PHY 205 Thermal Physics Lecture Notes PDF

FEDERAL UNIVERSITY OF TECHNOLOGY, AKURE SCHOOL OF PHYSICAL SCIENCES DEPARTMENT OF PHYSICS P. M. B. 704, AKURE, NIGERIA MODULE TITLE: THERMAL PHYSICS MODULE CODE: PHY 205 Module Contents: Introduction to Thermodynamics The foundation of Classical Thermodyna...

FEDERAL UNIVERSITY OF TECHNOLOGY, AKURE SCHOOL OF PHYSICAL SCIENCES DEPARTMENT OF PHYSICS P. M. B. 704, AKURE, NIGERIA MODULE TITLE: THERMAL PHYSICS MODULE CODE: PHY 205 Module Contents: Introduction to Thermodynamics The foundation of Classical Thermodynamics Zeroth Law of Thermodynamics and Temperature The First Law of Thermodynamics Work, Heat and Internal Energy Carnot Cycles and Second Law of Thermodynamics Entropy and Irreversibility Thermodynamic Potentials and the Maxwell Relations Applications: Qualitative Discussion References 1. Robert P. Baumann, Modern Thermodynamics with Statistical Mechanics, Macmillan, New York, 1992. 2. Ralph Baierlein, Thermal Physics, Cambridge University Press, New York, 1999. 3. Herbert B. Callen, Thermodynamics, John Wiley & Sons, New York, 1960. 4. Stephen J. Blindell, Concepts in Thermal Physics 5. Federick Reif, Fundamentals of Statistical and Thermal Physics 6. Ashley H. Carter, Classical and Statistical Thermodynamics, Prentice Hall, 2000.  Zeroth Law of Thermodynamics When two bodies at different temperatures are brought into contact, after some time they attain a common temperature and are then said to exist in thermal equilibrium. Zeroth Law of Thermodynamics states that when a body A is in thermal equilibrium with a body B, and also separately with a body C, then bodies B and C will be in thermal equilibrium with each other. A B C 𝑖. 𝑒. 𝐴 ≜ 𝐵 𝑎𝑛𝑑 𝐴 ≜ 𝐶 𝑡ℎ𝑒𝑛 𝐵 ≜ 𝐶 This law is the basis of temperature measurement. Therefore, to measure temperature, a reference body is used and a certain physical characteristics of this body which changes with temperature is selected. The reference body is the thermometer and the selected characteristic is the thermometric property. A common thermometer is one which consist of a small amount of mercury in an evacuated capillary tube. Temperature Temperature of a system is a scientific property that determines whether or not a system is in thermal equilibrium with other systems. It is a scientific property that distinguishes thermodynamics from other sciences. It can distinguish hot from cold. Temperature is very important to thermodynamics as force bears an important relation to statics and velocity does to dynamics. Measurement of Temperature There are five (5) different kinds of thermometer, each with its own thermometric property. S/N Thermometer Thermometric Property Symbol 1 Constant Volume gas thermometer Pressure 𝑃 2 Constant Pressure gas thermometer Volume 𝑉 3 Electric Resistance thermometer Resistance 𝑅  4 Thermocouple Thermal emf 𝜀 5 Mercury-in-glass thermometer Length 𝐿 It should be noted that measuring with any of these thermometers gives the same result. Relationship between Temperature and Thermometric Property If X is the thermometric property, and a corresponding temp is given as 𝜃(𝑋). When another body in equilibrium is measured, and its thermometric property is 𝑋1 , then its corresponding temperature will be 𝜃(𝑋1 ). 