Thermodynamic Properties Property Table w Property Table -- from direct measurement w Equation of State w Equation of State -- any equations that relates P,v, and T of a substance

Ideal -Gas Equation of State w Any relation among the pressure, temperature, and specific volume of a substance is called an equation of state. The simplest and best-known equation of state is the ideal-gas equation of state, given as where R is the gas constant. Caution should be exercised in using this relation since an ideal gas is a fictitious substance. Real gases exhibit ideal-gas behavior at relatively low pressures and high temperatures.

Universal Gas Constant Universal gas constant is given on R u = kJ/kmol-K = kPa-m 3 /kmol-k = bar-m 3 /kmol-K = L-atm/kmol-K = Btu/lbmol-R = ft-lbf/lbmol-R = psia-ft 3 /lbmol-R

Example Determine the particular gas constant for air and hydrogen.

Ideal Gas “Law” is a simple Equation of State

Percent error for applying ideal gas equation of state to steam

Question …... Under what conditions is it appropriate to apply the ideal gas equation of state?

Ideal Gas Law w Good approximation for P-v-T behaviors of real gases at low densities (low pressure and high temperature). w Air, nitrogen, oxygen, hydrogen, helium, argon, neon, carbon dioxide, …. ( < 1% error).

Compressibility Factor w The deviation from ideal-gas behavior can be properly accounted for by using the compressibility factor Z, defined as Z represents the volume ratio or compressibility.

Ideal Gas Z=1 Real Gases Z > 1 or Z < 1

Real Gases w Pv = ZRT or w Pv = ZR u T, where v is volume per unit mole. w Z is known as the compressibility factor. w Real gases, Z 1.

Compressibility factor w What is it really doing? w It accounts mainly for two things Molecular structureMolecular structure Intermolecular attractive forcesIntermolecular attractive forces

Principle of corresponding states w The compressibility factor Z is approximately the same for all gases at the same reduced temperature and reduced pressure. Z = Z(P R,T R ) for all gases

Reduced Pressure and Temperature where: P R and T R are reduced values. P cr and T cr are critical properties.

Compressibility factor for ten substances (applicable for all gases Table A-3)

Where do you find critical-point properties? Table A-7 Mol (kg-Mol) R (J/kg.K) Tcrit (K) Pcrit (MPa) Ar28,97287,0(---) O2O2 32,00259,8154,85,08 H2H2 2, ,233,31,30 H2OH2O18,016461,5647,122,09 CO 2 44,01188,9304,27,39

Reduced Properties w This works great if you are given a gas, a P and a T and asked to find the v. w However, if you are given P and v and asked to find T (or T and v and asked to find P), trouble lies ahead. w Use pseudo-reduced specific volume.

Pseudo-Reduced Specific Volume w When either P or T is unknown, Z can be determined from the compressibility chart with the help of the pseudo- reduced specific volume, defined as not v cr !

Ideal-Gas Approximation w The compressibility chart shows the conditions for which Z = 1 and the gas behaves as an ideal gas: w (a) P R 1

Other Thermodynamic Properties: Isobaric (c. pressure) Coefficient v T P

Other Thermodynamic Properties: Isothermal (c. temp) Coefficient v P T

Other Thermodynamic Properties: We can think of the volume as being a function of pressure and temperature, v = v(P,T). Hence infinitesimal differences in volume are expressed as infinitesimal differences in P and T, using  and  coefficients If  and  are constant, we can integrate for v:

Other Thermodynamic Properties: Internal Energy, Enthalpy and Entropy

Other Thermodynamic Properties: Specific Heat at Const. Volume u T v

Other Thermodynamic Properties: Specific Heat at Const. Pressure h T P

Other Thermodynamic Properties: Ratio of Specific Heat

Other Thermodynamic Properties: Temperature s u v T 1

Ideal Gases: u = u(T) Therefore, 0

We can start with du and integrate to get the change in u: Note that C v does change with temperature and cannot be automatically pulled from the integral.

Let’s look at enthalpy for an ideal gas: w h = u + Pv where Pv can be replaced by RT because Pv = RT. w Therefore, h = u + RT => since u is only a function of T, R is a constant, then h is also only a function of T w so h = h(T)

Similarly, for a change in enthalpy for ideal gases:

Summary: Ideal Gases w For ideal gases u, h, C v, and C p are functions of temperature alone. w For ideal gases, C v and C p are written in terms of ordinary differentials as

For an ideal gas, w h = u + Pv = u + RT C p = C v + R

Ratio of specific heats is given the symbol, 

Other relations with the ratio of specific heats which can be easily developed:

For monatomic gases, Argon, Helium, and Neon

For all other gases, w C p is a function of temperature and it may be calculated from equations such as those in Table A-5(c) in the Appendice w C v may be calculated from C p =C v +R. w Next figure shows the temperature behavior …. specific heats go up with temperature.

Specific Heats for Some Gases w C p = C p (T) a function of temperature

Three Ways to Calculate Δu and Δh w Δu = u 2 - u 1 (table) w Δu = w Δu = C v,av ΔT w Δh = h 2 - h 1 (table) w Δh = w Δh = C p,av ΔT

Isothermal Process w Ideal gas: PV = mRT

For ideal gas, PV = mRT We substitute into the integral Collecting terms and integrating yields:

Polytropic Process w PV n = C

Ideal Gas Adiabatic Process and Reversible Work w What is the path for process with expand or contract without heat flux? How P,v and T behavior when Q = 0? w To develop an expression to the adiabatic process is necessary employ: 1. Reversible work mode: dW = PdV 2. Adiabatic hypothesis: dQ =0 3. Ideal Gas Law: Pv=RT 4. Specific Heat Relationships 5. First Law Thermodynamics: dQ-dW=dU

Ideal Gas Adiabatic Process and Reversible Work (cont) First Law: Using P = MRT/V Integrating from (1) to (2)

Ideal Gas Adiabatic Process and Reversible Work (cont) Using the gas law : Pv=RT, other relationship amid T, V and P are developed accordingly:

Ideal Gas Adiabatic Process and Reversible Work (cont) An expression for work is developed using PV  = constant. i and f represent the initial and final states

Ideal Gas Adiabatic Process and Reversible Work (cont)  The path representation are lines where Pv  = constant.  For most of the gases,   1.4 w The adiabatic lines are always at the righ of the isothermal lines. w The former is Pv = constant (the exponent is unity) P v Q = 0 T=const. i f f

Polytropic Process A frequently encountered process for gases is the polytropic process: PV n = c = constant Since this expression relates P & V, we can calculate the work for this path.

Polytropic Process w Constant pressure 0 w Constant volume  w Isothermal & ideal gas 1 w Adiabatic & ideal gas k=C p /C v Process Exponent n

Boundary work for a gas which obeys the polytropic equation

We can simplify it further The constant c = P 1 V 1 n = P 2 V 2 n

Polytropic Process