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1. Review of Classical Thermodynamics

   Schroeder Ch. 1; Ch. 3.1, 3.2, 3.4, 3.5; Ch. 4; Ch. 5.1 – 5.3 Gould and Tobochnik Ch. 2.1 – 2.20, 2.22, 2.24; Ch. 7.3, 7.4
2. The Zeroth Law of Thermodynamics The zeroth law introduces a new variable called the temperature. There exists a variable of the macroscopic state called the temperature, which determines thermal equilibrium between two systems. Two thermodynamics systems having the same temperature will not exchange energy by thermal interactions on the average if their temperatures are the same
3. 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.
4. The First Law of Thermodynamics The first law introduces a new variable called the internal energy. It says that there exists a state function called the internal energy and asserts that the total energy is conserved Here, is the heat exchanged between the system and its environment and is the work done by the system on the environment.
5. Definition of Heat Capacity 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 We define the specific heat as
6. Heat Capacities of Ideal Gases Two heat capacities are of interest CV: Heat capacity at constant volume CP: Heat capacity at constant pressure By kinetic theory, the internal energy of an ideal gas a function of temperature only, given by Using the first law of thermodynamics we have
7. 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
8. 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
9. Quasistatic Processes A thermodynamic process is any process that takes a macroscopic system from one equilibrium state to another. Aquasi-static processis any sufficiently slow thermodynamic process such thatany intermediate state of the system can be considered as an equilibrium state For quasistatic processes, the state variables (e.g. pressure, volume, temperature) are well defined in the thermodynamic phase space. Examples of quasi-static processes: Isochoric (constant volume) Isobaric (constant pressure) Isothermal (constant temperature) Adiabatic (no heat exchange)
10. Summary of Quasistatic Processes for an Ideal Gas
11. Thermodynamic Cycles A thermodynamic cycle is a process in which the thermodynamic system periodically returns to its original macrostate. Any system undergoing a cyclic process will either do work on the environment or have work done upon it by the environment. If work is done by the cycle then the energy for the work done must be extracted from some external source. Examples include: Carnot cycle Otto cycle Diesel cycle Brayton cycle
12. Heat Engines and Refrigerators We define a heat engine as any thermodynamic cycle which extracts energy from a reservoir in the form of heat and performs mechanical work. We define a refrigerator as any cyclic thermodynamic process which transfers energy in the form of heat from a reservoir at a lower temperature to a reservoir at a higher temperature. The efficiency of a heat engine is defined as The efficiency of a refrigerator is defined as
13. Carnot Cycle
14. The Second Law of Thermodynamics The second law introduces a new variable called the entropy. It says that there exists a state function called entropy and asserts that a thermodynamic process can occur if and only if for the corresponding closed or isolated system. It follows that thermally isolated systems achieve equilibrium at the maximum of the entropy
15. Entropy and the Second Law The entropy statement of the second law implies that no engine can be 100% efficient (the Kelvin-Planck statement). The entropy statement of the second law implies that no engine can be more efficient than the ideal Carnot engine (Carnot’s theorem). The entropy statement of the second law implies that no process exists that can spontaneously transfer heat from a cold body to a warm body (the Clausiusstatement).
16. Entropy and Equilibrium For an isolated composite system, the two subsystems will approach equilibrium in such a way that the total entropy will be maximized. It can be shown that both subsystems will be in equilibrium when their thermodynamic temperatures and pressures are the same, as the defined as Temperature can be interpreted as a measure of the tendency of an object to spontaneously give up energy to its surroundings.
17. The Fundamental Surface The content of the first and second laws of thermodynamics can be expressed in terms of thermodynamic identity for Thermodynamics requires that all possible states accessible to a given thermodynamic system must lie on the surface [also called the fundamental surface].
