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Derivation of thermodynamic equations Outline: – Derivation of single phase region P,V,T relationship Volume expansivity (or thermal expansion) Isothermal compressibility – Derivation of ideal gas equations for process calculations – Irreversible processes everything begins with mass and energy balances (a.k.a., continuity equations)

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PVT Single Phase Region PV diagram for CO2 Phase diagram implies a relationship between V,P,T In 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 dont believe this) Ex: From the definition of a partial derivative:

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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 heated Change in volume with change in pressure (compression) Ex: Compressed gas in a gas cylinder Define: Derive:

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Example 1. Between the temperatures of 1 o C and 4 o C, liquid water has a thermal expansion coefficient of =2*10 -5 T – 7*10 -5 K -1. If a puddle of water outside had a volume of 5L at 1 o C, calculate its volume at 4 o C.

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Ideal Gas Process Equations

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Ideal gas equations for process calculations We can combine PV = RT and W = -PdV and 1 st Law in many interesting ways Main equation: (must be able to derive this in order to understand what assumptions were made) Derive equations for: 1. Isothermal process 2. Isobaric process 3. Isochoric process

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Insert Derivations of Ideal Gas Process Equations Here

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Example 2. A monatomic gas at 25 o C and 1 atm is to be heated and compressed reversibly to 300 o C and 10 atm. Compute the heat and work required along each of these paths: 1.Isothermal compression to 10atm followed by an isobaric heating to 300 o C 2.Isobaric heating to 300 o C, followed by isothermal compression to 10atm 3.An adiabatic compression to 10atm, followed by either isobaric cooling or heating to the required temperature

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Irreversible Processes A reversible process is the best case scenario – Ex: where friction doesnt 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 energy Practically this means reversible processes involve integration For Q and W (unless they are equal to a state function), they are influenced by irreversibilities (but not U,H)

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Irreversible Processes For Example: W = force distance (universal definition) W sys = -P ext area distance If equilibrium position abruptly shifts by a large quantity, (like L or V), then final state is the same but more work was required for compression: (reversible P ext = P gas at equilibrium) (irreversible, F ext const and P ext constant but not at equilibrium) For an isothermal process, if equilibrium position shifts by a series of infinitesimally small steps, (like dL or dV), the process is reversible: Area under the curve

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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 losses The efficiency quantifies those losses – For compression: Must use common sense when calculating efficiencies (efficiency cannot be > 100%)

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Example 3. An ideal gas undergoes expansion in a piston from its initial volume to two times this volume at 25 o C. What is the work done if the process is reversible? Irreversible? What is the efficiency of the expansion process?

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