The Second Law: The Machinery Chater 5 of Atkins: The Second Law: The Concets Sections 3.7-3.9 8th Ed, 3.3 9th Ed; 3.4 10 Ed.; 3E 11th Ed. Combining First and Second Laws Proerties of the Internal Energy Proerties of the Gibbs Energy The Chemical Potential of a Pure Substance Real Gases: The Fugacity Definition of Fugacity Standard States of Gases Relation Between Fugacity and Pressure Last udated: Nov. 1, 2018, slides 13, 17
The Second Law: The Machinery The laws of thermodynamics we have learned thus far now enable us to begin finding relationshis between roerties that may have not been thought to be related with one another - there are many interesting hidden relationshis that can be extracted - lets now combine the laws: The First Law of Thermodynamics: du dq % dw For reversible change in constant comosition system with no nonexansion work dw rev &dv dq rev TdS Therefore: du TdS & dv Fundamental Equation du is an exact differential, indeendent of ath, so the same values for du are obtained regardless of change being reversible or irreversible (closed system, no non-exansion work)
Proerties of Internal Energy Reversible change: T ds same as dq, and - dv same as dw Irreversible change: T ds > dq (Clausius inequality), and - dv > dw For a system of constant comosition: dw + dq = T ds + - dv When S and V are changed, du % ds and du % dv, from the fundamental equation, suggesting that du should be written as a function of S and V du du TdS & dv MS V ds % MV S dv The equation above means that the change in U is roortional to changes in S and V, with the coefficients being sloes of lots of U against S at constant V, and U against V at constant S. For systems of constant comosition: Thermodynamic definition of temerature MS V T MV S & Thermodynamic definition of ressure
Maxwell Relations The state functions are exact differentials, meaning that they must ass the test that indicate their indeendence of ath taken: Mg df gdx% hdyis exact if Mh My x Mx y We know that du = T ds - dv is exact, then MT & M MV S MS We have generated a relationshi between quantities that would not seem to be related on first sight! In fact, the four Maxwell relations can be derived from the four state functions U, H, G and A: V MT MV S & M MS V MT M S MV MS MV & MS MT M T M MT V MS MV T
Variation of Internal Energy with Volume In Chater 3, we defined a coefficient called the internal ressure B T which can be written as a thermodynamic equation of state, in terms of and T, and can therefore be alied to any substance B T T MV M MT T V & This exression for B T is obtained by dividing by dv and substituting in: du MS V MV T ds % MS MV V S dv MS MV T % T MS MV T & MS MV S V T MV S & a Maxwell relation MS M MV T MT V
Proerties of the Gibbs Energy Same arguments that aly for internal energy, U, are alied to G. When a system has a change of state resulting in )G, this results from changes in H, S and T. Again, for infinitesimal changes: dg dh & TdS & SdT Since H = U + V: dh du % dv % Vd For a closed system doing no non-exansion work, du can be relaced by the fundamental equation, du = T ds - dv dg (TdS & dv) % dv % Vd & TdS & SdT thus, dg Vd & SdT Change in G is roortional to changes in ressure and temerature Because we control and T, G is a very imortant quantity in chemistry. G carries the combined consequences of the 1st and 2nd laws and we do not have to worry about )S univ to determine sontaneity!
