Statistical Physics Lecture Notes PDF

Statistical Physics Mr. P.R.S.Tissera Department of Physical Sciences & Technology Faculty of Applied Sciences Sabaragamuwa University of Sri Lanka  Thermodynamic Quantities  Classical or Quantum mechanics can be used to ca...

Statistical Physics Mr. P.R.S.Tissera Department of Physical Sciences & Technology Faculty of Applied Sciences Sabaragamuwa University of Sri Lanka  Thermodynamic Quantities  Classical or Quantum mechanics can be used to calculate microscopic properties such as; velocity, momentum, energy, moment of inertia, impulse and position of these particles  The properties such as temperature, pressure, specific heat, total energy etc. which depend on the overall or the most probable behavior of the system under given conditions are known as macroscopic properties  Macroscopic properties can usually be measured directly in the laboratory  The approach which used to treat the physical system consisting of large number of particles using average values of the microscopic quantities known as the statistical mechanics 2  Temperature and Thermal Equilibrium Figure 1 : (a) Systems A and B are separated by an adiabatic wall. The systems have different temperatures TA and TB (b) Systems A and B are separated by a diathermic wall. The systems, having come to thermal equilibrium, have the same temperature T 3  When two systems are placed in to contact through a diathermic wall, the exchange of energy causes the macroscopic properties of the two systems to change  If the systems are confined gases, for example, the pressure might be one of the macroscopic quantities that change  The change are relatively rapid at first, but become slower and slower as time goes on, until finally the macroscopic properties approach constant values  When this occurs, we say the two systems are in thermal equilibrium with each other 4  It might, however, be inconvenient or even impossible to move two systems so that they would be in contact with one another  The systems might be too bulky to move easily, or they might be separated by a very high distance.  The way to test whether such separated systems are in thermal equilibrium is to use a third system C  By placing C into contact with A and then with B, we could discover whether A and B are in thermal equilibrium without ever bringing A and B into direct contact  This is summarized as a postulate called the Zeroth law of thermodynamics 5  Zeroth Law If systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other This third system C may be identified as a thermometer  This Zeroth law in effect defines the concept of temperature, which is fundamental to the first and second laws of thermodynamics  The law that establishes the temperature should have a lower number, so it is called zeroth law 6  Temperature  When two systems are in thermal equilibrium, say that they have the same temperature  Conversely, temperature is the only property of a system which equals that of another system when the two systems are in thermal equilibrium  Another statement of the Zeroth law, more formal and more fundamental, is the following; There exists a scalar quantity called temperature, which is a property of all thermodynamic systems in equilibrium. Two systems are in thermal equilibrium if and only if temperatures are equal 7  Measuring Temperature  Temperature is one of the seven base units  However, temperature has a natural different from that of other SI base units, and so this scheme will not work in quite that simple form  For instance, if we define one period of vibration of the light emitted by a cesium atom as a standard of time, then two such vibrations last for twice the time and any arbitrary time interval can be measured in terms of the number of vibrations  But even if we define a standard of temperature, such as that of water boiling under certain conditions, we have no procedure to determine a temperature twice as large. Two pots of boiling water, after all, have the same temperature as one pot 8 Establish a Measuring Scale for Temperature  We find a substance that has a property that varies with temperature, and we measure that property  The substance we choose is called the thermometric substance, and the property that depends on temperature is called the thermometric property Example:  the volume of a liquid (as in the common glassbulb mercury thermometer),  the pressure of a gas kept at constant volume,  the electrical resistance of a wire,  the length of a strip of metal, or  the colour of a lamp filament. 