KINETIC MOLECULAR THEORY OF GAS

By Group 3

1. Fauzan

(1201506)2. Nofri Setiawan

(1201509)3. Retno Farah Delima(1201502)4. Wahyuni Mardi Wati(1201499)Lecture : 1) Ananda Putra, S.Si, M.Si, Ph.D

2) Deskiberi, S.Si, M.Si

DEPARTMENT OF CHEMISTRY

FACULTY OF MATHEMATICS AND SCIENCES

STATE UNIVERSITY OF PADANG

2013

CHAPTER I INTRODUCTION

a. Background

Kinetic molecular theory of gas attemps to explain properties macroscopic of gas such as temperature, pressure, and volume by considering the molecular composition and motion. Essentially from this theory is the pressure not caused by vibrational motion between molecules as expected by Isaac Newton, but caused by inter molecular colissions that move depending by different velocity. This theory also known as Kinetic Molecuar theory or colissions theory or Kinetic theory of gas. So, Kinetic theory of gas discuss about properties of gas associated with translation from atom and molecule in the gas state, and examine how the properties of the gas can be discussed by the free translational motion and continuous from components. to be discuss the properties of the gas with more perfect, then the kinetic theory of gases used ideal gas approach (Atkin, 2006, Castellan, 1983, and Mortimer, 2008).

b. Problem Formulation

1. What is Classical Mechanics of Imaginary Particle in a Container ?

2. What is Kinetics Particle in Container Equations, Calculation of a Particle Velocity ?

3. What is Boltzmann Distribution, and Determination of Boltzmann Constant ?

4. What is Maxwell-Boltzmann Statistic Mechanics and Derivation ? 5. What is Mean free-path ?

6. How Gas diffusion and Grahams law ?

7. What is application of Kinetic Molecular Theory Of Gas?

8. What Barometric equations ?

c. Destination

1. Can explain and understanding about Classical Mechanics of Imaginary Particle in a Container.

2. Can explain and understand Kinetics Particle in Container Equations,and can Calculate of a Particle Velocity

3. Knowing of Boltzmann Distribution, and Determination of Boltzmann Constant4. Understand about Maxwell-Boltzmann Statistic Mechanics and Derivation

5. Understanding about Mean free-path

6. Understanding how gas diffusion 7. Understanding about Grahams Law.8. Understanding the application of Kinetic Molecular Theory Of Gas.

9. Understanding about barometric equationCHAPTER II

EXPLANATION

KINETIC MOLECULAR THEORY OF GAS

Kinetic molecular theory of gas attemps to explain properties macroscopic of gas such as temperature, pressure, and volume by considering the molecular composition and motion. Essentially from this theory is the pressure not caused by vibrational motion between molecules as expected by Isaac Newton, but caused by inter molecular colissions that move depending by different velocity. This theory also known as Kinetic Molecuar theory or colissions theory or Kinetic theory of gas. So, Kinetic theory of gas discuss about properties of gas associated with translation from atom and molecule in the gas state, and examine how the properties of the gas can be discussed by the free translational motion and continuous from components. to be discuss the properties of the gas with more perfect, then the kinetic theory of gases used ideal gas approach (Atkin, 2006, Castellan, 1983, and Mortimer, 2008). Approaching of ideal gas based on some assumed: Gas consist by very small particles with certain mass Molecules move randomly very fast and there are collids with wall of container by constant Collisions between gas particle with wall of container is resilient perfect Interaction inter-molecules can degligible

Overall volume of individual gas molecules is negligible compared to the volume of the container. This is equivalent to stating that the average distance between gas particles large enough compared to the size of the gas molecules themselves.Shape of molecules like ball and elastic

Average of kinetic energy just depend on temperature of system

Relativistic effect and mechanics quantum effect can digligible

Distance between particle more than wavelenght calor deBroglie, and molecules can be object

Broglie dan molekul-molekul dapat diperlakukan sebagai objek klasik.

