Category: Physics Notes PPT

The kinetic theory states that the average kinetic energy of gas molecules of an ideal gas is directly proportional to the absolute temperature of the molecules.

It is independent of the pressure, volume, and nature of the gas.

Such a kind of interpretation of temperature is called the kinetic interpretation of temperature.

The entire structure of the kinetic theory is based on the following assumptions that were stated by Classius.

What are the Assumptions Made About Molecules in an Ideal Gas?

The major assumptions made about the molecules in an ideal gas are:

• There are a large number of molecules present in gas and all the molecules of gas are identical to each other and have the same mass.

• The molecules of gases obey Newton’s law of motion and are therefore in a state of continuous motion which is not only random but also isotropic, that is, the motion of the molecules is the same in all directions.

• The molecules of the gases are much smaller in size as compared to the average distance between the molecules of the gases. Also, the total volume of the gas is much less than the volume of the container in which they are contained.

• The molecules of gases make elastic collisions with not only each other but also with the walls of the container in which they are contained.

• Van der Waal forces like gravitation and attraction acting on the molecules of the gases are assumed to be negligible.

Here, We Considered Ideal Gas, Which is a Perfect Gas With the Characteristics:

1. The size of the molecule of a gas is zero i.e., it has a point mass with no dimensions.

2. The molecules of gas do not exert any force of attraction or repulsion on each other except during collision. 

 

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Consider one mole of an ideal gas at absolute temperature T, of volume V and molecular weight M. Let N be Avogadro’s number and m be the mass of each molecule of gas.

Then by the relation:  

                                [M=mtimes N]

If C is the r.m.s velocity of the gas molecules, then pressure P exerted by an ideal gas is

                     [P=frac{1}{3}frac{M}{V}C^2] or [PV=frac{1}{3}frac{M}{C^2}]

For a gas equation, PV = RT, and R is a universal gas constant 

So, 

 [frac{1}{3}frac{M}{C^2} = RT] or [frac{3}{2}RT]

  Since, M = m x N

      [frac{1}{2}mC^2] = [frac{3}{2} frac{R}{NT}] 

               [frac{1}{2}mN = frac{3}{2}kT]       

                                   (∵ R/N = k)

Kinetic Interpretation of Temperature

According to the kinetic theory of gases, the pressure P exerted by one mole of an ideal gas is given by,

[P=frac{1}{3}frac{M}{V}C^2]or[PV=frac{1}{3}frac{M}{C^2}]or[frac{1}{3}MC^2=RT]

           [=C^2=frac{3RT}{M} or C^2alpha T] (∵ R and M are constants)

           [=Calpha sqrt{T}alpha C]      

We can say that the square root of absolute temperature T of an ideal gas is directly proportional to the root mean square velocity of its molecules.

Also,

                        [frac{1}{3}frac{M}{NC^2}= frac{R}{NT}=kT]

Or,                [frac{1}{2}mC^2=frac{3}{2} kT or frac{1}{2}mC^2alpha T]

Where, 3/2 k is constant.

But, [frac{1}{2}mC^2] is the average translational kinetic energy per molecule of a gas.

Therefore, the average kinetic energy of a translational per molecule of a gas is directly proportional to the absolute temperature of the gas.

Hence, we can define absolute temperature as the temperature at which the root mean square velocity of the gas molecules reduces to zero, which means molecular motion ceases at absolute zero.

What is the Significance of Kinetic Interpretation of Temperature and Root Mean Square Speed of Gaseous Molecules?

Kinetic interpretation of temperature is very significant in the field of the kinetic theory of gases. The average kinetic energy of a molecule does not depend on the type of molecule under consideration.  The average translational kinetic energy is only dependent upon the absolute temperature of the surroundings. Compared to macroscopic energies, the kinetic energy is very small because of which we do not feel when an air molecule hits our skin. 