𝜃(𝑋) 𝜃(𝑋1 ) ∴ = ⋯ ⋯ ⋯ (1) 𝑋 𝑋1 𝜃(𝑋) ⟹ 𝜃(𝑋1 ) = ∙ 𝑋1 ⋯ ⋯ ⋯ (2) 𝑋 and if there is another body (known as body 2) in thermal equilibrium with body 1, then we can write: 𝜃(𝑋1 ) 𝜃(𝑋2 ) = ∙ 𝑋2 ⋯ ⋯ ⋯ (3) 𝑋1 Methods Used in Measuring Temperature Method in Use before 1954: Old Method The thermometer is first placed in contact with the system whose temperature 𝜃(𝑋) is to be measured, and then in contact with an arbitrarily chosen standard system in an easily reproducible state where the temperature is 𝜃(𝑋1 ). Thus, we have 𝜃(𝑋1 ) 𝜃(𝑋) = ⋯ ⋯ ⋯ (4) 𝑋1 𝑋 which can also be written as: 𝜃(𝑋1 ) 𝑋1 = ⋯ ⋯ ⋯ (5) 𝜃(𝑋) 𝑋 Then the thermometer at temperature 𝜃(𝑋) is placed in contact with another arbitrarily chosen standard system in another easily reproducible state where the temperature is 𝜃(𝑋2 ), it gives 𝜃(𝑋2 ) 𝑋2 = ⋯ ⋯ ⋯ (6) 𝜃(𝑋) 𝑋 From equations (5) and (6), we can have 𝜃(𝑋1 ) 𝜃(𝑋2 ) 𝑋1 𝑋2 − = − ⋯ ⋯ ⋯ (7) 𝜃(𝑋) 𝜃(𝑋) 𝑋 𝑋 𝜃(𝑋1 ) − 𝜃(𝑋2 ) 𝑋1 − 𝑋2 = 𝜃(𝑋) 𝑋 Or 𝜃(𝑋1 ) − 𝜃(𝑋2 ) 𝜃(𝑋) = ∙ 𝑋 ⋯ ⋯ ⋯ (8) 𝑋1 − 𝑋2 If we assign an arbitrary number of degrees to the temperature interval 𝜃(𝑋1 ) − 𝜃(𝑋2 ), then 𝜃(𝑋) can be calculated from the measurement of 𝑋, 𝑋1 and 𝑋2. An easily reproducible state of an arbitrarily chosen standard system is called a fixed point. These two fixed points are ice point and steam point. Therefore, the temperature interval 𝜃(𝑋1 ) − 𝜃(𝑋2 ) = 100°𝐶 between these two fixed points is 100°𝐶. However, the use of these two fixed points was found unsatisfactory and complex. Therefore, it was abandoned and a new method was derived. Method in Use After 1954: New Method This new method involves only one fixed point i.e. the triple point of water, which is the state at which ice, liquid water and water vapour co-exist in equilibrium. The temperature at which this state exists is arbitrarily assigned the value of 273.16 𝐾 or 373.16°𝐶. Let’s represent the triple point of water by 𝜃𝑡 , and 𝑋𝑡 being the value of the thermometric property when the body whose temperature 𝜃 is to be measured, is placed in contact with water at its triple point. Therefore, we can write a linear function of 𝜃𝑡 as follows: 𝜃𝑡 = 𝑎𝑋𝑡 ⋯ ⋯ ⋯ (9) 𝜃𝑡 273.16 ∴𝑎= = ⋯ ⋯ ⋯ (10) 𝑋𝑡 𝑋𝑡 Also, 273.16 𝜃 = 𝑎𝑋 = ∙𝑋 𝑋𝑡 Or 𝑋 𝜃 = 273.16 ⋯ ⋯ ⋯ (11) 𝑋𝑡 The temperature of the triple point of water, which is an easily reproducible state is now the standard fixed point of the thermometry. Comparison of Thermometers Applying the above principle obtained in eqtn (11) to the five thermometers listed in the table earlier given, the temperatures are given as: 𝑃 (a) Constant Volume gas thermometer - 𝜃(𝑃) = 273.16 𝑃𝑡 𝑉 (b) Constant Pressure gas thermometer - 𝜃(𝑉) = 273.16 𝑉 𝑡  𝑅 (c) Electric Resistance thermometer - 𝜃(𝑅) = 273.16 𝑅𝑡 𝜀 (d) Thermocouple - 𝜃(𝜀) = 273.16 𝜀𝑡 𝐿 (e) Liquid-in-glass thermometer - 𝜃(𝐿) = 273.16 𝐿𝑡 If the temperature of a given system is measured at the same time (simultaneously) with each of the five thermometers, it is found that there is considerable difference among the readings. The smallest variation is however observed among different gas thermometers. That is why a gas is chosen as the standard thermometric substance. Ideal Gas At low pressure, experimental observations showed that the P-V-T behavior of one mole of gas is closely given by the following relation: 𝑃𝑉̅ = 𝑅̅ 𝑇 where 𝑅̅ is the universal gas constant given as 8.3143 𝐽/𝑚𝑜𝑙𝐾 