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KINETIC THEORY OF GASES,DISTRIBUTION OF MOLECULAR VELOCITY -THERMODYNAMICS.pptx
1. NAME-SUBRAT KUMAR YASHWANT
REG NO-21631203
DEPT OF MECHANICAL
3RD YEAR/V SEM
SUBJECT-ADVANCED ENGINEERING THERMODYNAMICS
SUBMITTED TO-MR.MAYANK POKHRIYAL
2. Kinetic theory of gases
Kinetic theory of gases defines properties such as temperature, volume, pressure s well as transport
properties such as viscosity and thermal conductivity as well as mass diffusivity. It basically explains all the
properties that are related to the microscopic phenomenon.
3. Kinetic Theory of Gases Assumptions
Kinetic theory of gases considers the atoms or molecules of a gas as constantly moving
point masses, with huge inter-particle distance and may undergo perfectly elastic
collisions. Implications of these assumptions are:-
i) Particles
Gas is a collection of a large number of atoms or molecules.
ii) Point Masses
Atoms or molecules making up the gas are very small particles like a point(dot) on a paper
with a small mass.
iii) Negligible Volume Particles
Particles are generally far apart such that their inter-particle distance is much larger than
the particle size and there is large free unoccupied space in the container. Compared to the
volume of the container, the volume of the particle is negligible (zero volume).
iv) Nil Force of Interaction
Particles are independent. They do not have any (attractive or repulsive) interactions
among them.
4. v) Particles in Motion
The particles are always in constant motion. Because of the lack of interactions and the free space available, the
particles randomly move in all directions but in a straight line.
vi) Volume of Gas
Because of motion, gas particles, occupy the total volume of the container whether it is small or big and hence
the volume of the container to be treated as the volume of the gases.
vi) Mean Free Path
This is the average distance a particle travels to meet another particle.
vii) Kinetic Energy of the Particle
Since the particles are always in motion, they have average kinetic energy proportional to the temperature of the
gas.
viii) Constancy of Energy / Momentum
Moving particles may collide with other particles or containers. But the collisions are perfectly elastic. Collisions
do not change the energy or momentum of the particle.
ix) Pressure of Gas
The collision of the particles on the walls of the container exerts a force on the walls of the container. Force per
unit area is the pressure. The pressure of the gas is thus proportional to the number of particles colliding
(frequency of collisions) in unit time per unit area on the wall of the container.
5. Kinetic Theory of Gases Postulates
The kinetic theory of gas postulates is useful in the understanding of the macroscopic
properties from the microscopic properties.
Gases consist of a large number of tiny particles (atoms and molecules). These particles are
extremely small compared to the distance between the particles. The size of the individual
particle is considered negligible and most of the volume occupied by the gas is empty space.
These molecules are in constant random motion which results in colliding with each other and
with the walls of the container. As the gas molecules collide with the walls of a container, the
molecules impart some momentum to the walls. Basically, this results in the production of a
force that can be measured. So, if we divide this force by the area it is defined to be the
pressure.
The collisions between the molecules and the walls are perfectly elastic. That means when the
molecules collide they do not lose kinetic energy. Molecules never slow down and will stay at
the same speed.
The average kinetic energy of the gas particles changes with temperature. i.e., The higher the
temperature, the higher the average kinetic energy of the gas.
6. 1.Understanding Gas Laws of Ideal Gases:
i) Pressure α Amount or Number of Particles at Constant Volume:
The collision of the particles on the walls of the container creates pressure. Larger the
number of the particle (amount) of the gas, the more the number of particles colliding
with the walls of the container.
At constant temperature and volume, larger the amount (or the number of particles) of the
gas higher will be the pressure.
ii) Avogadro’s Law – N α V at Constant Pressure:
When there is a greater number of particles it increases the collisions and the pressure. If
the pressure is to remain constant, the number of collisions can be reduced only by
increasing the volume.
At constant pressure, the volume is proportional to the amount of gas.
ii) Boyle’s Law – Pressure
7. at Constant Temperature:
At a constant temperature, the kinetic energy of particles remains the same. If the volume is
reduced at a constant temperature, then the number of particles in unit volume or area increases
If there is an increased number of particles in the unit area then it increases the frequency of
collisions per unit area.
At constant temperature, the smaller the volume of the container, the larger the pressure.
ii) Amonton’s Law: P α T at Constant Volume:
The kinetic energy of the particle increases with temperature. When the volume is constant, with
increased energy, particles move fast and increase the frequency of collisions per unit time on
the walls of the container and hence the pressure.
At constant volume, the higher the temperature higher will be the pressure of the gas.
iv) Charles’s Law – V α T at Constant Pressure:
Change of temperature changes proportionately to the pressure. If the pressure also has to
remain constant, then the number of collisions has to be changed proportionately. At constant
pressure and a constant amount of substance, collisions can be changed only by changing the
area or volume.
At constant pressure, volume changes proportionally to temperature.
8. iv) Charles’s Law – V α T at Constant Pressure:
Change of temperature changes proportionately to the pressure. If the pressure also has to
remain constant, then the number of collisions has to be changed proportionately. At constant
pressure and a constant amount of substance, collisions can be changed only by changing the
area or volume.
