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)

PVT Single Phase RegionPV 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 don’t believe this)

• Ex:

• From the definition of a partial derivative:

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:

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.

Ideal Gas Process Equations

Ideal gas equations for process calculations

• We can combine PV = RT and W = -PdVand 1st 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 process2. Isobaric process3. Isochoric process

Insert Derivations of Ideal Gas Process Equations Here

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:

1. Isothermal compression to 10atm followed by an isobaricheating to 300oC

2. Isobaric heating to 300oC, followed by isothermal compressionto 10atm

3. An adiabatic compression to 10atm, followed by eitherisobaric cooling or heating to the required temperature

Irreversible Processes• A reversible process is the best case scenario– Ex: 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 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)

Irreversible Processes• For Example: W = force distance (universal definition)

Wsys = -Pext area distance

• If 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:

(reversible Pext = Pgas at equilibrium)

(irreversible, Fext const and Pext constant but not at equilibrium)

2

1

2

1

V

V

L

L

PdVFdLW

VPLFW

• 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

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%)

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