On the other hand, the gravitational potential energy acting on a molecule is comparatively negligible when the molecule moves from the top of a room to the bottom of a room. Therefore, with this observation, we can conveniently neglect gravitational energy in typical real-world situations. The RMS speed of the gaseous molecules is surprisingly large. These large molecular velocities do not allow macroscopic movement of air because the molecules move in all directions. The mean free path (that is, the average distance of molecules between collisions) of molecules in the air is very small, therefore, the molecules move rapidly but do not cover a very large distance in a second. 

What are the Properties Defined by the Kinetic Theory of Gases?

The kinetic theory of gases helps us understand and define various properties of gases like temperature, volume, pressure, viscosity, mass diffusivity, and thermal conductivity.

 RMS Speed of Gas

Consider an ideal gas contained in a cubical container such that the volume,

V = a3.

Let n be the number of molecules and mass of each molecule be m. 

So, [M=mtimes N]

 

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The random velocities of gas molecules are A₁, A₂, A₃,…, An, = (x₁, y₁, z₁), (x₂, y₂, z₂),…., (x_n, y_n, z_n) be the rectangular components of the velocities  c₁, c₂, c₃,…,c_n along with three mutually perpendicular directions OX, OY and OZ.

So, x₁2 + y₁2 + z₁2 = c₁2……  x_n2 + y_n2 + z_n2 = c_n2

If x₁ is the component of velocity of the molecule A₁ along OX, and the initial momentum of  A₁ along OX = mx₁,

Momentum after collision = – mx₁

So, total momentum = – mx₁ – mx₁ = – 2mx₁

However, according to the law of conservation of momentum in one dimension, momentum is transferred to the wall by molecule A₁ = 2mx₁.

Time between two successive collisions,

                  [T= frac{D}{S}=frac{2a}{x_1}]

So, momentum transferred by  A₁ = 2mx₁ * x₁/2a = m x₁2/a

Since,  P_x (Pressure along x-axis)[= frac{F}{a^2}=frac{m}{a^3}(x_1^2+x_2^2+x_n^2)]

Similarly, [P_y=frac{m}{a^3}(P_x+P_y+P_z)]

So, [P=frac{(P_x+P_y+P_z)}{3}]

[P=frac{1}{3}frac{m}{a^3}left((x_1^2+x_2^2+..+x_n^2)+(y_1^2+y_2^2+..+y_n^2)+(z_1^2+z_2^2+..+z_n^2) right )]

    [=frac{1}{3}frac{m}{V}(c_1^2+c_2^2+..+c_n^2)=frac{1}{3}frac{mn}{V}frac{(c_1^2+c_2^2+..+c_n^2)}{n}]

As, [C=frac{(c_1^2+c_2^2+..+c_n^2)}{n}]

We get,

       [P=frac{M}{3V}C^2]

 

RMS Velocity of Gas

The RMS or root means square velocity of a gas is defined as the square root of the means of the squares of the random velocities of the individual molecules of a gas.

Putting, [frac{M}{V}=rho ], we get,      

[P=frac{1}{3 rho}C^2]

Or [C=frac{sqrt{3P}}{ rho}]

Hence, knowing the values of P and ρ, the RMS velocity of the gas molecules at a given temperature can be determined.

RMS Speed of Gas Molecules

The RMS speed of gas molecules is the measure of the speed of the particles in a gas. It is the average squared velocity of molecules in a gas.

To know more about the kinetic interpretation of temperature and RMS Speed of gas molecules, seek deeper insights from the top experts of and develop a brilliant conceptual foundation. 

You know that an ant releases an invisible, fragrant chemical called Pheromone. The successive ants smell this chemical and march-like an army. So, don’t you think this is a laminar flow? Yes, that’s true.

A laminar flow in fluid mechanics is the straight or linear flow of all the particles of the fluid that is similar to the marching of an ant. If in this case ants deviate from their path and make two directions, then it is the non-laminar flow.

This page discusses laminar fluid flow, lamellar flow, non-laminar flow, lumina flow, and laminar flux in detail.

Laminar Fluid

In a fluid, for laminar flow, all the particles carry constant attributes like velocity, pressure, and speed. These particles have a smooth movement as we walk on a smooth road. 