9  Let us assume that our particular thermometer is based on a system in which we measure the value of the thermometric property X  The temperature T is some function of X, T(X). We choose the simplest possible relationship between T and X, the linear function given by TX  a X  b Where the constants ‘a’ and ‘b’ must be determined  To determine a temperature on this scale, we choose two calibration points, arbitrarily define the temperatures T1 and T2 at those points, and measure the corresponding values X1 and X2 of the thermometric property 10  The Celsius and Fahrenheit Scales  In nearly all countries of the world, the Celsius scale is used for all popular and commercial and most scientific measurements  The Celsius scale was originally based on two calibration points: the normal freezing point of water (defined to be 0C) and the normal boiling point of water (defined to be 100C)  These two points were used to calibrate thermometers, and other temperatures were then deduced by interpolation or extrapolation 11  The Fahrenheit scale, used in the United States, employs a smaller degree than the Celsius scale, and its zero is set to a different temperature  It was also originally based on two fixed points, the interval between which was set to 100 degree: the freezing point of a mixture of ice and salt, and the normal human body temperature  The relation between the Celsius and the Fahrenheit scale is 9 TF  TC  32 5  Temperature is expressed in Celsius and Fahrenheit scale, respectively, C and F whereas temperature interval is expressed as C and F 12  The Kelvin Scale  On the Kelvin scale, one of the calibration points is defined to be at a temperature of zero, where the thermometric property also has a value of zero Then TX  a X  To determine a temperature on this scale, we need only one calibration point P. At that point, the temperature is defined to be Tp and the thermometric property has the measured value Xp TP In this case a  XP T X   X and so TP XP 13  By general agreement, we choose our calibration temperature at which ice, liquid water, and water vapour coexist in equilibrium  This point, which is very close to the normal freezing point of water, is called the triple point of water  The temperature at the triple point has been set by international agreement to be Ttriple = 273.16 K 1  The Kelvin temperature scale is defined as of the 273.16 temperature of the triple point of water 14  With the choice of that calibration point, the above equation become TX   273.16 K  X X triple  The definition of the Kelvin temperature scale is done as stated above to keep the size of the degree on the Celsius and the Kelvin scales to be same  But the zero of the Celsius scale is not same as that of the Kelvin scale  The relationship between the Celsius temperature TC and the Kelvin temperature T is now set as TC  T  273.15 15 Table 1 : The Kelvin, Celsius, and Fahrenheit temperature scales compared Temperature Description Kelvin Celsius Fahrenheit Scale Scale (K) Scale (C) (K) Absolute zero 0 273.15 459.67 Boiling point of liquid nitrogen 77 196 321 Freezing point of water 273.15 0 32 Triple point of water 273.16 0.01 32.018 Normal body temperature 310.15 37 98.6 Normal boiling point of water 373.15 100 212 16  The Ideal Gas Temperature Scale  To obtain a definite temperature scale, we must select one particular kind of thermometer as the standard  The choice will be made, not on the basis of experimental convenience, but by inquiring whether the temperature scale defined by a particular thermometer proves to be useful in formulating the laws of physics  The smallest variation in readings is found among ‘constant volume gasthermometers’ using different gases, which suggests that we choose a gas as the standard thermometric substance 17   The bulb B is immersed in a bath whose temperature T is to be measured  The difference between the pressure of the gas in a bulb and atmospheric pressure is determined by the height h of the column of mercury Figure 2 : A constantvolume gas thermometer 18  The constantvolume gas thermometer uses the pressure of a gas at constant volume as the thermometric property  The bulb B containing some gas is put into the bath or environment whose temperature T is to be measured; by raising or lowering the mercury reservoir R, the mercury in the left branch of the Utube can be made to coincide with a fixed reference mark, thus keeping the confined gas at a constant volume  The difference between the pressure P of the confined gas on the left branch of the tube and the pressure P0 of the atmosphere on the right branch of the tube is indicated by the height h of the column of mercury, and thus P  P0   g h Where  is the density of the mercury in the manometer 19  Let a certain amount of gas, nitrogen; for instance, be put into the bulb so that when the bulb is surrounded by water at the triple point the pressure Ptriple is equal to a definite value  Now we immerse the bulb in the system whose temperature T we wish to measure, and with the volume kept constant at its previous value, we measure the gas pressure P and calculate the temperature T of the system using following equation TP   273.16 K  constant V  P , Ptriple 20 End of the Chapter 03 21