Properties of ideal gases

University of Babylon, College of Science,
Chemistry Department, under graduated Study, 3rd class
Study year 2018-2019,
Physical Chemistry
Main Syllabus items:
1- General properties of gases,
2- Kinetics theory of gases,
3- Chemical kinetics,
4- photochemistry
5- Electrochemistry
6- Surface chemistry
Main References:
1- Peter Atkins, and Julio de Paula, Physical Chemistry, 8th Addition, Oxford University, New York, 2008,
2- Thermodynamics of gases, R. Sotaq, C. Borgnakke, and G. Wylen, John Wiley and Sons, INC, 1998.
3- Thermodynamics and Kinetic Theory of Gases, Wolfgang Paul and Charles Enz, Courier Corporation, 2000.
4- Chemical kinetics , by Vivek Patel, Janeza Trdine, 9 edition, Crotia, 2012,
5- Essential of physical chemistry, electrochemistry, by Arun Bahal, 2006.
6- Some scientific websites.

Prof. Dr. Abbas Jasim Atiyah
Incharge of the subject


Properties of ideal gases
An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact with each other except when they collide elastically. The ideal gas concept is useful because:
1- it obeys completely the ideal gas law, a simplified equation of state,
2- and is amenable to analysis under statistical mechanics. One mole of an ideal gas has a volume of 22.414 L at STP( 273.16 K and 1 atm.).
At controlled conditions, most real gases behave qualitatively like an ideal gas( especially under high temperature and reduced pressure) . Many gases such as nitrogen, oxygen, hydrogen, noble gases, and some heavier gases like carbon dioxide can be treated like ideal gases under these conditions.
Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure, as the work which is against intermolecular forces becomes less significant compared with the particles kinetic energy, and the size of the molecules become less significant compared to the empty space between them.
The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size become important. It also fails for most heavy gases, and for gases with strong intermolecular forces, notably water vapor. At high pressures, the volume of a real gas is often considerably greater than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less(why)?
1-One of them is the well known ideal gas law
This equation is derived from Boyle s law: (at constant T and n); Charles s law: (at constant P and n); and Avogadro s law: (at constant T and P). By combining the three laws, it would demonstrate that which would mean that.
Under ideal conditions, that is, .

The ideal gas model depends on the following assumptions:
• The molecules of the gas are indistinguishable, small, hard spheres
• All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
• Newton s laws apply
• The average distance between molecules is much larger than the size of the molecules
• The molecules are constantly moving in random directions with a distribution of speeds
• There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions
• The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls
• In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.
The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are much related:
1- Molecules are spherical in shape,
2-Collisions are elastic, and
3- There are no inter-molecular forces.
The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures.
Heat capacity
Heat capacity at constant volume, including an ideal gas is:
This is the dimensionless heat capacity at constant volume, which is generally a function of temperature due to intermolecular forces. For moderate temperatures, the constant for a monatomic gas is R while for a diatomic gas it is R, for complex molecule Cv=3R.
E= 3/2RT and Cv= (?E/?T)
Cv= (?3/2RT)/?T
Cv=3/2R
And we have the following:
Cp-Cv=R for one mole of gas and Cp-Cv=nR for moles of gases and always Cp> Cv (prove that)?.
It is seen that macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules.
The heat capacity at constant pressure of 1/R mole of ideal gas is:
where is the enthalpy of the gas.
Sometimes, a distinction is made between an ideal gas, where and could vary with temperature, and a perfect gas, for which this is not the case. The ratio of the constant volume and constant pressure heat capacity is

For air, which is a mixture of gases, this ratio is 1.4.