1. Classical Mechanics of Imaginary Particle in a Container

According to classical mechanics, the state of the microscopic system is determined by determining the position and velocity of every particle. The number of particles in this model range from 1 to N. Vector r of the particle at i position can be written as the vector sum of the three Cartesian coordinates (Mortimer, RG, 2008):

a. Position

Each term in the equation is the result of scalar quantity (component) and the unit vector. A scalar quantity can be positive, negative, or zero but not have a certain direction in space. Position i is the unit vector in the positive direction on the x-axis, position the unit vector j is in the positive y-axis direction, and the unit vector k positions are in the positive direction of the z axis. This picture shows the ri vector, Cartesian axis, vector units, and the Cartesian components.

b. VelocityVelocity vector can be represented by geometrically as in the direction ofvector, velocity is the direction in which the particles move. Pace were of a magnitude of velocity, which is given by the Pythagorean theorem in three-dimensional position:

This equation shows that the magnitude of the vector is always non-negative (positive or zero).

Velocity of i particle determining by velocity vector vi:

Component of velocity vector vi is the number of changes xi, yi, and zi:

2. Kinetics Particle in Container Equations, Calculation of a Particle Velocity

Gas momentum change can be expressed by:p = final momentum - initial momentump =-m0vx - m0vx =-2m0vx.Interval of time for the trip can be calculated by:

The rate of change of momentum of molecules on a wall of the same in accordance with the second law Newton is force, with equation:

Where p is force per area, so:

If there are N number of gas molecules in a closed container with a velocity component in the x-axis is v1x, v2x, ,vNx, the total gas pressure at the wall is:

Where mean :

So:

Because container volume is 3 so :

Square of the speed of each gas molecule is:

v2 = vx2 + vy2 +vz2

In accordance with the assumption that each molecule moves randomly in all directions with constant velocity, then the square average velocity in the direction of x, y, and z are equal,Vx = Vy = Vz =3 Vxso that

If value of vx2 entered to pressure equation, so:

Where: P = Pressureof gas (atm) m= mass a particle (molecule) of gas (kg) v2= average square of the velocity (m2/s2) V = Volume of gas (m3)3. Boltzmann Distribution, and Determination of Boltzmann ConstantBoltzmann distribution in chemistry, physics, and mathematics is a certain distribution function or probability measure for the distribution of thestates of a system. Boltzmann distribution is used specifically to describe the particle velocity gas. This distribution was discovered in the context of classical statistical mechanics by J.W Gibss in 1901

An a special case of the Boltzmann distribution used for describing velocities of a gas particle, its also known as Maxwell-Boltzman distribution. Initial velocity of a molecule with other between collusion can be same or different. So,there are distribution of the number of molecules from zero until high velocityDetermination of Boltzmann ConstantThe Boltzmann constant (k or kB), named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature. It is the gas constant(R)divided by the Avogadro constant (NA ):

The Boltzmann constant can be determined from experimen had been done by scientiest or ,we can get it, if we have some data that will point us to get Boltzmann constant .

In this equation, N1 is the number of molecules in state 1 and 1 is the energy of state 1; N2 is the number of molecules in state 2. The state of a particle is defined in classical mechanics byspecifying its position and velocity to within infinitesimal ranges. This Equation is a result of statistical mechanics for a system in thermal equilibrium. If 2 is greater than 1, then is positive so depend on equation N2 is less than N1. The number of molecules in a state decreases with increasing energy of the state.4. Maxwell-Boltzmann Statistic Mechanics and Derivation

Each energy level can be occupied by any particle energy levels and each has the same probability to be occupied looking probability of particle placement is to find the number of ways how the particles are placed. Determining probability of particle If N is the total number of particles involved in this system, the way the particles placement are as follows: To put the first particles there are N ways (because there are N particles involved). To put the two existing particles (N - 1) ways (because after the first particle placement there are (N - 1) particles).

For placing third particle exist (N - 2) ways, and so on. N1 number of ways to place N particles in E1 levels are N (N -1) (N - 2) (N - 3) ...... (N - n1)

Maxwell-Boltzmann statistics, the partition function:

Number for all circumstances into account also that the particles with particle can be distinguished.Note the possible permutations, so that for N particles:

Here nr = 0, 1, 2, 3,... with restrictions:

Partition function can be written:

Which is nothing but a binomial Newton:

Or And then:

Final equation :

5. Mean Free Path Imagine gas leakingout of a pipe. Itwould takea while forthe gasto diffuse and spread into the environment. This is because gas molecules collide with each other, causing them to changein speed and direction.Therefore,they can never move in a straight path without interruptions. Between every two consecutive collisions, a gas molecule travels astraight path. The averagedistance of all the paths of a molecule is the mean free path.