You can think of the laminar flow of liquid as the straight road (free of turns). While driving when you encounter a turn on the left or right of the road, the laminar flow becomes a non laminar flow.

Laminar Flow in Fluid Mechanics

In fluid dynamics, laminar flow is a smooth or regular movement of particles of the fluid. In Laminar flow, the fluid flows in parallel layers with lesser lateral mixing and no disruption between the layers. We call the laminar flow a streamline or viscous flow.

The terminology ‘streamlined flow’ is descriptive of the laminar fluid flow because, in laminar flow, layers of water flow over one another at varying speeds with virtually no mixing between layers, fluid particles move in definite and observable paths or streamlines just like the marching of Indian Army.

When a fluid flows via a closed channel such as a pipe or between two flat plates, the laminar flow may occur depending on the velocity, viscosity of the fluid, and the size of the pipe. Laminar flow occurs at lower velocities and high viscosity. 

Let’s look at the visual representation of the lumina flow:

                        

 

Laminar Flow in Liquid

A laminar flow in a liquid depends on the two following factors:

1. Velocity

2. Viscosity

Whenever a fluid flows in a pipe, the velocity of the laminar fluid flow remains to change that’s what we can observe in the above figure. So, when there is a pressure difference in the below and the above layer of the laminar fluid, the velocity difference also occurs.

Real-life Application of Laminar Flow

In aerodynamics, the same concept applies to dynamic lift. We see an airfoil, which carries a pressure difference; however, the flow remains laminar, and this pressure difference allows the wings of an aeroplane to lift up. So, here, laminar flow is one of the applications in the mechanics of air.

However, when the viscosity of the liquid is higher like honey, peanut butter, milk butter; they all have a higher viscosity and when they flow after melting, their layers remain intact, and therefore, all the particles to remain in close vicinity with each other, without leaving their exact lattice point (imaginary) while making movement in the forward direction. 

This smooth forward movement is the result of the laminar flow. So, the more is the viscosity, the more is the laminar flow, and the more is the laminar flux. So, do you know how laminar flux varies and turns to turbulence? If you don’t know, let’s understand this in detail:

Laminar Flow to Non-Laminar Flow

Do you know why a laminar flow turns to a non-laminar flow? In the above example, we discussed an application of aerodynamics for the laminar flow. 

In another example, we discussed honey, ghee, peanut butter, milk butter, and so on. These all eating items have high viscosity. Let’s suppose that you kept an opened ghee packet under the Sun. Now, on melting, it starts flowing very quickly in varying directions. So, when this variation in its laminar fluid flow occurs, this is the turbulent flow or turbulence nature. 

Turbulence occurs when the velocity of the fluid increases with the decrease in its velocity. We consider the turbulence flow as a non-laminar flow.

Let’s consider another example to understand the same:

When you are hurrying to your office, and you encounter heavy traffic after making a turn, you adjust the flow of your driving to whatever direction you can take to cross the traffic, here, your driving has a turbulent or a non-laminar flow because at each adjustment to drive ahead of all the vehicles, the property of your vehicle in motion like velocity, acceleration is changing. 

There is a whole other world to the Earth than what we can see superficially. So if you could grab the Earth and cut it down the middle, you’d see that it has different layers. Obviously, the inside of our reality keeps on holding a few secrets for us. Indeed, even as we boldly investigate different universes and convey satellites into space, the internal openings of our planet stay off breaking point from us. Nonetheless, propels in seismology have enabled us to gain proficiency with a lot about the Earth and the numerous layers that influence it to up. Each layer has its very own properties, creation, and qualities that influence a significant number of the key procedures of our planet. They are, all together from the outside to the inside – the crust, the mantle, the outer core, and the inner core. We should investigate them and see what they have going on. Like every single earthbound planet, the Earth’s inside is separated. This implies its internal structure consists of layers, orchestrated like the skin of an onion. Strip back one, and you locate another, recognized from the last by its substance and topographical properties, just as tremendous contrasts in temperature and weight. Our advanced, logical comprehension of the Earth’s inside structure depends on deductions made with the assistance of seismic observation. Basically, this includes estimating sound waves produced by seismic tremors and analyzing how going through the distinctive layers of the Earth makes them moderate down. The progressions in seismic speed cause refraction which is determined (as per Snell’s Law) to decide contrasts in thickness. These are utilized, alongside estimations of the gravitational and attractive fields of the Earth and explores different avenues regarding crystalline solids at weights and temperatures normal for the Earth’s profound inside, to figure out what Earth’s layers resemble. Moreover, it is comprehended that the distinctions in temperature and weight are because of extra warmth from the planet’s underlying development, the rot of radioactive components, and the solidifying of the inner core because of exceptional weight.

