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87 Cards in this Set
- Front
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Adiabatic |
Extremely fast Ideal process Q = constant
P × V^gamma T×V^(gamma - 1) T×P^(1-gamma/gamma) Cv = R / (gamma - 1) Work W = mR (T1-T2)/gamma - 1 W= (P1V1 - P2V2) /gamma - 1 |
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Isenthalpic |
H = constant
Throttling Its an expansion through a narrow restricted passage, without doing any external work, under adiabatic condition |
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Isentropic |
S = constant
Adiabatic + reversible |
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Lewis Randell rule |
It states that fugacity of a component in an ideal solution is directly proportional to molar fraction of the component in the solution f"i proportional to xi f"i = xi fi f"i = fugacity of i in ideal solution xi = mole fraction of i the solution fi = fugacity of i in pure state |
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Raults law |
Ideal solution PMP of solution = MP of solution Law P"i = xi Psi Partial pressure of the component in the ideal solution is directly proportional to mole fraction in the solution P"i = yi P yi P = xi Psi |
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Henry's law |
For dilute ideal solution, the fugacity of component is directly proportional to the mole fraction of the component in the solution. f"i proportional xi f"i = xi Hi xi →0, f"i = xi Hi xi →1, f"i = xi fi xi =1, f"i = fi |
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Activity in solution Effect of T and P |
Ratio of fugacity of the component in solution to the fugacity of component in the standard state
ai = f"i/f°
üi -üi(id) = RT (lnai)
For ideal solution f"i = xi f°i ; So ,ai = xi
For real solution fi≠f°i ai proportional to xi ai = Yi xi ; Yi = Activity Coeff (indicates extent to which solution is non ideal)
For real solution f"i = Yi xi fi
For ideal solution Yi = 1 Effect of Temp on Yi d/dT (lnYixi) = (H"i - Hi)/RT²
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Gibbs Duhem eqⁿ general form And other derived form |
€ ni dM"i = 0 Others € ni dG"i = 0 or € ni düi = 0 x1 dü1/dx1 - x2 dü2/dx2 =0 x1 dV"1/dx1 - x2 dV"2/dx2 =0 x1 dlnf"1/dx1 - x2 dlnf"2/dx2 =0 x1 dlnY1/dx1 - x2 dlnY2/dx2 =0 |
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Property change of mixing general form |
∆M = € xi (M"i-Mi) |
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Free Energy change of mixing |
∆G = RT € xi lnxi For ideal solution, ai = xi ∆G = RT € xi lnai For pure solution; xi→1 ∆G = 0 |
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Volume change of mixing |
Below differentiation is partial ∆V = RT € xi d/dP (lnai) For ideal solution,ai = xi , ai = xi ∆V = RT € xi d/dP (lnxi) For pure solution ∆V = 0 |
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Enthalpy Change of mixing |
At constant P ∆H = - RT² €xi ( d/dT (lnai)) For ideal solution, ai = xi ∆H = - RT² €xi ( d/dT (lnxi)) For pure solution, xi →1 ∆H = 0 |
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Entropy change of mixing |
∆s = -R € xi dlnai/dlnT -R €xi lnai For ideal solution ai = xi = constant ∆s = - R € xi lnxi For pure solution ∆s = 0 |
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Excess Property |
M(E) = M(Real) - M(ideal) How much the real mixture deviating from ideality M(E) >0 +ve deviation from ideality M(E) <0 -ve deviation from ideality |
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Excess change of mixing |
∆M(E) = ∆M (real) - ∆M(id) ∆M(E) = M (real) - M(id) = M(E) |
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Excess Gibbs free energy |
∆G(E) = RT ln(Yi) |
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Extent of reaction £ |
yi = ni/n ni = nio + Vi£ Vi = coefficient of product - coefficient of reactant n = no + V£ no = €nio V = €Vi |
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Gibbs phase rule or phase rule |
F=C-P+2 C= no. of chemical species.. F<0 ; system is in disequilibrium F= C-P+2-r r = no. of independent chemical rxn |
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1st law |
In a closed system going any thermo cycle, cycle integral of work and cycle integral of heat are proportional to each other proportional = when expressed in their own unit Equal to = when expressed in consistent units |
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$Q - $W , $ = partial derivative |
Independent path Point function Thermo property |