Gibbs Energy, Pressure & Temerature From the relationshi below, we can derive two new artial derivatives dg Vd & SdT MG MT &S G decreases when the T is increased at constant, because S is ositive G decreases most sharly when S is large (G of gaseous hase is more sensitive to )T then liquids or solids) G increases when is increased at constant T (since V is ositive) MG M T V
Temerature Deendence of G In many chemical and hysical transformations, the equilibrium comosition of a system deends on the Gibbs energy - so we must know the resonse of the Gibbs energy to temerature changes Since MG we write MG &S G & H MT MT T and we can rearrange to MG MT & G T & H T The LHS can be shown to be the derivative of G/T wrt T: M MT (in 59-241, you will see that we want the Gibbs- Helmholz equation in this form, since the equilibrium constants of reactions are related to G/T rather than just G) G T 1 T 1 T 1 T MG % G d MT dt MG & G MT T 2 MG MT & G T 1 T from which follows the Gibbs-Helmholz equation (which relates G and H): M MT G T & H T 2
Pressure Deendence of G Calculate the )G m of (a) H 2 O (l) treated as an incomressible fluid, and (b) H 2 O treated as a erfect gas, when ressure is increased isothermally from 1.0 bar to 2.0 bar at 298 K Integrate dg Vd & SdT wrt at constant T f G m ( f ) & G m ( i ) V m m d (a) incomressible fluid, V m = constant, (b) erfect gas, V m = RT/ (a) (b) G m ( f ) & G m ( i ) V m d V m m ( f & i ) i f i = (18.0 10-6 m 3 mol -1 ) (1.0 10 5 Pa) = +1.8 J mol -1 G m ( f ) & G m ( i ) m f i RT d RT ln f i = (2.48 kj mol -1 ) ln (2.0) = +1.7 kj mol -1 )G = +ve, though the increase for gas is 1000 times that for liquid
Pressure Deendence of G, Solid & Liquid For liquids and solids )V is very small, and in a lot of cases may be neglected (under lab conditions, V m ) is small!) Hence, we can normally assume that Gibbs energies for solids and liquids are indeendent of ressure - however, for geohysical roblems which involve very high temeratures and ressures, the volume effect on Gibbs energies cannot be ignored! Suose that for a solid hase transition that ) trs V = +1.0 cm 3 mol -1, then at a high ressure of 3.0 Mbar (i.e., 3.0 10 6 bar), Gibbs energy of transition from ) trs G (1 bar) is calculated as: ) trs G (3 Mbar) = ) trs G (1 bar) + )V m ( f - i ) = ) trs G (1 bar) + (1.0 10-6 m 3 mol -1 ) (3.0 10 11 Pa-1.0 10 5 Pa) = ) trs G (1 bar) + (3.0 10 2 kj mol -1 )
Pressure Deendence of G, Gases Molar volumes of gases are very large, so Gibbs energy may be strongly deendent uon the ressure G m ( f ) & G m ( i ) m f i RT d RT ln f i If ressure is increased by 10 at room temerature, then molar Gibbs energy increases by about a factor of 10. As well, if we set = o (standard ressure of 1.0 bar), the molar Gibbs energy of erfect gas at some ressure is related to its standard value by G m () = G o + RT ln m o
Chemical Potential Chemical otential, µ, of a ure substance is defined as µ MG Mn Chemical otential is useful for demonstrating how the Gibbs energy of a system changes as a substance is added to it. For a ure substance, Gibbs energy is G = n G m, so µ MnG m Mn T, G m Here, the chemical otential is the same as the molar Gibbs energy. For examle, chemical otential of a erfect gas is: µ µ o % RT ln We give it a secial name because later we will aly it to mixtures and talk about the chemical otentials of individual comonents. However, remember that the chemical otential of some comound is just its molar Gibbs energy. o T, µ % ln, standard state at o
Chemical Potential µ µ o % RT ln o This equation you will use about a hundred times in the next month, so make sure you understand it. It simly says that we can define a molar Gibbs energy for any substance as just the molar Gibbs energy under standard conditions lus some term that deends on the temerature and the natural log of the relative amount of the substance (exressed as a ressure in the case of a gas or a concentration in the case of a solute in a liquid). For an ideal gas, the chemical otential at some ressure is just the chemical otential at the standard ressure lus the change in entroy associated with changing to the new ressure. In general, contributions to chemical otential are slit into (i) terms that are roerties of the molecules in question and (ii) terms that have to do with changing the total number of states that are available to the molecules.
