance through which the molecule travels in one sec- ond. Now consider a geometrical space shaped in the form of a cylinder of radius σ and length

V avg . There are n´ 2 πσ V avg molecules within this cylinder where n´ denotes the number of mole- cules per unit volume. A molecule traveling through the cylinder will collide with all of the

molecules contained within it, since the cylinder radius equals σ. Therefore, the number of collisions occurring per unit time Z

coll is n´ πσ V avg , and the time taken for a single collision is the inverse of this quantity. The average distance traveled by the molecule during this time is called its mean free path l mean , where

mean =V avg /(n´ πσ V avg ) = 1/(n´ 2 πσ ).

Another relation for the mean free path is l mean = 1/(2 1/2

πn´σ 2 ).

Typically, the number of collisions is of the order of 10 39 m –3 s –1 . The mean free path is also the average distance between adjacent molecules. For instance, consider a room consisting of N 2 molecules at 298K, 1 bar. Then n ’ =2.43x 10 25 molecules/m 3 , σ= 3.74 Å, and l mean = 0.0662 µm

or 662 Å or 66.2 nm. All of the molecules do not travel at the average velocity. The typical velocity distri- butions (also called the Maxwellian distributions) of helium molecules at different tempera- tures are illustrated in Figure 20 . The typical velocity distributions can be determined from the expression

-0.7 Min ip

l/l 0

Figure 18. Dimensionless potential and force field between molecules.

(a)

(b)

(c)

Figure 19. Illustration of the energy modes associated with a diatomic molecule. (a) Translational energy (TE). (b) Rotational energy (RE). (c) Vibrational energy (VE).

(1/N)(dN´ /dV) = 4 –1/2 V π (m/(2k B T)) 3/2 V 2 exp(–(1/2)mV 2 /(k B T))) (59) where N´ V represents the number of molecules moving with a velocity in the range V and

V+dV, N the total number of molecules, m the molecular mass (= M/N Avog ), with M denoting the molecular weight). Therefore, the translational energy varies among the molecules, and integration of Eq. (59) between the limits V = 0 and ∞ results in a number fraction of unity.

Microscopically, the molecules are in state from which the average energy is subject to pertur- bations of varying strengths. In Chapter 10 we will learn that these perturbations cause certain states to become stable, metastable, or unstable.

Equation (59) can be rewritten in terms of the energy e = mV 2 /2 and integrated to ob- tain the fraction of molecules possessing energy in the range from E to ∞, i.e.,

N´ E /N = 2 –1/2 ((E/( R T)) 1/2 π exp(–(E/( R T)) + (1 – erf((E/ R T) 0.5 )), (60) where E = e´N

2 Avog =MV /2, M denotes the molecular weight (or the mass of 1 kmole), and R = k B N Avog is the universal gas constant. As E →0, so does the error function and the first

term in Eq. (60), and, therefore, as is logical, the term (N 0≤E≥∞ /N) →1. This term becomes neg- ligibly small as E →∞, since the volume fraction of molecules associated with extremely large energies normally approaches zero. Since E/ R T is typically large, the value of the last term on

the RHS of Eq. (60) is negligibly small. Hence, the fraction of molecules with a velocity in the range V to ∞ (or E≤E≤∞) may be expressed as

V /N = 2 π (E/( R T) 1/2 exp(–(E/( R T)). (61) Equation (61) indicates that the fraction of molecules associated with an energy of value E and

N´

greater is proportional to exp(–(E/( R T)). Chemical reactions between reactant molecules oc- cur when the energy E exceeds the minimum activation value, which is required to overcome the molecular bond energies, thereby allowing the atoms to be rearranged in the form of prod- ucts.

The average molecular speed V avg is

(62) Where m is the mass of molecule and the expression for the most probable speed is

V avg 1/2 = (8/(3 π)] V = (8 k T/(

= (8 R T/(M 1/2 π)}

(63) The root mean square speed V rms can be expressed as

V 1/2

mps = (2/3) V rms = (2k B T/m ) = (2 R T/M ) .