and 𝑉̅ is the molar specific volume (in 𝑚3 /𝑔𝑚𝑜𝑙) given as 𝑉⁄𝑛. Thus, we can re-write this expression as follows: 𝑃 𝑉⁄𝑛 = 𝑅̅ 𝑇 It should be noted that this expression is in terms of the total volume (𝑉) of gas. Therefore, it has been established from experimental observations that the P-V-T behavior of gases is given by the following relation: 𝑃𝑉 = 𝑛𝑅̅ 𝑇 ⋯ ⋯ ⋯ (12) where 𝑉 is the total volume of the gas, 𝑛 is the number of moles and 𝑅̅ is the universal gas constant given as 8.3143 𝐽/𝑚𝑜𝑙𝐾. The number of moles 𝑛 is related to the mass of the gas and its molecular weight as follows: 𝑚 𝑛= 𝜇 where 𝑚 is the mass of the gas and 𝜇 is the molecular weight. Therefore, eqtn (12) can be written as: 𝑚 𝑃𝑉 = 𝑅̅ 𝑇 ⋯ ⋯ ⋯ (12) 𝜇 and 𝑅̅ 𝑅= 𝐽/𝑘𝑔𝐾 𝜇 where 𝑅 is the characteristic gas constant. Also, we can re-write equation (12) as: 𝑃𝑉 = 𝑚𝑅𝑇 ⋯ ⋯ ⋯ (13) For two states of the gas, equation (13) can be written as: 𝑃1 𝑉1 𝑃2 𝑉2 = ⋯ ⋯ ⋯ (14) 𝑇1 𝑇2 Therefore, eqtns (12) and (13) are called the ideal gas equation of state. It should be noted that at very low pressure or density, all gases and vapour approach ideal gas behavior. Ideal Gas Temperature Electrical Resistance Thermometer In the resistance thermometer, the change in resistance of a wire due to its change in temperature is the thermometric property. The wire which is usually a platinum, may be incorporated in a Wheatstone bridge circuit. The platinum resistance thermometer measures temperature to a high degree of accuracy and sensitivity, which makes it suitable for the calibration of other thermometers. In a restricted range, the following quadratic equation is often used: 𝑅 = 𝑅0 (1 + 𝐴𝑡 + 𝐵𝑡 2 ) where 𝑅0 is the resistance of the platinum wire when it is surrounded by melting ice, and A and B are constants. Thermocouple This is a circuit made up of joining two wires made up of dissimilar metals. Due to the See back effect, a net emf is generated in the circuit which depends on the difference in temperature between the hot and cold junctions. The choice of metals depends largely on the temperature range to be investigated. Example of metals that can be used are copper-constantan wire, chromel-alumel and platinum-platinum-rhodium. The results of such measurements on most thermocouples can be usually represented by a cubic equation of the form 𝜀 = 𝑎 + 𝑏𝑡 + 𝑐𝑡 2 + 𝑑𝑡 3 where 𝜀 is the emf and 𝑎, 𝑏, 𝑐 and 𝑑 are constants for different thermocouples. Example The emf in a thermocouple with the test junction at 𝑡°𝐶 and reference junction at at ice point is given by 𝜀 = 0.20𝑡 − 5 × 10−4 𝑡 2 𝑚𝑉. The millivoltmeter is calibrated at ice and steam points. What will the thermometer read in a place where the gas thermometer reads 50°𝐶? Solution At ice point, when 𝑡 = 0°𝐶, then 𝜀 = 0 𝑚𝑉 At steam point, when 𝑡 = 100°𝐶, then 𝜀 = 0.20 × 100 − 5 × 10−4 (100)2 𝜀 = 15 𝑚𝑉 At 𝑡 = 50°𝐶, then 𝜀 = 0.20 × 50 − 5 × 10−4 (50)2 𝜀 = 8.75 𝑚𝑉 Recall, 𝜃(𝑋) 𝜃(𝑋1 ) = 𝑋 𝑋1 Or 𝑋1 𝜃(𝑋1 ) = ∙ 𝜃(𝑋) 𝑋 when the gas thermometer reads 𝑡 = 50°𝐶, the thermocouple will read 8.75 𝑚𝑉 ∙ 100°𝐶 = 𝟓𝟖. 