Using the Principles of Classical and Statistical Thermodynamics To

Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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Using the Principles of Classical and Statistical Thermodynamics To Calculate the Melting and Boiling Points, Enthalpies and Entropies of Fusion and Vaporization of Water, and the Freezing Point Depression and Boiling Point Elevation of Ideal and Nonideal Aqueous Solutions Arthur M. Halpern*,† and Charles J. Marzzacco‡ Downloaded via UNIV OF LOUISIANA AT LAFAYETTE on October 19, 2018 at 22:21:31 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States Chemistry Department, Florida Institute of Technology, Melbourne, Florida 32901, United States

‡

S Supporting Information *

ABSTRACT: A spreadsheet-based project is presented that is designed to enhance and expand student understanding of phase transition properties of pure water and ideal and nonideal (electrolyte) aqueous solutions. Using fundamental principles of classical and statistical thermodynamics, students calculate the melting and boiling points, the enthalpies and entropies of fusion and vaporization of pure water, and the freezing point depression and boiling point elevation of ideal and nonideal aqueous solutions. The results of the calculations are in good agreement with literature values. Complete details are provided in the Supporting Information.

KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Computer-Based Learning, Phases/Phase Transitions/Diagrams, Thermodynamics

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INTRODUCTION In an earlier publication in this Journal, Castellan used schematic diagrams of the molar free energy curves, μ(T), to illustrate transition temperatures and to rationalize how changes in the applied pressure affected the melting and boiling points of a liquid, and also how the freezing point depression and boiling point elevation resulted from the presence of a solute in an ideal solution.1 In the preceding article, we presented the methods and tools that enable students to calculate the chemical potential, μ, of the solid, liquid, and gas phases of mercury, diiodine, and ammonia (at 1 bar) as a function of temperature.2 Students used these results to construct graphs of μ vs T and to calculate the intersection point of the μgas(T) and μliq(T) curves, enabling them to obtain the standard boiling point Tsbp. From this information, they also calculated the molar enthalpy and entropy of vaporization, ΔvapHm and ΔvapSm. Similarly, they found Tsmp, ΔfusHm, and ΔfusSm from the intersection of μliq(T) and μsolid(T) curves. In this article, we extend and expand that approach to provide students with the opportunity to conduct a more in-depth study of water that includes calculating the boiling point at higher pressure and the boiling and melting points of an ideal and a nonideal (electrolyte) solution. © XXXX American Chemical Society and Division of Chemical Education, Inc.

The work described here, readily carried out using a scientific spreadsheet application, such as Microsoft Excel, not only reinforces the fundamental thermodynamic principles of phase equilibrium but also helps students strengthen their understanding of solution theory. This study enables them to become directly engaged in the theory and application of these core topics. To assess their work, they can compare their calculated results with the corresponding literature values. The chemical potential of gaseous, liquid, and solid water must be calculated over a temperature range that encompasses the melting and boiling points. For this study, T spans between 240 and 410 K in 1 K increments. The basis of these calculations is the fundamental Gibbs equation, which, for phase α, is3 dμα = −Sm,α dT + Vm,α dP

(1)

where Sm,α and Vm,α are the molar entropy and molar volume of water in phase α (solid, liquid, or gas). For a constant-pressure process, eq 1 reads dμα = −Sm,α dT

(2)

Received: July 15, 2018 Revised: August 30, 2018

A

DOI: 10.1021/acs.jchemed.8b00561 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

Article

A first objective, therefore, is to determine Sm,α over the selected temperature range. For the gas phase, Sm,α can be calculated directly using statistical thermodynamics, as will be demonstrated below. For liquid and solid water, the molar entropies are calculated from the respective isobaric heat capacities, CP,m, using the relationship dSm =

CP,m T

dT

value of 188.834 J/(K mol).8 They will be encouraged by the good agreement. The chemical potential of the gas is calculated using the integrated form of eq 2 ° − μgas (T ) = μgas

and the subsequent application of eq 2. Calculation of Sm,gas

The translational, rotational, and vibrational contributions to the entropy of gaseous water are obtained as follows. [Note that the electronic entropy contribution is neglected since only the (singlet) ground state of water is populated at the temperatures in this study.] The translational entropy is calculated from the Sackur−Tetrode equation4,5

μn = μn + 1 − Sm, n(Tn − Tn + 1) for T < T ° μn = μn − 1 − Sm,n(Tn − Tn − 1) for T > T °

in which the units of M and P are g/mol and bar. Water is a nonlinear polyatomic molecule, and therefore, it possesses three principal moments of inertia. The rotational entropy, Sm,rot, is obtained using eq 56 ÄÅ ÉÑ| l 1/2 Ñ ÅÅ i o o o ÅÅ 1 jj πT 3 yzz ÑÑÑo o o Sm,rot = R m 1.5 + lnÅÅÅ jj zz ÑÑÑ} o o j z Å Ñ o σ Θ Θ Θ ÅÅ k A B C { ÑÑo o o (5) ÅÇ ÑÖ~ n