The Ideal Gas Law
Gas properties are determined by four parameters (P,V, n and T). Many chemists had dreamed of having an equation that describes relation of a gas molecule to its environment such as pressure or temperature. However, they had encountered many difficulties because of the fact that there always are other affecting factors such as intermolecular forces. Despite this fact, chemists came up with a simple gas equation to study gas behavior while putting a blind eye to minor factors. In order for a gas to be ideal, its behavior must follow the Kinetic-Molecular Theory whereas the Non-Ideal Gases will deviate from this theory due to real world conditions.
.In this issue, two well-known assumptions should have been made beforehand:
1. The particles have no forces acting among them, and
2. These particles do not take up any space, meaning their atomic volume is completely ignored.
An ideal gas is a hypothetical gas dreamed by chemists and students because it would be much easier if things like intermolecular forces do not exist to complicate the simple Ideal Gas Law. Ideal gases are essentially point masses moving in constant, random, straight-line motion. Its behavior is described by the assumptions listed in the Kinetic-Molecular Theory of Gases. This definition of an ideal gas contrasts with the Non-Ideal Gas definition, because this equation represents how gas actually behaves in reality.
The Ideal Gas Equation
Before we look at the Ideal Gas Equation, let us state the four gas variables and one constant for a better understanding. The four gas variables are: pressure (P), volume (V), number of mole of gas (n), and temperature (T). Lastly, the constant in the equation shown below is R, known as the the gas constant, which will be discussed in depth further later:
[ PV=nRT]
Another way to describe an ideal gas is to describe it in mathematically. Consider the following equation:
[ {PV}/{nRT}=1]
An ideal gas will always equal 1 when plugged into this equation. The greater it deviates from the number 1, the more it will behave like a real gas rather than an ideal. Pressure is directly proportional to number of molecule and temperature. (Since P is on the opposite side of the equation to n and T). Pressure, however, is indirectly proportional to volume. (Since P is on the same side of the equation with V)
Boyle s Law
Boyle’s Law describes the inverse proportional relationship between pressure and volume at a constant temperature and a fixed amount of gas. This law came from a manipulation of the Ideal Gas Law. Boyle s law states that at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in an equivalent manner, that the product of absolute pressure and volume is always constant.

Most gases behave like ideal gases at moderate pressures and temperatures. The technology of the 17th century could not produce high pressures or low temperatures. Hence, the law was not likely to have deviations at the time of publication. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.
V? 1/P (T and constant) so that P=k/V ad PV=constant
P1V1=P 2V2 …. Etc






Figure 1: A graph of Boyle s original data







Figure 2: A graph of Boyle s original data
This equation would be ideal when working with problem asking for the initial or final value of pressure or volume of a certain gas when one of the two factors is missing.
Relation with kinetic theory and ideal gases
Boyle s law states that at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in an equivalent manner, that the product of absolute pressure and volume is always constant. So long as temperature remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of k will remain constant
PV=constant ( at constant T and n)
where:
• P denotes the pressure of the system.
• V denotes the volume of the gas.
• k is a constant value representative of the temperature and volume of the system.
Charles Gay-Lusack Law
Charles s law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles s law is:
When the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be in direct proportion.[1]
This directly proportional relationship can be written as:
V? T (P and n are constants) , V=kT
or V/T=k , V1/T1=V2/T2 and so on
V1T2=V2T1
where:
V is the volume of the gas,
T is the temperature of the gas (measured in kelvins),
k is a constant.
This law describes how a gas expands as the temperature increases; conversely, a decrease in temperature will lead to a decrease in volume. The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion.












Figure 3: An animation demonstrating the relationship between volume and temperature(Charrls-Gay –Lusack law)
The volume of a fixed mass of dry gas increases or decreases by 1?273 times the volume at 0 °C for every 1 °C rise or fall in temperature. Thus:
VT= Vo(1+T/273.15)
where VT is the volume of gas at temperature T, V0 is the volume at 0 °C.

Relation of Charles –Gay Lusack law to kinetic theory
The kinetic theory of gases relates the macroscopic properties of gases, such as pressure and volume, to the microscopic properties of the molecules which make up the gas, particularly the mass and speed of the molecules. In order to derive Charles law from kinetic theory, it is necessary to have a microscopic definition of temperature: this can be conveniently taken as the temperature being proportional to the average kinetic energy of the gas molecules, Ek:
T? Ek,
Under this definition, the demonstration of Charles law is almost trivial. The kinetic theory equivalent of the ideal gas law relates PV to the average kinetic energy:
PV=2/3NEk

Charles s Law describes the directly proportional relationship between the volume and temperature (in Kelvin) of a fixed amount of gas, when the pressure is held constant.
V?T, (P and n are constants), so that, V= kT and V/T=k
[ {V1}/{T1}= {V2}/{T 2}] and so on
This equation can be used to solve for initial or final value of volume or temperature under the given condition that pressure and the number of mole of the gas stay the same.
Avogadro s Law
Volume of a gas is directly proportional to the amount of gas at a constant temperature and pressure.
V? n, (P and T are constants) V=kn and V/n=k
where
V is the volume of the gas;
n is the amount of substance of the gas (measured in moles);
k is a constant for a given temperature and pressure.
This law describes how, under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as follows:
V1/n1= V2/n2 and so on
Avogadro s law also means the ideal gas constant is the same value for all gases, so:
constant = p1V1/T1n1 = P2V2/T2n2
V1/n1 = V2/n2

V1n2 = V2n1
where p is pressure of a gas, V is volume, T is temperature, and n is number of moles
The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas.