Layers:

A spectral line is defined as a dark or bright line in an otherwise continuous and uniform spectrum, resulting from light’s absorption or emission in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used in the identification of molecules and atoms. These “fingerprints” are compared with the previously collected “fingerprints” of molecules and atoms and are therefore used to identify (which would otherwise be impossible) the molecular and atomic components of planets and stars.

Types of Line Spectrum

The association between a quantum system (usually electrons, but at times, atomic nuclei or molecules) and a single photon is the product of spectral lines. When the photon has up to the right amount of energy (connected to its frequency) to allow a change in the system’s energy state (in the case of an atom, this is generally an electron changing orbitals), the photon can be absorbed.

Then it will be re-emitted spontaneously at the same frequency as the cascade or the original, where the sum of the emitted photon energies will be equal to the energy of the absorbed photon (assuming that the system returns to the original state).

A spectral line can be observed either as an absorption line or an emission spline. The type of line observed will depend on the type of material and its temperature relative to the other source of emission. An absorption line can be formed when photons from a broad and hot spectrum source pass via cold material.

The intensity of light over a narrow frequency range can be reduced because of absorption by the material and re-emission in random directions. In comparison, when photons from some of the hot material are observed in the presence of a broad spectrum from a cold source, a bright emission spectrum line is also formed. The light intensity, over a narrow frequency range, gets increased because of the emission by the material.

Spectral lines are highly atom-specific and are used to define any medium’s chemical composition that can allow light to pass through it. Many elements were discovered by means of spectroscopy, including thallium, caesium, and helium. Also, the spectral lines depend on the physical conditions of the gas. Therefore, they can be widely used in the determination of the chemical composition of stars and the other celestial bodies that cannot be analyzed by other means and their physical conditions as well.

The mechanisms other than the atom-photon interaction can form spectral lines. The frequency of the involved photons will vary widely based on the exact physical interaction (with single particles, molecules, etc.), and lines are observed across the electromagnetic spectrum, ranging from radio waves to gamma rays.

Line Broadening and Shift

There are many effects which control spectral line shape. A spectral line extends over a frequency range but not a single frequency (it means it has a nonzero linewidth). Additionally, its centre can be shifted from its nominal central wavelength. There are many reasons for this shift and broadening.

These specific reasons are divided into 2 general categories. They are: broadening because of local conditions and broadening because of the extended conditions. Broadening occurs regardless of the local conditions due to the effects around the emitting element in a small area, typically enough to ensure local thermodynamic equilibrium. Broadening due to the extended conditions may result from changes to the spectral distribution of the radiation because it traverses its observer’s path. It also can result from the combining of radiation from more regions that are far from each other.

Broadening Due to Local Effects

Natural Broadening

The lifetime of the excited states will result in natural broadening, which is also called lifetime broadening. The principle of uncertainty relates the lifetime of the excited state (due to the Auger process or spontaneous radiative decay) to the uncertainty of its energy. A short lifetime will contain a large energy uncertainty and a broad emission spectrum. This specific broadening effect results in an unshifted Lorentzian profile. At the same time, the natural broadening is experimentally altered only up to the extent where decay rates are artificially enhanced or suppressed.

Broadening Due to Non-Local Effects

Some types of broadening are explained as the result of conditions over a large region of space, rather than just upon the conditions, which are local to the emitting particle.

