Chapter 2. The First Law 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The Basic Concepts 2.1 Work, heat, and energy 2.2 The internal energy 2.3 Expansion Work 2.4 Heat Transactions 2.5 Enthalpy 2.6 Adiabatic changes Thermochemistry 2.7 Standard Enthalpy changes 2.8 Standard Enthalpies of formation 2.9 The temperature dependence of reaction enthalpies State functions and exact differentials 2.10 Exact and inexact differentials 2.11 Changes in internal energy 2.12 The Joule-Thompson effect System Surroundings Universe Surroundings The basic concepts Surroundings • System : the part of the universe that we are studying • Surroundings : the rest of universe The basic concepts Open: energy and matter can be exchanged with the surroundings Closed: energy can be exchanged with the surroundings, matter cannot Isolated: neither energy nor matter can be exchanged with the surroundings 2.1 Work, heat, and energy Key points 1. Work is done to achieve motion against an opposing force 2. Energy is the capacity to work 3. Heating is the transfer of energy that makes use of disorderly molecular motion 4. Work is the transfer of energy that makes use of organized motion 2.1 Work, heat, and energy (a) Operational definitions 1. Work (w) : Work is done to achieve motion against an opposing force 2. Energy : The capacity of the system to do work 3. Heat (q) : The quantity of energy transferred as a result of a temperature difference 4. Exothermic process : releases energy as heat into its surrounding 5. Endothermic process : energy is required Adiabatic Diathermic 2.1 Work, heat, and energy (b) Molecular interpretation 1. Heating is the transfer of energy that makes use of disorderly molecular motion (thermal motion) in the surroundings 2. Work is the transfer of energy that makes use of organized motion in the surrounding 3. The distinction between work and heat is made in the surroundings 2.2 The internal energy Key points 1. Internal energy (U) is the total energy of a system 2. Internal energy is a state function 3. The equipartition theorem can be used to estimate the internal energy 4. The First law ΔU = U f −Ui U is the state function in the sense that its value depends only on the current state of the system 2.2 The internal energy (a) Molecular interpretation of internal energy 1. Equipartition theorem : the average energy of each quadratic contribution to the energy is kT/2 2. k : Boltzmann constant 3 U m (T ) = U m (0) + RT (monatomic gas; translation only) 2 5 U m (T ) = U m (0) + RT (linear molecule; translation and rotation only) 2 U m (T ) = U m (0) + 3RT (nonlinear molecule; translation and rotation only) • • • • • Um = U/n : molar internal energy Um(0) : molar internal energy at T N = nNA and R = NAk 3/2RT = 3.7 kJ/mol at 25°C U of a perfect gas is independent of V 2.2 The internal energy (b) The formulation of the First Law 1. The First Law of Thermodynamics : The internal energy of an isolated system is constant 2. Heat and work are equivalent ways of changing a system’s internal energy ΔU = q + w • • The change in internal energy of a closed system is equal to the energy that passes through its boundary as heat or work ‘acquisitive convention’ : q and w are positive if energy is transferred to the system as work or heat and negative if energy is lost from the system 2.3 Expansion Work Key points 1. Expansion work is proportional to the external pressure 2. Free expansion does not work 3. To achieve reversible expansion, the external pressure is matched at every stage dU = dq + dw 2.3 Expansion Work (a) The general expression dw = −Fdz dw = − pex dV w=−∫ Vf Vi pex dV massless, frictionless, rigid, perfectly fitting 2.3 Expansion Work (a) The general expression 2.3 Expansion Work (b) Free expansion pex = 0 ⇒ w = 0 (c) Expansion against constant pressure Vf w = − pex ∫ dV = − pex (V f −Vi ) Vi w = − pex ΔV Indicator Diagram 2.3 Expansion Work (d) Reversible expansion 1. A reversible change : a change that can be reversed by an infinitesimal modification of a variable 2. Systems at equilibrium are poised to undergo reversible change 3. pex = p should be set at each stage of the expansion to achieve reversible expansion dw = − pex dV = − pdV w=−∫ Vf Vi pdV Consider a sample of gas in thermal and mechanical equilibrium with the surroundings; i.e., with Tgas = Tsurroundings and pgas = pexternal. If the external pressure is decreased infinitesimally at constant T, the gas will expand infinitesimally; if the external pressure is increased infinitesimally at constant T, the gas will be compressed by an infinitesimal amount. Strictly speaking, a reversible process cannot be achieved, since to carry out a finite transformation in a series of infinitesimal steps would require infinite time. All real processes are therefore irreversible. A reversible process is an idealization (very useful) 2.3 Expansion Work (e) Isothermal reversible expansion 1. Isothermal, reversible expansion 2. More work is obtained when the expansion is reversible!!! 