18. Introduction to Legendre Transformations The representation of the fundamental surface as is awkward for many applications because neither the entropy nor the internal energy are laboratory variables. It is possible to transform into equivalent fundamental surfaces, which have characteristic dependencies on other sets of variables, such as , , and . This is accomplished by using a special type of coordinate transformation known as the Legendre transformation.
19. Generalized Legendre Transformations We may replace the dependence of a general function on the variable in favor of a dependence on the partial derivative . The transformation produces an equivalent function defined by
20. Thermodynamic Potentials Using Legendre transformations to transform into equivalent fundamental surfaces gives us the total set of thermodynamic potentials. The five thermodynamic potentials are Internal Energy Entropy Enthalpy Helmholtz Free Energy Gibbs Free Energy
21. Summary of Thermodynamic Potentials
22. Enthalpy Enthalpy can be pictured as the total energy of given thermodynamic system (which includes the system’s internal energy and work done on its environment). During a reversible isobaric process , For isobaric processes, is just the amount of energy that is transferred to a system by heat for an isobaric process. For an isobaric process, the specific heat at constant pressure is given as
23. The Joule-Thomson Process A very important application of enthalpy is the Joule-Thomson process, or throttling process, which is used in refrigeration. Here a gas is allowed to flow adiabatically and steadily from a region of constant high pressure to a region of constant low pressure through a porous plug. It can be shown that the Joule-Thomson process is a constant enthalpy process and that the change in temperature during the Joule-Thomson process is governed by the Joule-Thomson coefficient When (), the gas will cool (warm) upon expansion
24. Enthalpy and Refrigeration The efficiency of a real refrigerator (also called the coefficient of performance) can be expressed in terms of the enthalpy
25. Free Energy Helmholtz free energy measures the maximum amount of expansion work that a thermodynamic system can perform during an isothermal process. The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system. At constant temperature and volume, the Helmholtz free energy tends to decrease At constant temperature and pressure, the Gibbs free energy tends to decrease
26. Free Energy and Enthalpy
27. Chemical Potential and Internal Energy Let’s rewrite the first law to include thermodynamic systems that contain variable contents We see that Thus, the chemical potential is the amount by which a system’s energy changes when you change the number of molecules in a given system.
28. Chemical Potential and Gibbs Free Energy Rewriting the first law in terms of Gibbs free energy gives We see that Thus, for a homogeneous system, the chemical potential is Gibbs free energy per particle.
29. Gibbs Phase Rule Consider a system composed of p phases and n components. What are the number of independent variables of the system? The answer is given by the Gibbs phase rule. The degree of variability required to describe this system is
30. Gibbs Phase Rule – Examples A system of one phase and one chemically well-defined substance (e.g. liquid water) A system of two distinct, chemically well defined gases A system of one chemically well defined substance and two phases (e.g. Liquid-vapor system) A system of one chemically well-defined substance and three phases (e.g. triple point of water)
31. Phase Diagrams A phase transformation is a discontinuous change in the properties of a substance. A graph showing the equilibrium phases as a function of temperature and pressure is called a phase diagram.
32. Clausius-Clapeyron Relation Recall that the integrability condition for the Gibbs free energy is Thus, the entropy determines the temperature dependence of the Gibbs free energy, while the volume determines its pressure dependence. This implies that the shape of any phase boundary on a phase diagram is related in a very simple way to the entropies and volume of the two phases. This is called the Clausius-Clapeyron relation.
33. The van der Waals Model The simplest model of a liquid-gas phase transition is the van der Waals model of real gases. The main reason for the transformation of gas into liquid is through the interaction between the molecules Two ingredients of the model Weak long-range attraction Strong short-range repulsion
34. Phase Transformation for van der Waals gas For each temperature, there is a well-defined pressure called the vapor pressure, at which the liquid-gas transformation takes place. For high-temperature isotherms, the phase boundary disappears above a certain temperature called the critical temperature The vapor pressure and corresponding volume just at is called the critical pressure and the critical volume respectively. These values define the critical point, where the properties of the liquid and gas become identical.