Stuff ---onight: Lecture 4 July ---Assignment has been posted. ---Presentation Assignment posted. --Some more thermodynamics and then problem solving in class for Assignment #. --Next week: Free Energy and In-class problem solving session. 1st Law of hermodynamics du = du = q P ext d d +,n i d,n i du = C v d + d,n i q v = C v d = First Law Differential Form otal Differential,n otal Differential Enthalpy Function: H(,P) H = U + P dh =( q P d )+P d + dp dh = dh = q + dp dh = q p = d + P dh = C p d + d P dp P dp P,n i C p = Large changes otal Differential qp = p At dp = 0 Recasting P Heat capacity of gases depends on P and we define: from large change ===> small change C v = q v = U C p = q p = H q v = 1 C v d C v = C p = q p = qp 1,n P,n C p d Heat Capacity Constant olume Heat Capacity Constant Pressure KM and the equipartition of energy links internal energy U and heat capacity, C of ideal gases. U = U = nr C v = n moles of monoatomic gas ( = nr ) IMPORAN: his equation says that the internal energy of a gas only depends on and nothing else! We can take the derivative and link the heat capacity to bulk properties = nr If we look at the results of the equipartion theorem we find the internal energies for linear and non-linear molecules. U = R U = R + R +(N 5)R U = R + R +(N 6)R C v = R C v = R + R +(N 5)R C v = R + R +(N 6)R (monoatomic) (linear molecule) (non-linear molecule) (monoatomic) (linear molecule) U C v (non-linear molecule)
Using the energies arising from equipartition of energy, show that the heat capacity at constant volume, Cv are the given values on the previous slides. U= R U = R + R + (N 5)R U= U= (monoatomic) ---With calculus 1st law sets up experimental and theoretical parameters of interest to measure. spontaneous (linear molecule) R + R + (N 6)R (non-linear molecule) Spontaneity refers to a process that appears to proceed naturally from an initial state to a final state without outside intervention (does not mean now ). > 0 C not spontaneous ---But the 1st law does not give us predictive power on observation of naturally-occurring processes. In the 1870 s, it was thought that!h determined the spontaneity of a chemical reaction. his notion proved wrong. Examples of spontaneous reactions: CH4 (g) + O (g) < 0 C CO (g) + HO (l)!h0 = -890.4 kj < 0C HO (l) HO (s)!h0 = - 6.01 kj Freezing and melting of water Our goal in this Chapter is to identify those factors which determine whether a chemical reaction will proceed spontaneously. > 0C HO (s) HO (l)!h0 = 6.01 kj NH4NO (s) H O!Hrnx does not determine whether a reaction is spontaneous! NH4+(aq) + NO- (aq)!h0 = 5 kj he rusting of a nail What is the parameter or a bulk function that can predict whether A ==>B will occur spontaneously (irreversibly)? nd law of thermodynamics postulates that all processes that are spontaneous produce an increase in the entropy of the universe.!suniv =!Ssys +!Ssurr > 0 he 1st Law tells us that energy is conserved, and sets up a formal accounting system to track energy transformation as heat and work. 1. Criteria for Spontaneous change!!suniv =!Ssys +!Ssurr < 0. No spontaneous change!suniv =!Ssys +!Ssurr = 0. Equilibrium condition here are many ways to express the nd Law of hermodynamics Kelvin Plank Statement: It is impossible for a system (engine) to undergo a cyclic process whose sole effects are flow of heat from a reservoir and the performance of an equal amount of work by the system. Practical: Heat can not spontaneously pass from a cold body to a hot body. Deep: Work can be completely converted to heat, but heat can not be completely converted to work! (1st Law symmetry of work = heat is broken) Spontaneous processes increase the total entropy of the universe.