𝟑𝟑°𝑪 15 𝑚𝑉 Assignment 1. The resistance of a platinum wire is found to be 11,000 𝑜ℎ𝑚𝑠 at the ice point, 15,247 𝑜ℎ𝑚𝑠 at the steam point, and 28,887 𝑜ℎ𝑚𝑠 at the Sulphur point. Find the constants A and B, if the total resistance is given by the equation 𝑅 = 𝑅0 (1 + 𝐴𝑡 + 𝐵𝑡 2 ) 2. The emf in a thermocouple with the test junction at 𝑡°𝐶 and reference junction at ice point is given by 𝜀 = 0.5 + 0.40𝑡 − 7.5 × 10−4 𝑡 2 𝑚𝑉. The milli-voltmeter is calibrated at ice and steam points. When the gas thermometer reads 50°𝐶, determine the expected reading on the thermometer in this situation.  WORK, HEAT AND INTERNAL ENERGY A closed system and its surroundings can interact in two ways, that is by work transfer and heat transfer. These may be called energy interactions and these bring about changes in the properties of the system (such as internal energy). Thermodynamics mainly studies these energy interactions and associated property changes of the system. Work Transfer Work is one of the basic modes of energy transfer. In mechanics, the action of a force on a moving body is identified as work. A force is a means of transmitting an effect from one body to another. But a force itself never produces a physical effect except when coupled with motion and hence it is not a form of energy. An effect such as raising of a weight through a certain distance can be performed by using a small force through a large distance or a larges force through a small distance. The product of force and distance is the same to accomplish the same effect in terms of work. In mechanics, it can be said that work is done by a force as it acts upon a body moving in the direction of the force. In thermodynamics, work transfer is considered as occurring between the system and the surroundings. It should be noted that when work is done by a system, it is arbitrarily taken to be positive and when work is done on a system, it is taken to be negative as shown below. The unit of work is 𝑁. 𝑚 or Joule (1 𝑁𝑚 = 1 𝐽𝑜𝑢𝑙𝑒). Power The rate at which work is done by or upon the system is known as power. The unit of power is 𝐽/𝑠 or 𝑊𝑎𝑡𝑡𝑠. Therefore, work is one of the forms in which a system and its surroundings can interact with each other. There are various types of work transfer which is involved in system-surrounding interactions, such as PdV-work or displacement work, electrical work, flow work etc. PdV-Work or Displacement Work Piston moves a distance 𝑑𝑙, and 𝑎 is the area of the piston, then the force 𝐹 acting on the piston is given as 𝐹 =𝑃∙𝑎 Therefore, the amount of work done by the gas on the piston is given as: 𝑑𝑊 = 𝐹 ∙ 𝑑𝑙 = 𝑃𝑎𝑑𝑙 = 𝑃𝑑𝑉 As the piston moves from position (1) to (2), the volume is changing from 𝑉1 to 𝑉2 , then we can write  𝑊2 𝑉2 ∫ 𝑑𝑊 = 𝑊1−2 = ∫ 𝑃𝑑𝑉 𝑊1 𝑉1 2 ⟹ 𝑊1−2 = 𝑃 ∫ 𝑑𝑉 = 𝑃(𝑉2 − 𝑉1 ) 1 2 Note that: ∫1 𝑑𝑊 = 𝑊2 − 𝑊1 Because work done in a quasi-static process between two given states depends on the path followed. Therefore, we can write 2 ∫ 𝑑𝑊 = 𝑊1−2 𝑜𝑟 1 𝑊2 1 PdV Work in various Quasi-static Processes (a) Isobaric or Isopiestic Process: Constant pressure process 𝑉2 𝑊1−2 = ∫ 𝑃𝑑𝑉 = 𝑃(𝑉2 − 𝑉1 ) 𝑉1 (b) Isochoric Process: Constant volume process 𝑉2 𝑊1−2 = ∫ 𝑃𝑑𝑉 = 0 𝑉1 (c) Process in which 𝑃𝑉 = 𝐶 𝑉2 𝑊1−2 = ∫ 𝑃𝑑𝑉 , 𝑃𝑉 = 𝑃1 𝑉1 = 𝐶 𝑉1 𝑃1 𝑉1 𝑃= 𝑉 𝑉2 𝑑𝑉 𝑊1−2 = 𝑃1 𝑉1 ∫ 𝑉1 𝑉 𝑉2 = 𝑃1 𝑉1 ln 𝑉1 𝑃1 = 𝑃1 𝑉1 ln 𝑃2 Heat Transfer Heat is defined as the form of energy that is transferred across a boundary by the effect of a temperature difference. The temperature difference is the ‘potential’ or ‘force’ and heat transfer is the ‘flux’. The transfer of heat between two