Figure 4: plotting of Avogadro s law
Derivation from the ideal gas law
The derivation of Avogadro s law follows directly from the ideal gas law, i.e
PV=nRT
Where where R is the gas constant, T is the Kelvin temperature, and P is the pressure (in pascals). Solving for V/n, we thus obtain:
V/n=RT/P
Avogadro s Law can apply well to problems using Standard Temperature and Pressure (0 ?C ad 1 atm.), because of a set amount of pressure and temperature.
Molar volume
Taking STP to be 101.325 kPa and 273.15 K, we can find the volume of one mole of gas:
Vm= V/n= RT/P= (0.0821 atm.L.mol-1.K-1 x 273.15 K)/ 1 atm= 22.414 L/mol.
For 100.00 kPa and 273.15 K, the molar volume of an ideal gas is 22.712 dm3mol?1.
Amontons s Law
Amonton’s law was discovered in the late 1600s by a French physicist named Guillaume Amonton. According to Amonton’s law, if the volume of a gas is held constant, increasing the temperature of the gas increases its pressure. Why is this the case? A heated gas has more energy. Its particles move more and have more collisions, so the pressure of the gas increases. The graph in Figure below shows this relationship.Given a constant number of mole of a gas and an unchanged volume, pressure is directly proportional to temperature.



P ? T (V and n constants), p=kT and p/T=k
P1/ T1 =P2/T2 and P1T2=P2T1 and so on
Boyle s Law, Charles Law, and Avogradro s Law and Amontons s Law are given under certain conditions so directly combining them will not work. Through advanced mathematics (provided in outside link if you are interested), the properties of the three simple gas laws will give you the Ideal Gas Equation.

Units of P, V and T
The table below lists the different units for each property.
Take note of certain things such as temperature is always in its SI units of Kelvin (K) rather than Celcius(C), and the amount of gas is always measured in moles. Gas pressure and volume, on the other hand, may have various different units, so be sure to know how to convert to the appropriate units if necessary.
Pressure Units
Use the following table as a reference for pressure.
Common Units of Pressure
Unit Symbol Equivalent to 1 atm
Atmosphere atm 1 atm
Millimeter of Mercury mmHg 760 mmHg
Torr Torr 760 Torr
Pascal Pa 101326 Pa
Kilopascal kPa* 101.326 kPa
Bar bar 1.01325 bar
Millibar mb 1013.25 mb
The Gas Constant (R)
Here comes the tricky part when it comes to the gas constant, R. Value of R WILL change when dealing with different unit of pressure and volume (Temperature factor is overlooked because temperature will always be in Kelvin instead of Celcius when using the Ideal Gas equation). Only through appropriate value of R will you get the correct answer of the problem. It is simply a constant, and the different values of R correlates accordingly with the units given. When choosing a value of R, choose the one with the appropriate units of the given information (sometimes given units must be converted accordingly). Here are some commonly used values of R:
Values of R
0.082057 L atm mol-1 K-1
62.364 L Torr mol-1 K-1
8.3145 m3 Pa mol-1 K-1
8.3145 J mol-1 K-1*
1.987 Cal.mol-1.K-1
Kinetic theory of gases
The temperature of an ideal and real gas is a measure of the average kinetic energy of its atoms. The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic properties of gases, such as pressure, temperature, viscosity, thermal conductivity, and volume, by considering their molecular composition and motion. The theory posits that gas pressure is due to the impacts, on the walls of a container, of molecules or atoms moving at different velocities.
Assumptions of kinetic theory of gases
The theory for ideal gases makes the following assumptions: The gas consists of very small particles known as molecules. This smallness of their size is such that the total volume of the individual gas molecules added up is negligible compared to the volume of the smallest open ball containing all the molecules. This is equivalent to stating that the average distance separating the gas particles is large compared to their size.
• These particles have the same mass(zero mass point).
• The number of molecules is so large that statistical treatment can be applied.
• These molecules are in constant, random, and rapid motion.
• The rapidly moving particles constantly collide among themselves and with the walls of the container. All these collisions are perfectly elastic. This means, the molecules are considered to be perfectly spherical in shape, and elastic in nature.
• Except during collisions, the interactions among molecules are negligible. (That is, they exert no forces on one another.)
• The time during collision of molecule with the container s wall is negligible as compared to the time between successive collisions.
• Because they have mass, the gas molecules will be affected by gravity. More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, and small gradients in bulk properties.
Pressure and kinetic energy
Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container of cubical box with length of a. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=a3. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed with Cx which is equal to Cy and Cz (an elastic collision), then the momentum lost by the particle and gained by the wall is:

Number of collision per a second for one molecule= speed/distance
No of collision= Cx/2a
Change in moment= ?P = mCx-(-mCx)= 2mCx
Force= no of collision x ?P= mCx2/a
For N molecules , F= NmCx2/a
Pressure (P)= F/area = F/a2
P= NmCx2/a3
P=NmCx2/ V
PV= NmCx2
C-2 = Cx2+Cy2+Cz2 ( we have Cx=Cy=Cz)
C-2= 3Cx2 , this means that Cx2=1/3 C-2
By substituting this value we get:
[PV=1/3NmC-2] or PV= NmV-2/3

Temperature and kinetic energy

Rewriting the above result for the pressure as , we may combine it with the ideal gas law
PV= nRT
PV=nKBT


Where kB is the Boltzmann constant and the absolute temperature defined by the ideal gas law.
Which leads to the expression of the average kinetic energy of a molecule,
.

PV=2/3Ek


Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.
Molecular speeds of gases:
Speed of molecules:
Generally, there are three main types of molecular speeds, root mean squre seed (rms), average speed or arithmetic speed (V-) and most probable speed (?).
From the kinetic energy formula it can be shown that:
Vrms= (3RT/M)1/2 (molar speed of gas)

(speed of a single molecule of gas)

with v in m/s, T in kelvins, and m is the molecule mass (kg). The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (isotropic distribution of speeds).
Average speed C-= (8RT/?M)1/2 (molar arithmetic speed)
Average speed C-= (8kT/?m)1/2 ( arithmetic speed of single molecule only)
Most probable speed, ?= (2RT/M)1/2 (molar speed)
Most probable speed, ?= (2RT/m)1/2 (speed of single molecule)

Root-mean-square speed
Is the measure of the speed of particles in a gas that is most convenient for problem solving within the kinetic theory of gases. It is defined as the square root of the average velocity-squared of the molecules in a gas. It is given by the formula .

where vrms is the root mean square of the speed in meters per second, Mm is the molar mass of the gas in kilograms per mole, R is the molar gas constant, and T is the temperature in kelvin. Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds and they stop and change direction according to the law of density measurements and isolation, i.e. they exhibit a distribution of speeds. Some move fast, others relatively slowly. Collisions change individual molecular speeds but the distribution of speeds remains the same. This equation is derived from kinetic theory of gases using Maxwell–Boltzmann distribution function. The higher the temperature, the greater the mean velocity will be. This works well for both nearly ideal, atomic gases like helium and for molecular gases like diatomic oxygen. This is because despite the larger internal energy in many molecules (compared to that for an atom), 3RT/2 is still the mean translational kinetic energy. This can also be written in terms of the Boltzmann constant (k) as

where m is the mass of one molecule of the gas.


which is equivalent. The same result is obtained by solving the Gaussian integral containing the Maxwell speed distribution, p(v):





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The Kinetic Theory of Gases
Avogadro constant
The laws of classical thermodynamics do not show the direct dependence of the observed macroscopic variables on microscopic aspects of the motion of atoms and molecules. It is however clear that the pressure exerted by a gas is related to the linear momentum of the atoms and molecules, and that the temperature of the gas is related to the kinetic energy of the atoms and molecules. In relating the effects of the motion of atoms and molecules to macroscopic observables like pressure and temperature, we have to determine the number of molecules in the gas. The mole is a measure of the number of molecules in a sample, and it is defined as" the amount of any substance that contains as many atoms/molecules as there are atoms in:
a 12-g sample of 12C "
Laboratory experiments show that the number of atoms in a 12-g sample of 12C is equal to 6.02 x 1023 mol-1. This number is called the Avogadro constant, NA. The number of moles in a sample, n, can be determined easily:

Translational Kinetic Energy
The average translational kinetic energy of the molecule discussed in the previous section is given by
Using the previously derived expression for vrms, we obtain
(M x m=NA)
The constant k is called the Boltzmann constant and is equal to the ratio of the gas constant R and the Avogadro constant NA

The calculation shows that for a given temperature, all gas molecules - no matter what their mass - have the same average translational kinetic energy, namely (3/2)kT. So that, when we measure the temperature of a gas, we are measuring the average translational kinetic energy of its molecules
Graham s Law of Effusion
Graham s Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its density or its molar mass.
[ The rate of effusion of a gas is inversely proportional to the square root of its molecular weight at a constant temperature and pressure].


Effusion refers to the movement of gas molecules from an enclosed space into a evacuated space.



Effusion diffusion
The key concept to underlying Graham s Law is the following. The faster the speed of a molecule, the faster it will effuse. Let s compare two gases at the same temperature and pressure. Which gas will effuse faster?
It is important to define what we mean by "the faster it will effuse". We can measure the time it takes the gas to effuse or we can determine the rate of effusion by measuring the number of molecules per second or the volume of gas released per second. There is a difference between "rate" and "time". The rate of effusion can be expressed in units of molecules/sec, cm/sec or cm3/sec.
Because the density of a gas changes depending on the temperature, pressure and volume, it is more practical to use the molecular weight of a gas in effusion problem calculations. The density of a gas is related to its molecular weight or molar mass.
PV = nRT
M = m/n
n = m/M
PV = m RT/M
P = mass/V RT/M
P= dRT/M
The rate of effusion of a gas is inversely proportional to the square root of its molecular weight.

When comparing two gases at the the same pressure and temperature,
We can define the kinetic energy of a single gas molecule as
KE = (1/2)mv2molecule
At a given temperature, different gases have the same average kinetic energy.
average KEgasA = average KEgasB
(1/2)mAuA2 = (1/2)mBuB2
Since the mass of gas A is different than the mass of Gas B, in order for this relationship to be true, the speed of the lighter molecule must be greater than the speed of the heavier gas molecule. In this case, the speed of the gas molecules is the root-mean-square speed of the molecules.

Where the root-mean-square speed, u, is the square root of the average speeds of the molecules in a sample of gas at a specific temperature and pressure.( derive Grahams law))
Rate1/Rate2= U1/U2

Does it makes sense that rate of effusion is inversly proportional to the molecular weight of the gas molecule?
Diffusion
Diffusion refers to the movement of gas molecules (Gas A) from a space where the gas A molecule are concentrated to a space or through a space in which there other gas molecules (Gas B) are present (a less concentrated space for Gas A).

diffusion
Comparing two gases at the same pressure and temperature, lower molecular mass molecules diffuse faster than higher molecular mass molecules.

Dalton s Law of Partial Pressures
Dalton s law of partial pressures states that the total pressure exerted by a mixture of gases is the sum of partial pressure of each individual gas present. Each gas is assumed to be an ideal gas.
Ptotal = P1 + P2 + P3 + . .
Where P1 and P2 are the partial pressure of gas 1and gas 2 in the mixture. Since each gas behaves independently, the ideal gas law can be used to calculate the pressure of that gas if we know the number of moles of the gas, the total volume of the container, and the temperature of the gas.

Each gas exerts the same pressure they would exert if they were in the container alone. For example, if a mixture of gases at 298 K in a 1.00 L container consists of 1.00 g of H2, 1.50 g of N2, and 2.00 g of Br2. The partial pressure of each gas can be written

The total pressure can be expressed as

The total pressure in this example is 13.8 atm.
Note the following observations about gas mixtures are important to remember. Each gas occupies the entire volume of the container.
The gases will mix homogeneously. The gases should not react (no chemical reaction should occur between the gases in the mixture).
Maxwell-Boltzmann distribution of molecular speed of gases
The average of speed of gas is a constant at constant temp. but each gas molecule has its own special speed. So that, this leads to distribution of molecular speeds.

