Lecture 1: Preliminaries

1. Lecture 1: Preliminaries Schroeder Ch. 1 Gould and Tobochnik Ch. 2.1 – 2.7

2. What is Thermal Physics? • Thermal physics = Thermodynamics + statistical mechanics • Thermodynamics provides a framework of relating the macroscopicproperties of a system to one another. • It is concerned only with macroscopic quantities and ignores the microscopic variables that characterize individual molecules • Statistical Mechanics is the bridge between the microscopic and macroscopic worlds: it links the laws of thermodynamics to the statistical behavior of molecules.

3. Thermodynamic Systems • A thermodynamic system is a precisely specified macroscopic region of the universe together with the physical surroundings of that region, which determine processes that are allowed to affect the interior of the region. • A thermodynamic system can be classified in three ways: • Open systems can exchange both matter and energy with the environment. • Closed systems can exchange energy but not matter with the environment. • Isolated systems can exchange neither energy nor matter with the environment.

4. Thermodynamic State • A thermodynamic state is the macroscopic condition of a thermodynamic system as described by a suitable set of parameters known as state variables. • Examples of state variables are temperature, pressure, density, volume, composition, and entropy. • Thermodynamic variables are often divided into two categories: • Intensive variables • Extensive variables

5. Pressure and Mechanical Equilibrium • Let be an surface element of the surface of this piston where the direction is the outward normal. • Let be the force normal to the surface element. • We can define the pressure as • If the pressure is constant, then the pressure of the gas exerted on the piston is • We say that two systems in contact with one another are in mechanical equilibrium when their pressures are equal.

6. Temperature and Thermal Equilibrium • Consider two thermodynamic systems, A and B, that are brought into contact with one another. • Over a period of time, the net exchange of energy between both systems ceases and we say that they are in thermal equilibrium. • Thermal equilibrium is determined by a single variable called the temperature.

7. The Zeroth Law of Thermodynamics • Thermal equilibrium satisfies the zeroth law of thermodynamics which states • If two thermodynamic systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. • The zeroth law of thermodynamics ensures that thermal equilibrium is determined solely by temperature.

8. Thermometers • We can use thermometric properties to build thermometers by defining the scale of temperature in such a way that for any thermometric property X, • We can define the two constants, and , that define this linear scale by choosing two reproducible phenomenon that always occur at the same temperature. • We choose • The boiling point of pure water at sea level • Triple point of pure water

9. Temperature Scales • Using these properties, we have that • Choosing and gives the Celsius scale • Choosing and gives the Fahrenheit scale

10. Constant Volume Gas Thermometer • The gas flask is inserted into an ice–water bath, and mercury reservoir B is raised or lowered until the volume of the confined gas is at some value, indicated by the zero point on the scale. • The height h, the difference between the levels in the reservoir and column A, indicates the pressure in the flask at 0°C. • The flask is inserted into water at the steam point, and reservoir B is readjusted until the height in column A is again brought to zero on the scale, indicating the pressure in the flask at 100°C.

11. Constant Volume Gas Thermometer • The line connecting two points on the pressure vs. temperature curve serves as a calibration curve for measuring unknown temperatures. • The height of the mercury column tells us the pressure of the gas, and we could then find the temperature of the substance from the calibration curve.

12. The Kelvin Scale • If we now plot the pressure vs. temperature as measured by our thermometer for different gases, we obtain a linear curve. • Notice that the pressure is exactly zero at for all cases. • This is often called absolute zero and serves as the basis for a new temperature scale called the Kelvin scale.

13. Ideal Gas Law • An equation of state is an equation that relates macroscopic variables for a given substance in thermodynamic equilibrium. • The most famous equation of state is the ideal gas law • Here is the number of moles present in the gas and R is the ideal gas constant • The ideal gas law can also be written in terms of the molecules present in the gas • Here, is the number of molecules in the gas and k is the Boltzmann’s constant.

