Physical Chemistry 1 Notes PDF

Physical Chemistry 1 CHU11E05 Prof. Chris McAuley Thermodynamics a branch of science that quantifies Thermodynamics: the transfer of heat, work and energy. First law, internal energy, enthalpy Introducti...

Physical Chemistry 1 CHU11E05 Prof. Chris McAuley Thermodynamics a branch of science that quantifies Thermodynamics: the transfer of heat, work and energy. First law, internal energy, enthalpy Introduction to entropy In chemistry we use thermodynamics to quantify 2nd and 3rd Laws; criterion for chemical change heat release/intake during a reaction but we also use it to determine if a reaction can happen. Equilibrium constant for a chemical reaction Note: quantifying the rate of a reaction is the field of Gibbs free energy Chemical Kinetics not Thermodynamics. Terms Symbol Quantity Common Unit 𝑐 Specific Heat Capacity J K−1 g−1 𝐶p Heat Capacity (constant pressure) J K−1 mol−1 𝐺 Gibbs Energy kJ mol−1 𝐻 Enthalpy kJ mol−1 𝐾 Equilibrium Constant 𝑛 moles mol 𝑃 Pressure Pa 𝑞 Heat kJ mol−1 𝑄 Reaction Quotient 𝑆 Entropy J K−1 mol−1 𝑇 Temperature K 𝑈 Internal Energy kJ mol−1 𝑉 Volume m3 𝑤 Work kJ mol−1 𝜈 Stoichiometric Coefficient 1.1 The Laws These can be defined in more than one way but below are the clearest expressions for this course. Definition : 1st Law Energy can not be created or destroyed (conservation of energy). Δ𝑈 = 𝑞 + 𝑤 𝑈 = internal energy, 𝑤 = work and 𝑞 = heat Internal energy: total of all possible kinds of energy present in a substance or substances. For a monoatomic ideal gas, the internal energy of the system is the sum of all of the kinetic energy of the individual gas atoms. 1  Definition : 2nd Law The entropy of the universe always increases: Δ𝑆universe > 0 𝑆 = entropy Entropy is a measure of the dispersion of energy in a system. The second law gives us the so-called ‘arrow of time’. It tells us why hydrogen and oxygen can combust to form water but why water does not spontaneously fall apart to form hydrogen and oxygen. Definition : 3rd Law For a perfect crystal at zero Kelvin, the entropy of the system is zero. 𝑆(0 K) = 0 The third law puts entropy on an absolute scale, such that we can measure and define the standard molar enthalpy (𝑆 ). The entropy of a system is often expressed via the Boltzmann equation: 𝑆 = 𝑘 𝐵 ln 𝑊 where 𝑘 𝐵 is the Boltzmann constant (1.380649 × 10−23 J K−1 ), 𝑊 is the number of available microstates- a measure of the ways a system can be configured. The more dispersed the energy, the higher the number of available microstates. 1.2 Enthalpy and Entropy Enthalpy (H) sum of a system’s internal energy and the mathematical product of its pressure and volume. 𝐻 = 𝑈 + 𝑃𝑉 However, most of the time we are not interested in the absolute enthalpy (this is hard to measure) but changes in enthalpy. Enthalpy Change (Δ𝐻) heat released or absorbed by a system under constant pressure during a chemical or physical process. So at constant pressure: Δ𝐻 = 𝑞 If a reaction releases heat, Δ𝐻 -ve, the reaction is exothermic, if a reaction takes in heat, Δ𝐻 +ve, the process is endothermic. Definition : Entropy Change Δ𝑆 From classical thermodynamics 𝑞 rev Δ𝑆 = 𝑇 𝑞 rev is the reversible heat added to or removed from a system and T is the temperature. This expression can also be used to allow use to quantify the entropy change associated with a change of phase (vaporisation, melting etc.). For example for melting: Δ𝐻fusion Δ𝑆 = 𝑇m 2 1.3 Thermochemical Equations We use thermochemical equations to describe changes in both matter and energy. For example the combustion of hydrogen is given by: H2 (g) + 1 2 O2 (g) −−−→ H2 O (g) Δ𝐻 = -241 kJ mol−1 Here not only is the reaction balanced (the number of atoms on the RHS and LHS are equal) but the state (gas (g), liquid (l), solid (s) or aqueous solution (aq)) is given for each substance along with information regarding the energy change during the course of the reaction. Example 1.3.1 (Manipulating Equations) We can manipulate thermochemical equations, for instance, if we wrote the reaction the other way round and scaled the stoichiometry of the reaction by a factor of 2, the enthalpy changes accordingly: 2 H2 O (g) −−−→ 2 H2 (g) + O2 (g) Δ𝐻 = (−2 × −241) kJ mol−1 = 482 kJ mol−1 We can also combine thermochemical equations together, for example the combustion of carbon is: C(s) + O2 (g) −−−→ CO2 (g) Δ𝐻 = -393 kJ mol−1 Combined together we get: C(s) + O2 (g) + 2 H2 O (g) −−−→ CO2 (g) + 2 H2 (g) + O2 (g) Δ𝐻 = (-393 + 482) kJ mol−1 C(s) + 2 H2 O (g) −−−→ CO2 (g) + 2 H2 (g) Δ𝐻 = 89 kJ mol−1 This final reaction is related to the steam reformation process, as used in the industrial production of hydrogen. This is an example of using Hess’s law... regardless of the multiple stages or steps of a reaction, the total enthalpy change for the reaction is the sum of all changes. 1.4 Determining Δ𝐺 𝑟𝑥𝑛 , Δ𝐻𝑟𝑥𝑛 and Δ𝑆𝑟𝑥𝑛 The enthalpies, entropies and Gibbs energies associated with the formation of many compounds from their elemental forms have been tabulated for standard conditions ( , 298 K, 1 bar). From these tabulated values (click here) we can readily calculate the energy changes associated with a reaction. Definition : Standard Enthalpy, Entropy and Gibb’s Energy Changes For a given reaction: Õ Õ Δ𝐻𝑟𝑥𝑛 = (𝜈Δ𝐻 𝑓 (products)) − (𝜈Δ𝐻 𝑓 (reactants)) Õ Õ Δ𝑆𝑟𝑥𝑛 = (𝜈𝑆 𝑓 (products)) − (𝜈𝑆 𝑓 (reactants)) Õ Õ Δ𝐺 𝑟𝑥𝑛 = (𝜈Δ𝐺 𝑓 (products)) − (𝜈Δ𝐺 𝑓 (reactants)) where 𝜈 is the stoichiometric coefficient. This again is an application of Hess’s law, essentially when looking at the energies associated with a reaction we only need to look at the start and end points to work out the change. Example 1.4.1 (Calculating the Standard Molar Entropy for a Reaction) Take the reaction: 1 H2 (g) + 2 O2 (g) −−−→ H2 O 3  determine the standard molar entropy for the reaction (Δ𝑆rxn ) 1 Δ𝑆rxn = 𝑆 (H2 O) − 𝑆 (H2 ) − 𝑆 (O2 ) 2 Δ𝑆rxn = 69.9 − 131.0 − 0.5 × 205.0 JK−1 mol−1 Δ𝑆rxn = −164 JK−1 mol−1 Alternatively, when quantifying the energy changes during a reaction we can consider the differences in the bond dissociation energies (D) of the products and reactants. Õ Õ Δ𝐻𝑟𝑥𝑛 = 𝐷bonds broken − 𝐷bonds formed Note the order in the above expression, when a bond is formed heat is released. 1.5 Gibb’s Energy Definition : Gibb’s Energy Δ𝐺 = Δ𝐻 − 𝑇Δ𝑆 If a reaction is thermodynamically favorable then it has a negative Gibb’s energy and vice versa. The change in Gibb’s energy is the maximum work a system can do – the work that it actually does is always less, as some will be released as heat. It can be interpreted as representing the difference between the energy produced by the process, Δ𝐻, and the energy lost to entropy, 𝑇Δ𝑆. The Gibb’s energy was derived in the lectures from the second law of thermodynamics, where we showed at constant pressure Δ𝐺 = −𝑇Δ𝑆universe. Because the Gibb’s