3. It is because matching the external pressure to the internal pressure at each stage of the process ensures that none of the system’s pushing power is wasted w=−∫ w = −nRT ∫ Vf Vi Vf Vi pdV Vf dV = nRT ln V Vi 2.4 Heat transactions Key points 1. The energy transferred as heat at constant volume is equal to the change in internal energy of the system 2. The heat capacity at constant volume is the slope of the internal energy with respect to temperature dU = dq + dwexp + dwe dU = dq at constant volume ΔU = qV 2.4 Heat transaction (a) Calorimetry 1. Calorimetry : the study of heat transfer 2. Calorimeter : a device for measuring energy transferred as heat 3. Calorimeter constant may be measured electrically by passing a current, I, of potential difference Δϕ for time, t. q = CΔT calorimeter constant q = ItΔφ I : current t : time Δϕ : potential diff. Bomb Calorimeter 2.4 Heat transaction (b) Heat Capacity 1. Heat capacity : the slope of the tangent to the curve at any temperature 2. Heat capacity is extensive properties 3. Molar heat capacity at constant volume is intensive property 4. Specific heat capacity is the heat capacity per unit mass 5. Heat capacities depend on the temperature 6. Over small ranges of T at and above room T, the variation is quite small " ∂U % CV = $ ' # ∂T &V CV ,m = CV / n dU = CV dT (at constant volume) ΔU = CV ΔT (at constant volume) qV = CV ΔT 2.4 Heat transaction (b) Heat Capacity 1. Large heat capacity : there will be only a small increase in T for a given quantity of energy transferred as heat 2. An infinite heat capacity : there will be no increases in T 3. At a phase transition, T does not rise as energy is supplied as heat 4. At a phase transition, the heat capacity of a sample is infinite " ∂U % CV = $ ' # ∂T &V CV ,m = CV / n dU = CV dT (at constant volume) ΔU = CV ΔT (at constant volume) qV = CV ΔT 2.5 Enthalpy Key points 1. Energy transferred as heat at constant pressure is equal to the change in enthalpy of a system 2. The heat capacity at constant p is equal to the slope of enthalpy with T 2.5 Enthalpy (a) The definition of enthalpy 1. Enthalpy (H) is defined as H = U + pV 2. H is a state function because U, p, and V are state functions 3. The change in enthalpy is equal to the energy supplied as heat at constant p dH = dq ΔH = q p 2.5 Enthalpy (b) The measurement of an enthalpy change 1. Isobaric calorimeter : monitors the temperature change at constant pressure 2. Adiabatic flame calorimeter 3. When a process involves only solids or liquids, the values of ΔH and ΔU are almost identical because pVm is so small. 4. Differential scanning calorimeter (DSC) 5. For a perfect gas: H = U + pV = U + nRT ΔH = ΔU + Δng RT 2.5 Enthalpy (c) The variation of enthalpy with T 1. The heat capacity at constant p, Cp : the slope of the tangent to the a plot of enthalpy against T at constant p 2. T rises at constant p less than when heating occurs at constant V because systems at constant p do work on the surroundings 3. Cp is larger than Cv " ∂H % Cp = $ ' # ∂T & p dH = C p dT (at constant pressure) ΔH = C p ΔT (at constant pressure) q p = C p ΔT 2.6 Adiabatic changes Key points 1. For the reversible expansion of a perfect gas, p and V are related by an expression that depends on the ratio of heat capacities 2.6 Adiabatic changes 1. 2. Consider the changes that occur when a perfect gas expands adiabatically The overall change in internal energy arises solely from the second step ΔU = CV (Tf − Ti ) = CV ΔT wad = CV ΔT 2.6 Adiabatic changes 1. The physical reason for the difference between adiabats and isotherms is that in an isothermal expansion, energy flows into the system as heat and maintains T; as a result, p does not fall as much as in an adiabatic expansion !V Tf = Ti ## i " Vf 1c $ && % c = CV ,m R ViTi c = V f T cf p f V γf = piViγ γ = C p,m CV ,m