Introduction into thermodynamics Solid-state thermodynamics, J. Majzlan Chemical thermodynamics deals with reactions between substances and species. Mechanical thermodynamics, on the other hand, works with engines and their performance. In this class, we will explore chemical thermodynamics. We will ask and nd the answer to questions such as why does ice melt at room temperature, why can diamond burn, and why some substances dissolve but others apparently do not. It is very important to keep in mind that thermodynamics deals with models of reality, not reality itself. Reality is too complicated and therefore we need such models. First, we need a few denitions. System : A part of the universe we consider. For example, water in the bottle, a piece of rock. Surrounding : Everything outside of the system. Equilibrium : A state of a system in which none of the properties of the system change with time. Phase : A mechanically separable part of a system. Minerals are phases ; water is also a phase. Sodium chloride dissolved in water is not a phase because it cannot be mechanically separated (for example, by ltration) from the water. Component : An algebraic operator. The composition of any phase in the system must be expressed as a linear combination of the components. One should try to select the smallest number of components. Example. A system is made of K-feldspar (KAlSi 3 O 8 ), sillimanite (Al 2 SiO 5 ), and quartz (SiO 2 ). Dene the components. Components for this system are K 2 O, Al 2 O 3, and SiO 2. Each phase can be expressed as a linear combination of these components : 1 2 K 2O + 1 2 Al 2O 3 + 3SiO 2 = KAlSi 3 O 8 0K 2 O + 1Al 2 O 3 + 1SiO 2 = Al 2 SiO 5 0K 2 O + 0Al 2 O 3 + 1SiO 2 = SiO 2 Thermodynamic variables 1
Temperature. In everyday life, temperature is measured in o C. In thermodynamics, however, we always express temperature in Kelvin, K. The relationship between these two units is T [K] = T [ o C] + 273.15 Pressure. The SI unit for pressure is Pascal, Pa. One Pascal is a small pressure ; the atmospheric pressure is approximately 10 5 Pa. Variables that were previously in use are also 1 bar (= 10 5 Pa) and 1 atmosphere (1 atm 1 bar). Internal energy. Internal energy U is a fundamental property of every system. It can be related to the rest energy E r dened by the well-known equation as E r = mc 2 E r = U + constant where the value of the constant is unknown. The consequence is that the internal energy has no absolute scale. All variables which are dened via internal energy have no absolute scale. The importance of this statement will become soon obvious. Internal energy can be transferred in two ways, by transfer of heat (q) and transfer of work (w). Mathematically, this is expressed in the equation U = q + w Under work, we usually understand mechanical work in thermodynamics. Without going into detail, the mechanical work is the "pressure-volume work", that is Heat and entropy : w = P V There is another thermodynamic function which relates heat and temperature. q = T S The equation for internal energy can be rewritten as U = T S P V U = T S P V 2
if we consider the extensive or intensive internal energy of a system, respectively. The dierential form of this equation is a fundamental equation of thermodynamics. du = T ds P dv Thermodynamic variables in equilibrium Entropy is a variable that will be maximized in a system in equilibrium. Internal energy will be minimized in systems held at constant entropy and volume : du = T ds P dv Such conditions, that is, constant entropy and volume, are impossible to maintain in the laboratory or expect in the nature. Therefore, although the internal energy if a fundamental variable used to derive other important thermodynamic functions, it is of little importance for practical use. How about enthalpy? Let's recall the denition of the enthalpy and dierentiate to H = U + P V dh = du + P dv + V dp Inserting the fundamental equation of thermodynamics, we get dh = du + P dv + V dp = T ds P dv + P dv + V dp = T ds + V dp Therefore, enthalpy is a variable that will be minimized in a system in equilibrium at constant entropy and pressure. Such conditions are also dicult to maintain or imagine. We need yet another thermodynamic variable. Gibbs energy. We can dene another variable and dierentiate to G = U + P V T S dg = du + P dv + V dp T ds SdT = T ds P dv + P dv + V dp T ds SdT = SdT + V dp 3
This variable, Gibbs energy or Gibbs free energy, will be minimized in equilibrium in a system at constant temperature and pressure. Such conditions can be maintained in a laboratory or attained in nature. Gibbs free energy is therefore of central importance in thermodynamics. It is derived from internal energy and therefore has no absolute scale. Gibbs free energy is dened as which can be reduced to G = U + P V T S G = H T S Because we do not know the absolute values of those energies, this equation can be actually used in the form G = H T S Let's consider a general chemical equation mm + nn = xx + yy If the phases involved in this reaction are pure solids or liquids in their standard state, the change in the Gibbs free energy associated with that reaction is r G o = xg o X + yg o Y mg o M ng o N This is, however, a problem, because we do not know the absolute values of G. We solve these problems by introducing reference state and write r G o = x f G o X + y f G o Y m f G o M n f G o N What happens, when the phases that participate in this reaction are not pure solids and liquids? We have to know how does the Gibbs free energy of these phases change with their composition. We dene a new variable closely related to Gibbs free energy, the chemical potential µ ( ) G µ i = n i T,P,n j or for small (innitesimal) changes in n i 4
( ) G µ i = n i T,P,n j For our chemical reaction mm + nn = xx + yy The change of chemical potential for this reaction, r µ, is r µ = x µ X + y µ Y m µ M n µ N r µ = x( µ o X + RT lna X ) + y( µ o Y + RT lna Y ) m( µ o M + RT lna M ) n( µ o N + RT lna N ) This equation can be re-arranged r µ = r µ o + RT ln ax X ay Y a m M an N This expression is valid for the general reaction above. For any chemical reaction, the equation can be further generalized r µ = r µ o + RT lnπa νi i = r µ o + RT lnq Greg Anderson (2005 : Thermodynamics of Natural Systems) calls this equation "the most useful equation in thermodynamics". It relates the change in Gibbs free energy, via the chemical potential, to the thermodynamic activities of the species involved. In chemical equilibrium, r µ is equal to zero. The equation reduces to r µ o = RT lnq In this, and only in this situation, the stoichiometric quotient Q gets a special name equilibrium constant and the symbol K r µ o = RT lnk Variation of G with temperature Heat capacity. Heat capacity is the amount of heat that has to be added to 1 mole of a substance to increase its temperature by 1 K. It is designated as C p and dened 5
C p = ( ) dh dt P Heat capacity of substances is measured and then tabulated as polynomials which were tted to the data. A common C p function is the Maier-Kelley heat capacity polynomial C p = a + bt + ct 2 Other functions were also proposed and some will be discussed in the class. The change in entropy can be also dened via heat capacity as S = Cp T dt 6