Analogy

Imagine a ball traveling in a box (Figure 1) wherethe ball represents a moving molecule. Everytime it hits the wall, a collision occurs andthe direction of the ball changes. In figure 1, the ball hits the wall five times, causing five collisions. Between every two consecutive collisions, the ball travels an individual path.Ittravelsa total of four paths between the five collisions;each path has a specific distance, d. The mean free path of this ball is the average distance of allfour paths.

Figure 1. This figure shows a ball traveling in a box.

Every time it hits the wall, a collision occurs and the ball changes direction.The ball travels an individual path between every two consecutive collisions. Each path traveled by the ball has a distance, d.

Free Mean Path () = [d(1) + d(2) + d(3) + d(4)] / 4

CalculationsIn reality, the mean free path ( )cannot be calculatedby taking the average ofall the pathsbecause it is impossible to know the distance of each path traveled by a molecule. However,we can calculate itfrom the average speed ( c ) of the moleculedivided by the collision frequency (z). The formula for this is:

Since is equal to 1 / (average time between collisions), the formula can also be:

= c/ z [1/ (average time between collisions)]

= c x (average time between collisions)

ZIn addition, since z is also equal to d2 c (N/v), where d is the diameter of the molecule and (N/v) is the density,the formula can be further modified to:

= c / [ d2 c (N/v)]

= 1 / [ d2 (N/v)]

Factors affecting mean free path

Density:As gas density increases, themoleculesbecome closer to each other. Therefore, they aremore likely to run into each other, so the mean free path decreases.

Increasing the number of molecules or decreasing the volume willcause density to increase. This will decrease the mean free path.

Radius of molecule: Increasing the radiusof the molecules will decrease the space between them, causing them to run into each other more. Therefore, mean free path decrease.

Pressure, Temperature, and other factors that affect density can indirectly affect mean free path.

6. Gas Diffusion and Grahams Law

Gas diffusion

Diffusion is a simple process that can be explained by kinetic theory.When you open a bottle of perfume, it can very quickly be smelled on the other side of the room. This is because as the scent particles drift out of the bottle, gas molecules in the air collide with the particles and gradually distribute them throughout the air. Diffusion of a gas is the process where particles of one gas are spread throughout another gas by molecular motion.

We can explain the diffusion of gases by using the average quantity of the average distance squared : .If no other gas flow disturbing the movement of gas molecules, so apparently proportional to the travel time t.

6D is a proportionality constant value. Where D is the diffusion constant of the molecule. Distance of the square root of the mean is

This equation also shows that the diffusion constant of the gas is directly proportional to the mean free path and with an average velocity of molecules c . However, the constant of proportionality is difficult to count. The simplest example is the constant of proportionality of the gas components with value , so:

Grahams Law of Effusion

Effusion is a similar process. Effusion is the process where gas molecules escape from an evacuated container though a small hole. It is assumed that while a molecule is exiting, there are no collisions on that molecule.

Note how the lighter molecules are the first to exit because they have a faster speed. You can see in Figure 4 above, the smaller green molecules exit at a faster rate. This is where Graham's law of effusion comes in. It tells us the rate at which the molecules of a certain gas exit the container, or effuse. To see how Graham's law of effusion is derived from kinetic theory, consider the equation for the kinetic energy of a gas (ignoring rotation).

Because temperature is a measure of the average kinetic energy of a gas, two gases at the same temperature will also have the same kinetic energy. Thus,

Simplify by multiplying both sides by two,

By rearranging terms, we get

Taking the square root of both sides gives the equation,

wherev1is the average velocity of the molecules in gas 1,v2is the average velocity of the molecules in gas 2, andm1andm2are their respective molar masses. According to Graham's law, the molecular speed is directly proportional to the rate of effusion. You can imagine that molecules that are moving around faster will effuse more quickly, and similarity molecules with smaller velocities effuse slower.Because this is true, we can substitute the rates of effusion into the equation below. This yields Graham's law of effusion.