bodies in direct contact is called conduction. Heat transferred between two bodies separated by empty space or gases is radiation. The transfer of heat between a wall and a fluid system in motion is known as convection. It should be noted that the heat flow into a system is taken to be positive and heat flow out of a system is taken to be negative. Internal Energy The internal energy of a thermodynamic system is the energy contained or accumulated within it, and associated with random or disordered motions of the molecules or particles. It is also the total kinetic energy due to the motion of molecules and the potential energy associated with the vibration motion and electric energy of atoms within the molecules. It should be noted that the internal energy (𝑈) is more related to microscopic quantities.  First Law of Thermodynamics The first law of thermodynamics provides a relationship between the types of energy transfer (i.e. work and heat transfer) and the internal energy of a thermodynamic system. The first law of thermodynamics states that the net change in total internal energy of a closed system (∆𝑈) is equal to the heat added to the system (𝑄) minus work done by the system (𝑊). ∆𝑈 = 𝑄 − 𝑊 The first law of thermodynamics is also known as the law of conservation of energy, which states that energy can neither be created nor destroyed, but can be transferred or change from one form to another. For example, turning on a light would seem to produce energy, however, it is electrical energy that is converted. Alternatively, the first law of thermodynamics states that any change in the internal energy (∆𝑈) of a system is given by the sum of the heat (𝑄) that flows across its boundaries and the work (𝑊) done on the system by its surroundings: ∆𝑈 = 𝑄 + 𝑊 Therefore, this law states that there are two kinds of processes or transfers i.e. heat and work, that can lead to a change in the internal energy of a system. Since both heat and work can be measured and quantified, this is the same as saying that any change in the energy of a system must result on a corresponding change in the energy of the surroundings outside the system. In other words, energy can neither be created nor destroyed. If heat flows into a system or the surroundings do work on it, the internal energy increases and the sign of 𝑄 and 𝑊 are positive. Conversely, heat flow out of the system or work done by the system (on the surroundings) will be at the expense of the internal energy, and 𝑄 and 𝑊 will therefore be negative. Revision Questions 1. A mass of 3.5 𝑘𝑔 of air is compressed in a quasi-static process from 0.5 𝑀𝑃𝑎 to 0.7 𝑀𝑃𝑎 for 𝑃𝑉 = 𝐶. The initial density of air is 1.16 𝑘𝑔/𝑚3. Find the work done by the piston to compress the air. 2. A hydrogen gas is compressed from a pressure 𝑃1 , volume 𝑉1 to a volume 𝑉2 with a process path generally described by 𝑃𝑉 = 𝐶. Simply sketch the process path on a single P-V diagram. Calculate the work for a compression process starting from a gas volume of 0.01 𝑚3 , pressure 1.5 𝑏𝑎𝑟 to a pressure of 20 𝑏𝑎𝑟. State the implication of the obtained work done. 3. How much total work is done by the system if the state of gas is taken from state A to B through a process in which the net heat absorbed by the system is 9.35 𝑐𝑎𝑙, and the internal energy is 22.3 𝐽.