14. The Kinetic Theory of Gases • In the previous section, we discussed the macroscopic properties of an ideal gas. • Now, we consider the ideal gas model from a microscopic point of view using kinetic theory. • The kinetic theory of gases makes the following assumptions • All molecules in the gas are identical • The molecules interact only through short-range forces during elastic collisions • The molecules obey Newton’s laws of motion • The number of molecules in the gas is large • The average separation between molecules is larger compared with their dimensions

15. The Kinetic Theory of Gases • Consider a one-dimensional gas in a one-dimensional box of length L. • The change in momentum after the molecule collides with the wall is • Since the molecule must travel a distance 2L before returning to the same wall, the rate at which the molecules imparts momentum to the wall is

16. The Kinetic Theory of Gases • If there are N molecules in the box, then the force on the wall is • The pressure on the wall is given by • Since the molecules are equally probable to move in all three directions of space, then we have

17. The Kinetic Theory of Gases • Comparing our previous result with the ideal gas law, we see that temperature is associated with the mean kinetic energy of the molecules • We can also obtain a relationship between the pressure of a gas, its density, and the root mean square speed .

18. Thermodynamic Processes • A thermodynamic process is any process that takes a macroscopic system from one equilibrium state to another. • We will be concerned with energy conservation in thermodynamic processes and thus it will be important to define two important variables • Work • Heat

19. Definition of Work • Let be an surface element of the surface of this piston where the direction is the outward normal. • Let be the net force exerted by the system on the surface element of the boundary. • Suppose that the boundary experiences a deformation so that the surface element is displaced by . • The work done by the system on the boundary is

20. Quasistatic Processes • Quasi-static (quasi-equilibrium) processes –sufficiently slow processes, any intermediate state can be considered as an equilibrium state • For quasistatic processes, the state variables (e.g. pressure, volume, temperature) are well defined. • Examples of quasi-equilibrium processes: • Isochoric (constant volume) • Isobaric (constant pressure) • Isothermal (constant temperature)

21. Thermodynamic Diagrams • The evolution of a thermodynamic system can be given by a thermodynamic diagram. • Because there is one equation of state, all processes will occur in a two-dimensional plane, which can be spanned by any of the three possible pairs: (p,V), (p,T), and (V,T). • The area under the graph in a PV diagram is equal in magnitude to the work done on the gas.

22. Work in Thermodynamic Processes • For an isochoric process, no work is done since . • For an isothermal process, the work done is • For an isobaric process, the work done is

23. Heat • Heatis energy transferred into or removed from a macroscopic system on the molecular level, as opposed to the direct application of mechanical work on the system by deformations of its macroscopic parameters. • The unit of heat is the calorie and it is defined as the energy necessary to raise the temperature of 1 g of water from 14.5⁰C to 15.5⁰C. • Note that 1 calorie is equal to 4.186 J.

24. Heat Capacity and Specific Heat • We define the heat capacity of a thermodynamic system as the amount of heat required to raise the temperature of the system by one degree Kelvin • Here, is the heat absorbed and is the change of temperature. • We define the specific heat as • In terms of specific heat,

25. Latent Heat • Energy may be absorbed or released from a system during isothermal processes through phase transitions. • We define latent heat as the heat required to change the phase of one gram of a substance

26. Calorimetry • Heat capacities, specific heats and latent heats of substances are measured using calorimetry. • Acalorimetric experiment involves the transfer of energy between two or more thermodynamic systems while the combination of the systems is kept isolated from the rest of the universe. • Devices in which the exchange of energy occurs are called calorimeters, whose main function is to isolate whatever is placed inside. • Since the combination of thermodynamic systems is kept isolated, the calorimetric process satisfies

27. Mechanisms of Heat Transfer • Heat transfer occurs primarily through three processes: conduction, convection, and radiation. • Conduction: the energy transfer by molecular contact • Convection: the energy transfer by macroscopic motion of fluids • Radiation: energy transfer by emission/absorption of electromagnetic radiation.

28. Conduction • Energy transfer between two macroscopic systems due to a difference in temperature between them and which does not involve the gross movement of matter is called conduction. • Conduction can be understood on the microscopic scale as the direct exchange of mechanical energy from a region of higher temperature to a region of lower temperature by molecular collisions.

29. Newton’s Law of Heat Conduction • Consider a solid material with a cross-sectional area S of constant temperature within it. • Newton’s law of heat conduction says that, assuming there is a temperature gradient across S, the rate at which energy is transferred across S is given by