Presentation on theme: "Derivation of thermodynamic equations"— Presentation transcript:
1 Derivation of thermodynamic equations everything begins with mass and energy balances (a.k.a., continuity equations)Outline:Derivation of single phase region P,V,T relationshipVolume expansivity (or thermal expansion)Isothermal compressibilityDerivation of ideal gas equations for process calculationsIrreversible processes
2 PVT Single Phase Region PV diagram for CO2Phase diagram implies a relationship between V,P,TIn the single phase region, an equation of state (EOS) can be solved for any one of V,P,T as a function of the other two (check back with the phase rule if you don’t believe this)Ex:From the definition of a partial derivative:
3 PVT Single Phase Region The partial derivative terms actually have physical meanings!Change in volume with change in temperature (thermal expansion)Ex: Metals expand when heatedChange in volume with change in pressure (compression)Ex: Compressed gas in a gas cylinderDefine:Derive:
4 Example 1.Between the temperatures of 1oC and 4oC, liquid water has a thermal expansion coefficient of b=2*10-5T – 7*10-5 K-1. If a puddle of water outside had a volume of 5L at 1oC, calculate its volume at 4oC.
6 Ideal gas equations for process calculations We can combine PV = RT and W = -PdV and 1st Law in many interesting waysMain equation:(must be able to derive this in order to understand what assumptions were made)Derive equations for: 1. Isothermal process2. Isobaric process3. Isochoric process
7 Insert Derivations of Ideal Gas Process Equations Here
8 Example 2.A monatomic gas at 25oC and 1 atm is to be heated and compressedreversibly to 300oC and 10 atm. Compute the heat and work required along each of these paths:Isothermal compression to 10atm followed by an isobaric heating to 300oCIsobaric heating to 300oC, followed by isothermal compression to 10atmAn adiabatic compression to 10atm, followed by either isobaric cooling or heating to the required temperature
9 Irreversible Processes A reversible process is the best case scenarioEx: where friction doesn’t play a role, chemical kinetics are fast, wear and tear is assumed to be non-existent…More accurately: The equilibrium position only changes by infinitesimal amounts and can be returned to its original state without a loss of energyPractically this means reversible processes involve integrationFor Q and W (unless they are equal to a state function), they are influenced by irreversibilities (but not U,H)
10 Irreversible Processes For Example: W = force distance (universal definition) Wsys = -Pext area distanceFor an isothermal process, if equilibrium position shifts by a series of infinitesimally small steps, (like dL or dV), the process is reversible:(reversible Pext = Pgas at equilibrium)Area under the curveIf equilibrium position abruptly shifts by a large quantity, (like DL or DV), then final state is the same but more work was required for compression:(irreversible, Fext const and Pext constant but not at equilibrium)
11 Irreversible Processes The reversible process is the best case scenario in terms of maximum work produced during expansion or minimum work required for compression.The irreversible process contains some lossesThe efficiency quantifies those lossesFor compression:Must use common sense when calculating efficiencies (efficiency cannot be > 100%)
12 Example 3.An ideal gas undergoes expansion in a piston from its initialvolume to two times this volume at 25oC. What is the workdone if the process is reversible? Irreversible? What is the efficiency of the expansion process?