This equation allows us to compare the rates of effusion for two different gases under the same conditions of temperature and pressure. It is important to note that when solving problems for effusion, the gases must contain equal moles of atoms. You can still solve the equation if they are not in equal amounts, but you must account for this. For example, if gas A and gas B both diffuse in the same amount of time, but gas A contains 2 moles and gas B contains 1 mole, then the rate of effusion for gas A is twice as much.

7. Some Application of Kinetic Molecular Theory Of Gas

Dinamic of Reaction ( Collisions Theory and affecting of Temparature to Rate of Reaction)

Based on the Bimolecular Collision Theory, Rate of Reaction is the result of collision frekuensi times collision fractions that have enough energy.

If the activation energy for this reaction is Ea, then only partially [exp (-Ea/RT)] of the collision would have enough energy to produce a product (subsequently referred to as the frequency of collisions). Each collision of a pair of molecules which effectively resulted in a decrease in the number of molecules of A in the reaction mixture twice, So, the rate of change of the number of molecules A unity volume is

Where Mr = molar mass = relative molecular mass

Because the price of the reaction rate constant (kr) involves the number of moles per volume, ie [A], not the number of molecules of A, then both (number of molecules of A and [A]) are linked by: the NAV is Avogadro's number. By taking the definition of the reaction rate for a second order reaction above:

For reactions; 2A product, the speed law:

Based on two equation, we can make new formula which is (1) equation = (2) equation

Become,

8. Barometric EquationThe barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude.Starting at some point in midair, the change in pressure associated with a small change in height can be found in terms of the weight of the air.

The change in pressure depends on density, but depends on the pressure as follows:

The change in pressure depends on density, but depends on the pressure as follows:

The solution for the change from the ground ( P0) to height h ( Ph ) gives

Ph = P0e-mgh/RT

The development of the barometric formula makes use of a number of concepts from kinetic theory, such as the ideal gas law and the associated molecular constants. In the exponential, the two terms have the units of energy. The numerator mgh is gravitational potential energy and the term kT is thermal energy.

Having shown that the rate of change of pressure with height has the form

it is necessary to take the limit as the change in height approaches zero, putting it in the form of a derivative.

This type of equation can be solved for P by making a substitution of the type

fitting the boundary conditions givesPh = P0e-mgh/RT

The equation for the variation of barometric pressure with height has the formwhich has the formal solution

Substituting the solution gives

Since this equation must be valid for all values of h, forcing the solution to fit the physical boundary conditions yields:

C= 0 since other term vary in h

by setting cofficients equal

A= Po

since that must be the value of h = 0CHAPTER III

CLOSING

Conclusion :

a. Kinetic molecular theory of gas attemps to explain properties macroscopic of gas such as temperature, pressure, and volume

b. Classical Mechanics of Imaginary Particle in a Container can determind :

Positition

Velocityc. Boltzmann distribution in chemistry, physics, and mathematics is a certain distribution function or probability measure for the distribution of thestates of a system. Boltzmann distribution is used specifically to describe the particle velocity gas. This distribution was discovered in the context of classical statistical mechanics by J.W Gibss in 1901.d. The averagedistance of all the paths of a molecule is called as The mean free path.

e. Diffusion is a simple process that can be explained by kinetic theory.f. Effusion is a similar process. Effusion is the process where gas molecules escape from an evacuated container though a small hole.g. The Barometric Formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude.REFERENCEAtkins, P.W.,, 2006. Physical Chemistry, 8th Ed. Oxford University Press. New York.

Castellan, G.W., 1983. Physical Chemistry, 3th Ed. Addison-Wesley Publishing

Company. Singapore.

Mortimer, R.G., 2008. Physical chemistry. 3th Ed. Elsevier Academic Press. London.

Moore, W.J., 1972. Physical Chemistry. Printice-Hall Inc. New Jersey.

Oxtoby, D.W., et al., 2008. Principles of Modern Chemistry, Sixth Edition. Thomson

Brooks/Cole, a part of The Thomson Corporation. USA.

Wikipedia, 2011. Teori Kinetik. Tersedia pada http://id.wikipedia.org/wiki/Teori_kinetik.

Diakses pada tanggal: 19 September 2013.Mendikbud: Kurikulum 2013 Bukan Hanya Berkaitan Pendidikan

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