is too often described as positionally disordered. his is not correct, in the chemical sense, but it is useful device to predict. It appears that entropy focuses on the most statistically favored position distribution entropy is more concerned with the fact that energy levels become more closely spaced and more occupied in that most favored state. How many ways can you have your room or a deck of cards organized vs disorganized? 1.0 atm evacuated Gas expands spontaneously into larger volume Quantum mechancis dictates closer energy level spacing as increases. 0.5 atm Energy level Same amount of energy is dispersed or spread among more energy levels. Energy level here is a thermodynamic state function called entropy, S, that is a measure of the energy dispersal that occurs when a change of state occurs. is a measure of the magnitude of energy dispersal over available quantum states in a chemical system. he more dispersed or spread out the energy is the higher the entropy. Macroscopic Microscopic E E6 E9 1. Bulk defintion (it needs no molecules).!s = -qsys reversible Change in. Molecular Description!S = k (ln Wf - ln Wi) Increasing the volume of a system decreases energy level spacing increasing entropy by populating more. L E E1 L E5 E4 E E E1 L E8 E7 E6 E5 E4 E E E1 Boltzmann founded a thermodynamic state function called entropy, S, that is a measure of the dispersion or spread of energy that occurs when all spontaneous reactions occur. # of microstates A system can be described by bulk or macroscopic that we can observe and measure, and the microscopic or molecular that we can t see but can model statistically. You must see both! Consider 4 labeled molecules A,B,C,D S = k ln W Microstate is particular distrubution that corresponds to some macrostate. Boltzman s constant 1.8 " 10 # J/K.!S = k ln Wf - k ln Wi Change in Boltzmann s omb In ienna, Austria When W f > W i then!s > 0 When W f < W i then!s < 0 Multiplicity is the number of microstates that give a specific macrostate. Macrostate is the observed state of the system that represents on a molecular level that microstate with the highest probability or number. 5-observable macrostates
Microstate Left Side A,B,C,D A,B,C A,B,D A,C,D B,C,D Right Side - D C B A Multiplicity 1 4 Probability 1/16 4/16 Macrostate Bulk Property Boltzman s statistical interpretation of entropy (dispersal or spread of energy) is connected to a macroscopic defintion of heat flow per unit temperature. Macroscopic 1. Bulk defintion (it needs no molecules). Microscopic. Molecular Description A,B A,C A,D B,C B,D C,D C,D B,D B,C A,D A,C A,B 6 - A,B,C,D 1 16 6/16 1/16!S = -qsys reversible Change in!s = k (ln Wf - ln Wi) Increasing entropy or the dispersal or spread of energy occurs in all spontaneous processes. It provides humans with an arrow of time as the reverse situation never happens. Example: two block of different temperatures are brought together. hermal energy flows from the higher occupied energy levels in the warmer object into the unoccupied levels of the cooler one until equal numbers of states are occupied. 1 100 C 10 C 100 C 10 C HO COLD COMBINED here are many chemical reactions that lead to an increase in dispersal of energy spread over a larger number microstates entropy (!S > 0) Dissolution!S > 0 Dissolution!S > 0 Mixing!S > 0 Increasing Can We Determine Mathematically Criteria For Spontaneous Change Using What We Know So Far? Usual approach is Carnot Cycle Carnot analysis is long and the result unsatisfying Another approach is purely mathematical based on the Euler criteria and gives the same result. LE S RY I KM and the equipartition of energy links internal energy U and heat capacity, C of ideal gases. U = nr U = C v = n moles of monoatomic gas KE gives this ( = nr ) IMPORAN: his equation says that the internal energy of a gas only depends on and nothing else! We can take the derivative and link the heat capacity to bulk properties = nr
Criteria For Spontaneous Change? Assume monoatomic ideal gas U = / nr start with 1st law du = q rev + w rev differential form q rev = du + P ext d rearrange ( du = d = nr ) d = from KM nrd q rev = nr d + nr d substituting Is the above equation a total differential? How do we know or not know? Prove it is not. q rev = nr d + nr d here is a math theorem that says we can transform an inexact differential to an exact one by multiplying by and integrating factor. Can we find one? 1 1 1 nr ds = q rev = nr d + d Show that this function is an exact differential using Euler s criteria. S = ds = q rev Only true for reversible path! Does a change in entropy predict spontaneous processes? -Isolated -compartment system. Each has its own emp, olume, U not at equilibrium. Our conditions impose the following: U A + U B = constant U A = U B du A = q rev = A ds A du B = q rev = B ds B No work, no energy or matter in or out, rigid container d = 0. Diathermal wall between compartments. S = S A + S B ds = ds A + ds B δs > 0 predicts spontaneous processes. U A + U B = constant U A = U B ds = ds A + ds B = du A A = du B A + du B B + du B B ds = du B 1 B 1 A Case 1: B > A ds > 0 Case : A > B (-) (-) (+)(+) ds > 0 S = S A + S B Analyze ds For: ds increases for spontaneous flow of heat Problems Involving I Nearly all problems start with the 1st Law or the enthalpy equations and involve restricting the path to eliminate variables. du = q P ext d du = C v d + S,n i d Problems Involving II Nearly all problems start with the 1st Law or the enthalpy equations and involve restricting the path to eliminate variables. du = q P ext d du = C v d + S,n i d A. Changes With emperature @ Constant Case I. Changes With emperature, d = 0 ds = C v ( ) 1 d ds = C v d Is Cv constant? B. Isothermal Expansion of An Ideal Gas Case I. Changes ====> d = 0, du = 0 du = q P ext d = 0 ds = P nr d = d 1 P P 1
Problems Involving III Nearly all problems start with the 1st Law or the enthalpy equations and involve restricting the path to eliminate variables. du = q P ext d du = C v d + A. Isothermal Expansion of An Ideal Gas S,n i d Case I. Changes ====> d = 0, du = 0 du = q P ext d = 0 ds = P nr d = d 1 P P 1