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dU= |
$Q - $W All function are perfect differential |
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Isothermal |
T = constant Extremely slow process Ideal process ∆U = 0 Q=W ∆H = 0 Work W = P1V1 ln (V2/V1) = RT ln(V2/V1) = RT ln(P1/P2) |
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Iso baric |
P = constant Q = ∆H Work W = P (V2-V1) |
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Isochoric |
V = constant W =0 Q = U |
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Polytropic |
PVⁿ = constant TV^(n-1) = c TP^ (1-n)/n = c Work = (P1V1-P2V2)/ n-1 |
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For air Cp, Cv and R |
Cp = 1.005 kJ/kgK Cv = 0.718 kJ/kgK R = 0.287 kJ/kgK |
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For compression |
W(irreversible) = W(reversible)/n |
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For expansion |
W(irreversible) = W(reversible) × n n = efficiency |
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U and H |
H used in open system Sum of internal energy and flow energy U used in closed system |
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For open system 1st law |
dH + d(K.E.) +d(P.E.) = $q - $Ws |
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WD in open system and adiabatically |
W (a to b) = int (a to b) {-VdP} W (1 to 2) = gamma (P1V1- P2V2) / gamma - 1 |
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Relationship b/w open system and closed system WD |
WD(ops) = gamma WD(cls) WD(ops) = WD(cls) + net flow of system - int (1to2)VdP = int (1to2)PdV +(P1V1-P2V2) |
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Clausis statement |
Heat can't flow from low T to high T naturally |
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Kelvin Planck's |
It is impossible to have 100℅ n of a engine |
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Heat engine |
n (efficiency) = o/p÷i/p n = energy aimed at / energy that costs = Net WD / Heat Supplied Net WD = Qh - Ql Net WD = W(by turbine) - W(by pump) n = Qh - Ql / Qh Its an expansion process, work is obtained heat must be supplied Its a compression process, work is required heat should be rejected out |
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Carnot engine |
WD = Qh - Ql = P1V1ln(V2/V1) - P3V3 ln(V3/V4) n(carnot) = Th-Tl/Th = Qh - Ql/Qh |
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C.O.P. of Refrigerator and Heat pump |
Reciprocal of efficiency |
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Clausis Inequality |
If Th = constant
cyclic integration $Q/T ≤ 0
For actual system ie Irreversible system due to friction cyclic integration $Q/T = 0 For ideal system ie Rev system or frictionless cyclic integration $Q/T ≥ 0 Impossible |
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Entropy 's' |
ds = $Q/T ( Extensive property ) ds' = $q/T (Intensive property )
s/m = s' & Q/m = q
Its a thermodynamic property that ↑ when heat is supplied, expansion takes place, work is obtained,T↓, ↓ when heat is rejected & vice versa & remains constant if no heat is supplied or rejected
s'2 - s'1 = Cv ln (T2/T1) + R ln (V2/V1)
s'2 - s'1 = Cp ln (T2/T1) - R ln (P2/P1)
s'2 - s'1 = Cv ln (P2/P1) + Cp ln (V2/V1)
Isobaric s'2 - s'1 = Cp ln (V2/V1) = Cp ln (T2/T1) Slope +
Isochoric Process s'2 - s'1 = Cv ln (P2/P1) = Cv ln (T2/T1) Slope +
Isothermal process s'2 - s'1 = - R ln (P2/P1) = R ln (V2/V1) Slope 0
Adiabatic .s'2 - s'1 = 0, s = constant Slope ∞ |
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Maxwell relation |
V A T -U -G S H P |
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Volume Expansivity ß or Coeff. of thermal expansion And Isothermal Compressibility K |
Differentiation is partial ß = 1/V (dV/dT) at constant P K = -1/V (dV/dP) at constant T Cp - Cv = TVß²/ K For incompressible fluid ß = K = 0 For an ideal gas ß is fⁿ of T and ß = 1/T K is fⁿ of P and K =1/P h is fⁿ of T For real gas h is fⁿ of P,V |
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Joule-kelvin or jolue Thomson coefficient |
Differentiation is partial
ü = (dT / dP) at constant H
ü = (dT / dP) at constant H = 1/Cp [T dV/dT(at constant P) - V]
If ideal gas is throttled ü = (dT / dP) at constant H = 0
Expansivity ü = (dT / dP) at constant H = V/Cp(Tß-1) ü + ; cooling ü - ; heating
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Clapeyron eqⁿ |
dP/dT = s'g-s'f/Vg-Vf = hfg /{Ts(Vg-Vf)}
hfg = latent heat of vaporisation dP(saturation)/ dT = ∆H/T∆V