Opacity Broadening

As it journeys through space, electromagnetic radiation emitted at a particular point in space is reabsorbed. This absorption is based on wavelength. The line is broadened due to the photons at the line centre holding a greater reabsorption probability compared to the photons at the line wings. Indeed, in contrast to the wings at the middle of the line, the reabsorption at the centre of the line may be so great as to induce a self-reversal, where the amplitude is poor.

What is Luminosity?

We have an almost intuitive understanding of the relationship between light and temperature like if we look at the stove when it is turned on, then there would be no difference whether it is turned on or off especially when the temperature is not specifically high. As the temperature increases, we would see that a red glow of the flame blows about the stove and when the temperature continues to increase, the flame turns to a bright yellowish-white color, as you can see in the two images below.

Now, if we look at other sources, like iron being heated turns blue, stars emit light, the electric bulbs emit energy in the form of small packets called photons.

Since the light has energy, therefore, the stove and the candle emit light energy.

So the light energy emitted by it per second is called its luminosity, denoted by symbol L.

The unit used for measuring the luminosity is Joules per second or J/s.

What Does Luminosity Mean?

Luminosity is described as an inherent property of objects that emit light such as a star, flame of a burning candle, iron rod on getting heated, electric bulbs. They all give off light energy in all directions every second.

So, this is the same as saying that the more the object emits light, the more it gives off the power in watts, the more will be its luminosity.

What is the Definition of Luminosity?

We define luminosity as the total amount of energy given off by an object every second.

The graph below shows that the wavelength of the yellow light is at the peak.

Let’s take a real-life example:

If an electric bulb gives off 100 watts of power, it means it gives off 100 Joules of energy every second.

We can see that the luminosity of the sun is 3.85 x 1026Watts which is a very high luminosity.

As you compare between the Sun and the bulb, the sun is four trillion times trillion more powerful light is being radiated by the Sun every second.

What is the Meaning of Luminosity?

Let’s talk about an astronomical body like a star (an imperfect blackbody); they are spherical objects, primarily made up of very hot elements .i.e., hydrogen and helium that emits light continuously.

We can’t measure the temperature of a star, all we can do is just observe the light with maximum wavelength and use the Wein’s displacement law which states that the temperature can be found at the point where the radiation curve peaks, i.e. 入peak = x 10[^{m}] nm (nano microns).

As you can see, this graph shows the spectrum of a star and through this, we can determine the wavelength where the emission peaks are given by, 入peak.

By Wein’s formula:

               入peak = (0.29 cm K)/T

Here, T = 5500 K

Putting the value in the above equation, we get:

                 入peak = 5.27 x 10[^{-5}] m

Then, compare to the spectra of computer models of stellar spectra of different temperatures (as you can see in Fig.2), and develop an exact color-temperature relation.

Luminosity Theory

In this theory, we will discuss the relationship of luminosity to temperature and the surface area of an object.

Let’s talk about the factors here:

1.  Luminosity: The total amount of energy emitted per second in Watts.

2. Apparent brightness: It determines how bright a star appears to be; the power per meter squared as measured at a distance from the star.

Its unit is Watt/meter[^{2}].

Luminosity is denoted by L.

So, LSUN = 3.85 x 10[^{26}] J/s or watts.

In the CGS system, it is 3.8 x 10[^{33}] erg/s.

So, the surface area of a star can be calculated as A = 4πR² 

Where R is the radius of a star.

If we consider a star to be a completely black body, the radiation emitted per second can be calculated by using the Stefan- Boltzmann law.

Stefan-Boltzmann states that the total energy emitted by a body is directly proportional to the fourth power of temperature.

     

    

Here s= Stefan’s constant whose value = 5.7 x 10[^{-8}] Wm[^{-2}] K[^{-4}], 

               A = Surface area of a star, and

               T = Absolute temperature of a star.

