Physics 334Modern PhysicsCredits: Material for this PowerPoint was adopted from Rick Trebinos lectures from Georgia Tech which were based on the textbook Modern Physics by Thornton and Rex. Many of the images have been used also from Modern Physics by Tipler and Llewellyn, others from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc), and contributions are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class, as a study aid only.

Why StatisticsProbability DistributionGaussian Distribution FunctionsTemperature and Ideal GasThe Maxwell-Boltzmann DistributionDensity of States

Chapter 2 Statistics and Thermodynamics

What has Statistics to do with Modern PhysicsIn physics and engineering courses we talk about motion of baseballs, airplanes, rigid bodies etc. These descriptions are from a macroscopic point of view

In Kinetic theory of gases, the macroscopic properties of gases for example, pressure, volume and temperature are based on microscopic motion of molecules and atoms that make up the gas.

All matter is made of atoms and molecules, so it make sense to talk about macroscopic quantities in light of microscopic quantities. Therefore it is necessary to combine mechanics (classical or quantum) with statistics. This is called Statistical Mechanics.

The main concept that will be used is the concept of probabilities, since in quantum mechanics the interest is in measuring the probabilities of a physical quantity (position, momentum, energy etc.)

Probability Distribution FunctionA distribution function f(x) or (x) is a function that gives the probability of the particle found in a certain range of allowed values of an event.

An event is an act of making a measurement of an observable. Say if our measurement is of position x than the probability function of finding the particle in the range x and x+dx is

Probability Calculations The probability of finding the particle is given by

Normalization The particle has to be some where. Normalization means that the sum or integral should be equal to 1.

NormalizationExample: Find the normalizing constant A for the function

Expectation (Average) ValueAverage value of a physical quantity is called expectation value. If several measurements are made, the value that is expected on the average is the expectation value even though no single measurement may be the expectation value. This is also the mean value.

Expectation ValueExample: Find the expectation value of the function

Standard DeviationStandard deviation () tells us how much deviation to expect from the average value. Variance (2) also gives us the spread.

Gaussian DistributionBinomial distribution, Gaussian distribution and Poisson distribution are the most common distributions encountered in physics.

We will discuss only the Gaussian distribution as it occurs widely and is used in many different fields. Gaussian distribution is also called Bell curve or standard distribution.

Consider a drunk undergoing a random walk.

Startx

Gaussian DistributionDisplacement is the sum of several random steps. The probability of small steps to cancel each other out is greater than the probability of large steps which might be in the same direction.

The distribution function is given as

Bell CurveThe solid green line shows thatThe most probable value (peak) is at x=25The median (50% level) is at x=25The mean is at x=25

= 6Red dash linex-

Thermal PhysicsThermal physics is the study of TemperatureHeatHow these affect matterHow heat is transferred between systems and to the environmentHeatIt is a process in which energy is exchanged because of temperature differences.

Thermal ContactObjects are said to be in thermal contact if energy can be exchanged between them.

Thermal EquilibriumEnergy cease to exchange between objects

Zeroth Law of ThermodynamicsIf objects A and B are separately in thermal equilibrium with a third object, C, then A and B are in thermal equilibrium with each other.Allows a definition of temperature

Temperature is the property that determines whether or not an object is in thermal equilibrium with other objects

ThermometersUsed to measure the temperature of an object or a systemMake use of physical properties that change with temperatureMany physical properties can be usedvolume of a liquidlength of a solidpressure of a gas held at constant volumevolume of a gas held at constant pressureelectric resistance of a conductorcolor of a very hot object

Thermometers, contA mercury thermometer is an example of a common thermometerThe level of the mercury rises due to thermal expansionTemperature can be defined by the height of the mercury column

Temperature ScalesThermometers can be calibrated by placing them in thermal contact with an environment that remains at constant temperatureEnvironment could be mixture of ice and water in thermal equilibriumAlso commonly used is water and steam in thermal equilibrium

Celsius ScaleTemperature of an ice-water mixture is defined as 0 CThis is the freezing point of waterTemperature of a water-steam mixture is defined as 100 CThis is the boiling point of waterDistance between these points is divided into 100 segments or degrees

Fahrenheit ScalesMost common scale used in the USTemperature of the freezing point is 32Temperature of the boiling point is 212180 divisions between the points

Kelvin ScaleWhen the pressure of a gas goes to zero, its temperature is 273.15 CThis temperature is called absolute zeroThis is the zero point of the Kelvin scale273.15 C = 0 KTo convert: TC = TK 273.15The size of the degree in the Kelvin scale is the same as the size of a Celsius degree

Pressure-Temperature GraphAll gases extrapolate to the same temperature at zero pressureThis temperature is absolute zero

Modern Definition of Kelvin ScaleDefined in terms of two pointsAgreed upon by International Committee on Weights and Measures in 1954First point is absolute zeroSecond point is the triple point of waterTriple point is the single point where water can exist as solid, liquid, and gasSingle temperature and pressureOccurs at 0.01 C and P = 4.58 mm Hg

Modern Definition of Kelvin Scale, contThe temperature of the triple point on the Kelvin scale is 273.16 KTherefore, the current definition of the Kelvin is defined as 1/273.16 of the temperature of the triple point of water

Some KelvinTemperaturesSome representative Kelvin temperaturesNote, this scale is logarithmicAbsolute zero has never been reached

Comparing Temperature Scales

Converting Among Temperature Scales

Temperature and Kinetic EnergyThe average kinetic energy of a molecule in thermal equilibrium with its surrounding is given by

ExampleCalculate the average kinetic energy of a gas molecule at room temperature T=20 degree Celsius

It is useful to remember that at room temperature of 300 K the value of kT is

Ideal GasA gas does not have a fixed volume or pressureIn a container, the gas expands to fill the containerMost gases at room temperature and pressure behave approximately as an ideal gas.

Exercise: Consider a container of gas that has a volume V in thermal equilibrium at a temperature T. Show that the pressure P is given by PV=NkT, where N is the Avogadro's number and k is the Boltzmann constant.

Example: Calculate the volume occupied by one mole of molecules at a pressure of one atmosphere (1.01x105N/m2) and a temperature of 273 K. This condition is called Standard Temperature and Pressure (STP)

Ideal GasExample: Calculate the volume occupied by one mole of molecules at a pressure of one atmosphere (1.01x105N/m2) and a temperature of 273 K. This condition is called Standard Temperature and Pressure (STP).

*Photo: http://www.spaceandmotion.com/quantum-theory-max-planck-quotes.htm***********