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Clausius Clapeyron Eqⁿ 3 forms |
1. dP/P = hfg / RT² × dT 2. ln(P2/P1) = ( hfg / R) × (1/T1-1/T2) For liq. and vapor For liq. and vapor 3. d(lnP)/d(1/T) = - hfg/R Slope = -hfg/R = -ve |
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Dryness Factor |
X = Mass of dry & sat. steam / (Mass of dry & sat. steam + Mass of saturated water ) |
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Degree of super heat |
= Tsup. h - Ts |
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Degree of super heat |
= Tsup. h - Ts |
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hg, sg, and h(wet) |
hg = hf+hfg s'g = s'f + s'fg h(wet) = hf+ xhfg hg = enthalpy of sat. vapor hf = enthalpy of sat. liq. (At exit) h(wet) = enthalpy of sat. liq. (At enter) |
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hg, s'g, and h(wet) |
hg = hf+hfg
s'g = s'f + s'fg
h(wet) = hf+ xhfg
hg = enthalpy of sat. vapor hf = enthalpy of sat. liq. (At exit) h(wet) = enthalpy of sat. liq. (At enter) |
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Virial EOS Compressibility factor = Z |
Z = volm of real gas / volm of ideal gas Z = V’ /(RT/P) = PV’ /RT Z = 1 + B°(T)/V’ + B¹(T)/V’² + ……… B°; 0 is not a power only denotation Z = 1 + B°(T)/V’ B° = (RTc/Pc) (B'+WB") W = accentric factor (Molecule shape, density, sphericity of molecule) B' = 0.083 - (0.422/Tr ^1.6) B" = 0.139 - (0.172)/Tr^4.2 Tr = Reduced temp.= T/Tc |
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Vander wall EOS a, b, V'c, Zc |
P = [RT/(V'-b)] - (a/V'²) b → figure volume occupied by molecule a/V'² → attractive a = 3 Pc V'c² b = V'c /3 a = 27/64 R² Tc²/ Pc b = ⅛ R Tc/Pc Zc = Pc V'c/R Tc = ⅜ Zc = compressibility at critical point. |
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Chemical potential |
Change in entropy per unit mass is chemical potential of any system üA = dS(T)/dnA
Differentiation is partial T n A are in suffix |
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0th law |
If A n B system are in thermal equilibrium with C system then A n C are in thermal equilibrium also |
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0th law |
If A n B system are in thermal equilibrium with C system then A n C are in thermal equilibrium also |
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1st law |
For any cyclic process Change in IE is equal to the heat supplied to system - work obtained by the system |
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2nd law |
HT can't be done from a cold region to hot region naturally, spontaneously (irreversible) Reversible → quasi static → almost at rest |
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3rd law |
At absolute zero temp , Abs entropy is minimum for all the systems |
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Chemical potential ü |
It's a partial molar free energy It is equal to change in total Gibbs fee energy power unit change of moles of any component. ü = (dGt/dna) T,P,n≠A |
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Reversible |
Ideal case Net change in ü is 0 ∆G = 0 |
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Irreversible |
Spontaneous process in forward direction ∆G < 0 |
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Not possible |
∆G >0 Spontaneous process in backward directions |
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Thermodynamic equilibrium |
Mechanical equilibrium Thermal equilibrium Chemical equilibrium dG = 0 dP = 0 dT =0 |
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dU |
= Tds - PdV |
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dH |
= Tds + VdP |
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dA |
= -PdV - sdT |
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dG |
= VdP - sdT |
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Irreversible |
Spontaneous + Naturally |
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For open system |
Fundamental relation for whole thermodynamics dG= VdP - sdT + € üi dni
For n moles dG= VdP - sdT + € üi dxi
Here xi = ni/n € = summation |
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Phase equilibrium |
üi (alpha) = üi (ß) |
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Molar property of i in pure substance (Mi) and Partial molar property of ith in mixture (M'i) |
1. Pure Substance ∆Mt = ∆ni × Mi 2. Mixture ∆Mt ≠ ∆ni × Mi ∆Mt = ∆ni × M'i M'i = d(Mt) / d(ni) At constant P,T,ni≠i Denotation Pure Vi, ui, si, Hi, Gi Mixture V'i, u'i, s'i, H'i, G'i |