Let’s consider radius of a star like the Sun = 6.96340 x 10[^{8}] 

Surface area of star = 4πR² = 4 x 3.14 x(6.96340 x 10[^{8}])[^{2}]

TSUN = 6000 K

On calculating, we get

A = 6.09 x 10[^{18}] m[^{2}] 

Now using the equation(1):

  E = (5.7 x 10[^{-8}]) * (6.09 x 10[^{18}]) * (6000)[^{4}]

Therefore, we get the energy radiated by the sun = 4.4998 x 10[^{26}] J.

Magnetic fields are usually produced by moving electric charges and the intrinsic magnetic moments of particles that are the elementary ones which are associated with a fundamental quantum property of their spin.  The Magnetic fields and the electric fields are said to be interrelated and are both components of the forces which are electromagnetic and one of the four fundamental forces of nature.

The part which is of the magnet in a material that arises from a current which is applied externally and is not intrinsic to the material itself. Here, we are going to discover a few of the most important things related to it. The current is said to be expressed as the vector which is denoted by letter  H and is measured in units of amperes per metre. 

The definition of letter  H is given as H = B/μ − M, where letter B is said to be the magnetic flux density. That is a measure of the actual magnetic field which is inside a material considered as a concentration of magnetic field lines or the flux and that too per unit cross-sectional area. The symbol μ denotes the magnetic permeability, and the capital letter M is the magnetization. The magnetic field also denoted by H might be thought of as the magnetic field which is produced by the flow of current in wires and the magnetic field B as the total magnetic field which is also the contribution to the field M made by the magnetic properties of the materials in the field. 

Magnetic Field Intensity

A magnetic field is a field of the vector that describes the magnetic influence which is applied on moving electric charges or an electric current and magnetic materials. A moving charge that is in a magnetic field generally experiences a force which is perpendicular to its own velocity and to the magnetic field as well. A permanent magnet′s magnetic field usually is pulled on ferromagnetic materials for example as iron and repels or attracts other magnets. In addition to this, we can say that a magnetic field that varies with location will exert a force which is on a range of non-magnetic materials by affecting the motion of their outer electron atoms. The magnetic fields which generally surround magnetized materials and are created by electric currents such as those which are used in electromagnets and by the electric fields that are varying in time. Since, both the direction and strength of a magnetic field may vary with the location so they are described as a ma. A kind of map which is assigning a vector to each point of space is because of the way the magnetic field transforms under mirror reflection that is as a field of pseudovectors.

Unit of Magnetic Field Strength

In electromagnetics, the term “magnetic field” is generally used for two distinct but closely related fields of vectors which is generally denoted by the symbols denoted by B and H. In the International System of Units that is the SI unit,  H magnetic field strength is measured in the SI base units that are of ampere per meter that is A/m.  The symbol which is denoted by letter B is said to be the magnetic flux density that is said to be measured in tesla which in SI base units is kilogram per second per ampere. 

Both the terms that are H and B differ in how they account for magnetization. In a vacuum, we can see that the two fields are generally related through the vacuum permeability, In a magnetized material the terms that generally differ by the material’s magnetization at each point.

The magnetic fields that are usually used throughout modern technology particularly in electrical engineering and electromechanics as well. The magnetic field which is rotating is used in both electric motors and generators. We can say that the interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as magnetic circuits. Here we can say that the magnetic forces generally give information about the charge carriers in a material through the Hall effect. The planet Earth generally produces its own magnetic field which shields the planet that is earth’s ozone layer from the solar wind and is important in navigation using a compass.

Magnetic Field Intensity Formula

The force that is said to be on an electric charge that generally depends on its location and the speed and direction which is two vector fields that are used to describe this force. The first is the electric field which generally describes the force that is acting on a charge that is stationary and that gives the component of the force that is independent of motion. The magnetic field which we have seen describes the component of the force that is proportional to both the direction and speed of charged particles. The magnetic field is generally defined by the law of the Lorentz force and is at each instant said to be perpendicular to both the motion of the charge and the force it experiences.

There are two different but we can say very closely related fields which are both sometimes known as the “magnetic field” written as letter B and H.