Figure 5: Boltzmann- Maxwell distribution of molecular speeds as a function of temperature


Figure 6: Boltzmann- Maxwell distribution of molecular speeds as a function of temperature.

Collision theory of gases

Collision theory is a theory proposed independently by Max Trautz in 1916 and William Lewis in 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. The collision theory states that when suitable particles of the reactant hit each other, only a certain percentage of the collisions cause any noticeable or significant chemical change; These successful changes are called successful collisions. The successful collisions have enough energy, also known as activation energy, at the moment of impact to break the preexisting bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant particles or raising the temperature, thus bringing about more collisions and therefore many more successful collisions, increases the rate of reaction.
When a catalyst is involved in the collision between the reactant molecules, less energy is required for the chemical change to take place, and hence more collisions have sufficient energy for reaction to occur. The reaction rate therefore increases
One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time. Assuming an ideal gas, a derivation results in an equation for total number of collisions per unit time per area:
Z11=?2??2C- n (homogeneous collision)
Z12= 1/?2??2C-n2 (heterogeneous collision)


Types of collisios:
There are three main type among gas molecules ( for both Z11 and Z12), these are ,
1-glancing collisions which lead to reduce speeds for both collided molecules in comparison with their original speeds (i.e C0> C-, C0 is the speed before any collision, and C- is the speed after making collision).



2-The second type is the head collision in this case the mass centre of both molecules at straight line (?= 180?). In this case speed of molecules increase by doubling (C-=2C0),
3-Mixed collisions: This type is more common and it is a mixture of the above two types (C-=?2 C0).

Mean Free Path
The mean free path or average distance between collisions for a gas molecule may be estimated from kinetic theory. Serway s approach is a good visualization - if the molecules have diameter d, then the effective cross-section for collision can be modeled by

using a circle of diameter 2d to represent a molecule s effective collision area while treating the "target" molecules as point masses. In time t, the circle would sweep out the volume shown and the number of collisions can be estimated from the number of gas molecules that were in that volume.

The mean free path could then be taken as the length of the path divided by the number of collisions.

The problem with this expression is that the average molecular velocity is used, but the target molecules are also moving.

L= C-/Z11 or L= C- /Z12
L11= 1/?2??2C-n,
L12=?2/??2C-n2

Rate constant
The rate constant for a bimolecular gas phase reaction as predicted by collision theory is:
.
where:
• Z is the collision frequency.
• is the steric factor.
• Ea is the activation energy of the reaction.
• T is the temperature.
• R is gas constant.
The collision frequency is:

whereas:
• NA is the Avogadro constant
• ?AB is the reaction cross section
• kB is Boltzmann s constant
• ?AB is the reduced mass of the reactants.
Quantitative insights
Consider the reaction:
A + B ? C
In collision theory it is considered that two particles A and B will collide if their nuclei get closer than a certain distance. The area around a molecule A in which it can collide with an approaching B molecule is called the cross section (?AB) of the reaction and is, in principle, the area corresponding to a circle whose radius ( ) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume (? r2 AB C-A) per a second as it moves, where CA is the average velocity of the particle.
From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of,

where is Boltzmann constant and is the mass of the molecule.
The solution of the two body problem states that two different moving bodies can be treated as one body which has the reduced mass of both and moves with the velocity of the center of mass, so, in this system must be used instead of .
Therefore, the total collision frequency of all A molecules, with all B molecules, is:

From Maxwell- Boltzmann distribution, it can be deduced that the fraction of collisions with more energy than the activation energy .Therefore the rate of a bimolecular reaction for ideal gases will be:

Where: Z is the collision frequency, is the steric factor, Ea is the activation energy of the reaction, T is the absolute temperature., R is gas constant.


Effect of temperature, pressure and density on collision frequency
Temperature is evident from the collisional frequency equation, when temperature increases, the collisional frequency increases. Density from a conceptual point, if the density is increased, the number of molecules per volume is also increased. If everything else remains constant, a single reactant comes in contact with more atoms in a denser system. Thus if density is increased, the collisional frequency must also increase.
Size of Reactants
Increasing the size of the reactants increases the collisional frequency. This is directly due to increasing the radius of the reactants as this increases the collisional cross section. This in turn increases the collisional cylinder. Because radius term is squared, if the radius of one of the reactants is doubled, the collisional frequency is quadrupled. If the radii of both reactants are doubled, the collisional frequency is increased by a factor of 16.