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Relation Btw PMP and MP |
dMt = €(iton) M'i dni
Mt = € M'i ni
M = € M'i xi (similarly for H,U and V)
M = M'1 x1 + M'2 x2
M = Mt /n
Below differentiation is in partial form M'1 = M + (n1+n2) dM/dn1 M'1 = M - x2 dM/dx2
M'2 = M + (n1+n2) dM/dn2 M'2 = M - x1 dM/dx1
x1 dM'1/dx1 + x2 dM'2/dx2 = 0
When chemical potential x1 dG'1/dx1 + x2 dG'2/dx2 = 0
At ∞ dilution of 1, x1→0 then H'1 = H'1^∞
At ∞ dilution of 2, x2→0 then H'2 = H'2^∞ |
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Chemical potential and G
Effect of T on ü and Effect of P on ü |
Here ü (act as PMP) & G (act as MP)
dG = € üi dxi
dG = € (dG/dni) dxi { inside brackets differentiation is partial } at constant P,T,ni≠i Effect of T dü/dT = -(dst/dni) at constant P = - s'i
Effect of P dü/dP = (dVt/dni) at constant T = V'i |
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Ideal gas and real gas |
Id : there is no interaction between molecule Real : molecular interaction is there but behaves as ideal at low P and high T |
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Fugacity |
Ideal gas dG = RT d(lnP) P = pressure of ideal gas
Real gas dG = RT d(lnf) f=effective pressure exerted in case of real gas
Ideal gas ∆G = G2 - G1 = RT ln(P2/P1)
Real gas ∆G = G2 - G1 = RT ln(f2/f1)
For ideal gas f=P For real gas f≠P
Standard state for fugacity Low pressure P° f°=P°
G - G° = RT ln(f/f°)
If p°→0 f/p = 1 If p↑,f↑ real condition exist f≠p |
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Fugacity coefficient ∅ |
∅ = fugacity of gas / pressure of ideal gas ∅ = f/p Real gas approaching towards ideality ∅→1 f→p p° →0 Gas is far from ideality ∅ → 0 f↑ , p°↑ If p→0 f°=p° ∆G = -∞ |
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Calculation of fugacity using compressibility factor Z |
Z= volm of real gas / volm of ideal gas
d(lnf/p) = (Z-1)/P dP
ln∅ = Int (0toP) (Z-1)/P dP |
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Calculation of fugacity using Residual Molar Volm (alpha) |
Alpha = volume of gas - volume occupied by ideal gas Alpha = V - RT/P V = ZRT/P Alpha = (Z-1) RT/P As P↑,Z low & P↓,Z high Int(0toP) alpha dP = RT ln(f/p) |
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Calculation of fugacity using H and s |
ln(f/f°) = (H-H°)/RT - (S°-S)/R
f=∅p, f°=p°
H → real case H° → ideal case
Effect of T dlnf/dT = H°-H/RT² Here f° not comes because its constant
T↑,H↑,(H°-H) ↓, f↓
Effect of p dlnf/dP = V/RT |
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Standard case of fugacity |
G-G° = RTln(P/P°) + RT ln∅ Change in Gibbs free energy for real gas = Change in Gibbs free energy for ideal gas + RT ln∅ RT ln∅ → Due to intermolecular forces ∅ = 1, f=p RT ln∅ = 0 ∆G=∆G° ∅ <1 ∆G < ∆G° p→0, ∅→1,real gas → ideality p→∞, ∅→0,real gas far from ideality |
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Fugacity in solid and liquid |
∆G = Gs - Gv = RT ln (fs/fv) At equilibrium ∆G = 0 Gs =Gv (fs=fv) For liquid (f l=fv) |
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Calculation of fugacity in solids and liquids |
At low VP P→ low Pv → low At equilibrium P=Pv fv = Pv = f l At low VP, ln(f/fs) = V/RT (P-Ps) |
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Activity or Relative fugacity
ai |
Ratio of fugacity to the fugacity in standard state for pure substances
ai = f(i)/f°(i)
∆G = RT ln(a)
Effect of T dlna/dT = H°-H/RT² Effect of P dlna/dP = V/RT Or lna = V/RT(P-P°) |
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Fugacity in solution f" = f cap G' = G bar |
For pure solution dG = RT d(lnf) For real mixture dG' = RT d(lnf") f"i = dfti/dni fi = fugacity in pure solution f"i = fugacity in real mixture düi = RT d lnf"i, for real solution düi = d G"i = RT dlnf"i dlnf"i = V/RT dp"i On integrating ln f"I = P/RT Int (0to1) Vdyi dln(f"i/P) = (Vyi/RT-1/p) dp ------(1) ln(f"i/P) = Int (0top) (Vyi/RT-1/p) dp From eq (1) P = P"i/y and V = V"i/yi Then dln(f"i/P"i) = (Vi/RT-1/p) dp |
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For ideal mixture |
Pure mixture V"i = Vi Ideal pure mixture Vi = RT/P ln ( f"i/p'i) = 0 f"i = p'i fugacity of i in mixture = partial pressure of i for ideal mixture |
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Volumetric property of fluid |
dV/V (differentiation is partial) = ßdT - KdP |
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Actual flame temp |
<Theoretical Flame Temp <Adiabatic Flame Temp |
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Troutons ratio |
K= ∆Hn/RTn ≈ 10 Here 10 is troutons no. |