V rms = (3k B T/m ) 1/2 = (3 R T/M ) 1/2 . (64)

2 2 2 where 2 V rms = V x + V y + V z is based on the three velocity components, and

m V 2 = m ( V 2 + V 2 + V 2 )/ 2 = (/) 32 kT

rms

From Eq. (64) note that average te per molecule 3k B T/2 where k B = R /N Avog . It is customary to assume three velocity components to equal each other in magnitude, i.e., each translational degree of freedom contributes energy equivalent to (1/2)k B T to the molecule. At standard conditions V

rms ≈ 1770, 470, and 440 m s , respectively, for H 2 , N 2 and O 2 , and is typically of the same magnitude as the sound speed in those gases. Recall that for an ideal gas the sound speed c = (k R T/M) 1/2 , where 1 ≤k≤5/3. For gaseous N 2 and H 2 , respectively, at stan-

×10 –26 kg and 0.34 ×10 kg; σ = 3.74 Å and 2.73 Å; l = 650 Å and 1230 Å; and Z coll =7

dard conditions V avg ≈ 475 and 1770 m s –1 ; m = 4.7

9 ×10 9 and 14.4 ×10 collisions s –1 . Recall that for an ideal gas the sound speed

c = kRT / M , where 1 < k < 5/3. The sound speed is compa- rable to average molecular velocity.

i. Monatomic Gas The only molecular energy mode in monatomic gases is translational. Helium, argon,

and other noble gases are examples of monatomic gases. The energy per molecule u´ in a monatomic gas is

u´ = e´ T = (3/2)k B T.

(65) where energy per degree of freedom is given by (1/2) (k B T) and at 298 K energy per degree of

freedom is given as 0.5*1.38x10 -26 kJ/molec. K * 298 = 2.05x10 -24 kJ/molec. Monatomic gas has 3 degrees of freedom. For a mass containing Avogadro’s number of molecules N Avog ,

(66) T = 2/3( u / R ). If an ideal monatomic ideal gas is placed in a rigid container and heated, the

u = (3/2)N Avog k B T = (3/2) R T, i.e.,

intermolecular spacing remains unchanged and, as shown in Figure 18 , the potential energy is still negligible. However, due to a rise in the translational energy, the internal energy increases.

ii. Diatomic Gas There are three translational and two rotational modes for a diatomic gas. At low

temperatures the vibrational modes can be neglected so that

(67) At higher temperatures there are (3n–5) = 1 vibrational modes. If a diatomic molecule is visu-

u´ = e´ T + e´ R = (5/2)k B T.

alized as two atoms attached by a spring, each vibrational mode for this combination has two degrees of freedom, i.e., due to the potential energy (that is similar to the energy stored in a spring), and to the kinetic energy of the atoms with respect to the center of mass. Each degree

of freedom contributes an energy equivalent to (1/2)k B T, and

e´ V,diatomic = 2(1/2)k B T=k B T.

(68) At higher temperatures, since u´= (e´ T +e´ R )+e´ V , its value equals (7/2)k B T. Therefore, for dia-

tomic gases

(69) T = 2/7( u / R ).

u = (7/2)N Avog k B T = (7/2) R T, i.e.,

Comparing Eqs. (66) and (69) it is seen that for similar increase in u, the temperature change for the diatomic molecule gas is smaller compared to a monatomic gas due to the higher en-

ergy storage capacity of the diatomic molecule. iii.

Triatomic Gas We have seen that each vibrational mode has two degrees of freedom for linear mole-

cules containing δ number of atoms. Therefore, linear triatomic molecules each have (3+2+(3δ –5) ×2), i.e., (6δ–5) degrees of freedom, while nonlinear molecules have (3+3+(3δ–6)×2) or

(6 δ –6) degrees of freedom. Each mode contributes (1/2)k B T of energy. The molecular energy in a linear polyatomic molecule is

(70) Likewise, for a nonlinear molecule

u´ = (6 δ–5) (1/2)k B T, i.e., u = (6 δ–5) (1/2) R T.

u´ = (6 δ–6) (1/2)k B T, i.e., u = (6 δ–6) (1/2) R T.

This simplified theory suggests that the internal energy per mole is proportional to the tem-

perature. The translational energy e´ T ≈ 0 for liquids, while for solids both e´ T and e´ R are neg- ligible.