At constant pressure, volume changes proportionally to temperature.
v) Graham Law of Diffusion –
Two gases with molecular weights M1 and M2 will have the same kinetic energy at the same temperature. Then,
The velocity of the molecules are inversely proportional to their molecular weights.
9. 2.Understanding Non-ideal Gas Behaviour:
All the gas molecules obey the ideal gas laws only under special conditions of low
pressures and high temperatures. The deviations of the real gases, from the ideal gas
behaviour, is traced mainly to wrong or incorrect assumptions in the postulates.
They are,
• The particles are point charges and have no volume: Then, it should be possible to
compress the gases to zero volume. But, gases cannot be compressed to zero volume
indicates that particles do have volume though small and cannot be neglected.
• Particles are independent and do not interact: Particles do interact depending upon
their nature. The interactions affect the pressure of the gas. Volume and the interactions
differ from gas to gas. Many gas laws have been developed for the real gases
incorporating correction factors in the pressure and volume of the gases.
• Particles collisions are not elastic: Particle collisions are elastic and they exchange
energy. The particles hence do not have the same energy and have a distribution of
energy.
10. 3.Maxwell – Boltzmann Molecular Distribution of Energy and
Velocity:
The kinetic theory of gas postulates predicted the particles as always in motion and that
they have kinetic energy proportional to the temperature of the gas.
This concept was used by Maxwell – Boltzmann, to find the distribution of gaseous
particles between energy zero to infinity and calculate the most probable, average and
root mean square velocity of the particles. we must define a distribution function of
molecular speeds, since with a finite number of molecules, the probability that a
molecule will have exactly a given speed is 0.
We define the distribution functionf(v) by saying that the
expected number N(v1,v2) of particles with speeds
between v1 and v2 is given by
N (v1,v2) =N ∫ f(v) dv.
v2
v1
11. (Since N is dimensionless, the unit of f(v) is seconds per meter.) We can write this equation
conveniently in differential form:
dN=N f(v) dv.
In this form, we can understand the equation as saying that the number of molecules
with speeds between v and v+dv is the total number of molecules in the sample
times f(v) times dv. That is, the probability that a molecule’s speed is between v and
v+dv is f(v)dv.
MAXWELL-BOLTZMANN DISTRIBUTION OF SPEEDS
The distribution function for speeds of particles in an ideal gas at temperature T is
12. That the curve is shifted to higher speeds at higher temperatures, with a broader range of
speeds.
We can use a probability distribution to calculate average values by multiplying the distribution
function by the quantity to be averaged and integrating the product over all possible speeds.
(This is analogous to calculating averages of discrete distributions, where you multiply each
value by the number of times it occurs, add the results, and divide by the number of values. The
integral is analogous to the first two steps, and the normalization is analogous to dividing by the
number of values.) Thus the average velocity is as in Pressure, Temperature, and RMS Speed. The
most probable speed, also called the peak speed vp, is the speed at the peak of the velocity
distribution.
13. Similarly,
(In statistics it would be called the mode.) It is less than the rms speedvrms. The most
probable speed can be calculated by the more familiar method of setting the derivative of the
distribution function, with respect to v, equal to 0. The result is
which is less than vrms. In fact, the rms speed is greater than both the most probable speed
and the average speed.
The peak speed provides a sometimes more convenient way to write the Maxwell-Boltzmann
distribution function:
14.
15. PROBLEM-SOLVING STRATEGY
Speed Distribution
Step 1. Examine the situation to determine that it relates to the distribution of molecular speeds.
Step 2. Make a list of what quantities are given or can be inferred from the problem as stated
(identify the known quantities).
Step 3. Identify exactly what needs to be determined in the problem (identify the unknown
quantities). A written list is useful.
Step 4. Convert known values into proper SI units (K for temperature, Pa for pressure, m3 for
volume, molecules for N, and moles for n). In many cases, though, using R and the molar mass
will be more convenient than using kB and the molecular mass.
Step 5. Determine whether you need the distribution function for velocity or the one for energy,
and whether you are using a formula for one of the characteristic speeds (average, most
probably, or rms), finding a ratio of values of the distribution function, or approximating an
integral.
16. Step 6. Solve the appropriate equation for the ideal gas law for the quantity to be
determined (the unknown quantity). Note that if you are taking a ratio of values of the
distribution function, the normalization factors divide out. Or if approximating an integral,
use the method asked for in the problem.
Step 7. Substitute the known quantities, along with their units, into the appropriate
equation and obtain numerical solutions complete with units.
17. Transport properties of gases
Transport properties of a substance:
the ability of the substance to transfer matter, energy or some
other properties, from one place to another.
• Diffusion: matter flows in response to gradient of concentration.
• Thermal conduction: migration of energy because of a temperature gradient
• Electric conduction: migration of electric charges along an electric potential
gradient.
• Viscosity: migration of linear momentum results from a velocity gradient.
• Transport properties are expressed by phenomenological equations (empiric
equations obtained from a summary of experiments and observations)
18. Phenomenological equations:-
Flux, J represents the quantity of a property passing through a given area in a given
interval of time.
Types of flux:
• Matter flux: if the matter is flowing through the area in the interval of time > number
of molecules/(m2 s).
• Energy flux: if energy is flowing through the area in the interval of time > J/(m2 s).
Experimental observations on transport properties show that the flux of a property is
is proportional to the first derivative of a related property (gradient of the property)