energy is temperature dependent if Δ𝐻 and Δ𝑆 are of the same sign then the Gibb’s energy for a process can change from being spontaneous to non-spontaneous as a function of temperature. Example 1.5.1 (Switching Temperature) Ammonium Nitrate dissolves in water, this reaction is endothermic but causes an increase in entropy. ↽ NH4 NO3 (s) − −−−⇀ −− NH4 + (aq) + NO3 – (aq) Δ𝐻 = 25.41 kJ mol−1 , Δ𝑆 = 108.7 J K−1 mol−1 Assuming the entropy and enthalpy changes are relatively insensitive to change in temperature. At what temperature does the reaction switch from being thermodynamically favorable to unfavorable? To find this point we need to determine at what temperature Δ𝐺 = 0. Δ𝐻 𝑇switch = Δ𝑆 𝑇switch = 234 K (i.e. − 39◦ C) 1.6 Chemical Equilibrium Many reactions are a balance between the reactants and the products, we use the reaction quotient to quantify whether, for a given set of conditions, we have more products or reactants. 4  Definition : Reaction Quotient mA + nB ↽ −⇀ −−−−− xC + yD where m, n, x and y are stoichiometric coefficients and A, B, C and D are chemical species. The reaction quotient is: [𝐶]𝑥 [𝐷] 𝑦 𝑄= [𝐴]𝑚 [𝐵]𝑛 remember that to calculate Q we need to use the standard units mol dm−3 for concentrations and bar for pressures. At equilibrium 𝑄 = 𝐾, where 𝐾 is the equilibrium constant. The equilibrium constant is directly related to the Gibb’s energy for the reaction. Δ𝐺 = −𝑅𝑇 ln 𝐾 where 𝑅 is the gas constant and 𝑇 is temperature. 1.7 Heat Capacities The amount of heat that needs to be added to a system to increase its temperature is an important parameter in thermodynamics. For example, the specific heat capacity of water is 4.184 J K−1 g−1. The heat capacity is central in calorimetry; the experimental technique that is used to accurately and precisely determine energy changes in a reaction. Even more fundamentally, the heat capacity for a material can be related to its change in enthalpy and entropy as a function of temperature. It is important to recognise that here, we are not talking about reaction enthalpies and entropies but the change in the energy associated with a substance. ∫ 𝑇𝑓 Δ𝐻 = 𝐶 𝑝 𝑑𝑇 𝑇𝑖 ∫ 𝑇𝑓 𝐶𝑝 Δ𝑆 = 𝑑𝑇 𝑇𝑖 𝑇 To quantify the change in enthalpy of a reaction as a function of temperature we need to consider the difference in the heat capacities of the products and reactants. ∫ 𝑇2 Δ𝐻𝑟𝑥𝑛,𝑇2 = Δ𝐻𝑟𝑥𝑛,𝑇1 + Δ𝐶 𝑝 𝑑𝑇 𝑇1 1.8 Fundamental Thermodynamic Relationship We can only get so far in Thermodynamics by considering finite (Δ) changes in energy, to harness the full power of the subject we need to deal with the differential forms of the expressions. For instance, if we want to derive the relationship between a material’s heat capacity and its change in entropy as a function of temperature we need to use the differential form. Definition : Fundamental Relationship 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 relatedly, we can define the fundamental relationship in terms of an infinitesimal change in enthalpy. 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉 𝑑𝑃 1.9 Gases The ideal gas law relates the volume and pressure of a gas to its temperature and number of moles. 𝑃𝑉 = 𝑛𝑅𝑇 5 𝑅 = 8.314 J K−1 mol−1 = 8.314 m3 Pa K−1 mol−1 Definition : Entropy change of an Expanding gas 𝑉𝑓 Δ𝑆 = 𝑛𝑅𝑇 ln 𝑉𝑖 derivation of this expression requires consideration of the differential expression describing the work done as a gas is compressed or expands. 6