Compressibility factor
The compressibility factor (Z), also known as the compression factor, is the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. It is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound specific empirical constants as input.
The compressibility factor is defined as

For an ideal gas the compressibility factor is per definition. In many real world applications requirements for accuracy demand that deviations from ideal gas behaviour, i.e., real gas behaviour, is taken into account. The value of generally increases with pressure and decreases with temperature. At high pressures molecules are colliding more often.
Generalized compressibility factor graphs for pure gases

Generalized compressibility factor diagram.
The unique relationship between the compressibility factor and the reduced temperature, , and the reduced pressure, , was first recognized by Johannes Diderik van der Waals in 1873 and is known as the two-parameter principle of corresponding states. The principle of corresponding states expresses the generalization that the properties of a gas which are dependent on intermolecular forces are related to the critical properties of the gas in a universal way. That provides a most important basis for developing correlations of molecular properties. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, , and reduced pressure, , should have the same compressibility factor.
The reduced temperature and pressure are defined by
and
Here and are known as the critical temperature and critical pressure of a gas. They are characteristics of each specific gas with being the temperature above which it is not possible to liquefy a given gas and is the minimum pressure required to liquefy a given gas at its critical temperature. Together they define the critical point of a fluid above which distinct liquid and gas phases of a given fluid do not exist.
Determination of gas compressibility values
The ideal gas law is defined as:
and the ideal gas law corrected for non-ideality is defined is:

where:
P = pressure
Vm = molar volume of the gas

Z = compressibility factor
R = Universal gas constant

T = temperature
and thus:
which is the simplest and most widely used real gas equation of state (EOS). The major limitation of this equation of state is that the gas compressibility factor, Z, is not a constant but varies from one gas to another as well as with the temperature and pressure of the gas under consideration. So that It must be determined experimentally.
Where experimental data is available for specific gases, that data may be used to produce graphs (such as in Figure 1) of Z versus pressure at a constant temperature or of Z versus pressure for various temperatures for those specific gases. Such graphs are useful for readily obtaining interpolated values of Z between the experimentally determined values. The compressibility factor, as mentioned earlier, may also be expressed as:


The van der Waals equation
The van der Waals equation was developed in 1873 and may be expressed as

Where a is a measure of the strength of attraction between the gas molecules
b accounts for the volume occupied by gas molecules, which decreases the available open volume, The van der Waals equation may be re-arranged as:

and the compressibility factor can be written as:

and we now have an equation for determining Z by using the van der Waals parameters a and b:

Although a and b are referred to as "the Van der Waals constants", they are not truly constants because they vary from one gas to another; they are, however, independent of P, V and T. In other words, they are constant for the gas being considered. Given the critical temperature and pressure for a specific gas, a and b can be obtained for that specific gas from these equations:
Deviations from Ideal Gas Law Behavior
Term contained a second constant (a) and has the form: an2/V2. The complete van der Waals equation is therefore written as follows.

This equation is something of a mixed blessing. It provides a much better fit with the behavior of a real gas than the ideal gas equation. But it does this at the cost of a loss in generality. The ideal gas equation is equally valid for any gas, whereas the van der Waals equation contains a pair of constants (a and b) that change from gas to gas.
The ideal gas equation predicts that a plot of PV versus P for a gas would be a horizontal line because PV should be a constant. Experimental data for PV versus P for H2 and N2 gas at 0?C and CO2 at 40C are given in the figure below. Values of the van der Waals constants for these and other gases are given in the table below.
Vader Waals equation and internal pressure of real gase
The general form of Vader Waals equation including volume correction (b) and pressure correction (a) takes the following form:
(p+ an2/V2)(V-nb)=nRT
For one mole of a gas it takes the form:

(p+a/Vm2)(Vm-b)=RT
P= RT/(Vm-b)- a/Vm2
The term (a/Vm2) known as internal pressure of gas (IP)

P=RT/(Vm-b)- IP
IP= RT/(Vm-b)-P
This type of pressure results from attraction forces among real gas molecules and so that it equal to zero in case of an ideal gas.