Real Gases & Fugacity Consider the ressure deendence of the chemical otential of a real gas To adat to the case of a real gas, we relace the true ressure by an effective ressure, f, called the fugacity µ µ o % RT ln f o Fugacity, from Latin for fleetness ; refers to the tendency to escae. Fugacity has the same units as ressure, and is a bit of a fudge factor for treating real gases In hysical chemistry, since many roerties of materials are derived from chemical otentials, fugacities are used to describe ressures As 6 0, µ coincides with a erfect gas. At intermediate, attractive forces dominate (f < ), and at high ressures, reulsion gives f >
Standard States of Real Gases A erfect gas is in standard state when ressure is o (1 bar): ressure arises from kinetic energy, with no interactions taken into account The standard state of a real gas is a hyothetical state in which the gas is at o and behaving erfectly # We choose a hyothetical standard state in order to standardize the interactions between gas articles (for different tyes of gases) by setting all interactions to zero. # We do not choose the standard state as a gas at ressure aroaching zero, since the chemical otential of a gas will aroach negative infinity at this oint: (µ 6-4 as 6 0) # Standard chemical otential of gases, µ o, arises solely from internal molecular structure and roerties of gases, and not from the interactions between articles
Fugacity is written as: f N Fugacity vs. Pressure where N is the dimensionless fugacity coefficient, which deends on the nature of the gas, the temerature and the ressure. Thus, µ µ o % RT ln o % RT ln N where µ o refers to the standard state gas influenced only by kinetic energy The second term describes the erfect gas, and the third term exresses the effect of the molecular forces of a real gas: all gases become erfect as aroaches 0, so f 6 as 6 0, and N 6 1 as 6 0 We can rove that the fugacity coefficient of a gas can be calculated from Z & 1 ln N d m 0 where Z = V/nRT = V m /RT is the comression factor of the gas. Let s do this on the next slide:
For real gases Fugacity and Comression Factor, Z G m () & G m ( ) ) m ) V m d µ & µ ) RT ln f is fugacity at ressure and f & is fugacity at ressure &. Perfect gas: the difference of the equations is ) which is rearranged to: V m erfect,m d µ erfect & µ erfect RT ln ) (V m & V erfect,m )d RT ln ln f ) f ) ) 1 RT If & 6 0, the gas behaves erfectly, and f & 6 & ln f ln N 1 RT m 0 ) f f ) & ln ) ) (V m & V erfect,m )d (V m & V erfect,m )d V erfect, m = RT/, V m = RTZ/, and substituting in gives us roof, q.e.d. f f )
Fugacity and van der Waals Gases To evaluate N, exerimental data on Z is needed from low ressures u to the ressure of interest - sometimes, the fugacity can be obtained from the virial coefficients of a gas: ln N B ) % 1 2 C ) %... The full VdW equations of state are shown in terms of fugacity coefficients as functions of reduced ressure (and labelled as curves at reduced temeratures, T/T c ), roviding estimates of fugacities for a wide range of gases
Fugacity Summary Most gases: Z < 1 at moderate ressures, attractive forces Z > 1 at higher ressures, reulsive forces Z < 1, moderate Z > 1, high ln N m 0 Z & 1 d < 0 ln N m 0 Z & 1 d > 0 f < f > µ < µ o µ > µ o articles stick together escae tendency lessened articles driven aart escae tendency heightened
Fugacity Calculation Suose attractive interactions for ammonia gas can be neglected at 10.00 atm and 298.15 K. Find an exression for fugacity of a van der Waals gas in terms of the ressure. We neglect a from the VdW equation (for attractive forces) and write RT V m & b, Z 1 % b RT Evaluating the integral for ln N: m 0 Z & 1 Then solving for f we have d m 0 f e b/rt b RT d b RT with b = 3.707 10-2 L mol -1, then b/rt = 1.515 10-2, giving f = (10.00 atm) e 0.01515 = 10.2 atm The reulsive term, b, increases the fugacity above the ressure, so the escaing tendency is greater than if it were erfect