Scheme 1. Thermochemical Steps Used To Calculate μgas ° of Water Vapor

The symmetry number σ = 2 because the water molecule has a 2-fold rotational axis of symmetry. The three characteristic rotational temperatures, ΘA, ΘB, and ΘC, are obtained from the rotational constants A0, B0, and C0 (cm−1 units) using the conversion factor hc/k (1.43761 K/cm−1). The constants A0, B0, and C0 are available from literature sources, or they can be calculated, if desired, using an ab initio quantum chemistry application. The water molecule has 3N − 6, or three (nondegenerate), vibrational modes, and the vibrational entropy, Svib,m, is obtained from the sum of the contributions of each mode

and 11 to obtain μgas(T) is demonstrated in the Water.xlsx workbook. The need for two algorithms, eqs 10 and 11, to obtain the μ values is explained in the Supporting Information. Calculation of μgas °

Under standard conditions (298.15 K and 1 bar), the liquid is the lowest free energy phase of water, and thus, μ°liq is set to equal 0. To obtain μgas ° the following three isothermal steps are used. The molar free energy change of the overall process is equal to μ°gas and is equal to the sum of the Δμ values of the three steps. Pvap is the vapor pressure of water at 298.15 K, which is 3.1698 × 10−2 bar.9 Step 1 describes the isothermal reduction of pressure above liquid water from 1 bar to 3.1698 × 10−2 bar. The molar volume of water, Vm, at 298.15 K is 1.8064 × 10−5 m3 mol−1 and may be regarded as independent of pressure.10 Thus,

3

(6)

i=1

where the vibrational entropy of the ith mode is ÄÅ ÉÑ ÅÅ Θvib, i Ñ 1 −Θvib,i / T Ñ ÑÑ Sm,vib, i = RÅÅÅÅ − ln(1 − e ) ÑÑ ÅÅÇ T e Θvib,i / T − 1 ÑÑÖ 7

Pvap

Δμ1 = ∫ Vm,liq dP = (1.8064 × 10−5 m3 mol−1)(3.1698 × 1 10−2 bar − 1 bar)(105 Pa bar−1) = −1.75 J mol−1. Step 2 expresses the vaporization of liquid water under equilibrium conditions, and hence, Δμ2 = 0. Step 3 represents the isothermal compression of gaseous water from 3.1698 × 10−2 bar to 1 bar. Since the gas is assumed

(7)

The vibrational frequencies are also available from the literature, or they may be obtained from a quantum chemical calculation. They are converted to the respective characteristic vibrational temperatures Θvib,i using the factor 1.43761 K/cm−1. The rotational constants and vibrational frequencies and their use in calculating the entropy contributions are available in the Excel workbook (Water.xlsx) available in the Supporting Information (SI). The total entropy of the gas is the sum of the three components Sm,tot = Sm,tr + Sm,rot + Sm,vib

(11)

T° is the standard temperature (298.15 K) at which μgas = μgas °. In eqs 10 and 11, the index n denotes a spreadsheet row number and a specific temperature. The determination of μgas ° is described below in Scheme 1. The implementation of eqs 10

(4)

∑ Svib,i

(10)

and

Sm,tr = R[1.5 ln(M /g mol−1) + 2.5 ln(T /K) − ln(P /bar)

Sm,vib =

(9)

where μ° is the chemical potential of gas phase water at 298.15 K and 1 bar. Because Sm,gas values vary slowly between one T value and the adjacent ones (increments of 1 K), Sm,gas in eq 9 can be regarded as constant between adjacent temperatures, and thus, μgas(T) can be calculated using the algorithm

(3)

− 1.15169]

T

∫T °=298.15 Sm,gas dT

1

to be ideal, Δμ3 = ∫ Vm,gas dP = RT ln (1 bar/3.1698 × 10−2 P vap

bar) = 8,556.13 J mol−1. Thus, μ°gas is the sum of Δμ1 to Δμ3 and is 8,554.38 J mol−1. Calculation of μliq(T)

(8)

Guided by eq 2, students will calculate μliq by first obtaining values of Sm,liq between 240 and 410 K. From Sm,liq(T), they then use the expression analogous to eq 9 and the algorithm, eqs 10

At this point, students can compare their calculation of Sm,tot at 298.15 K from eqs 4−8 [188.719 J/(K mol)] with the literature B



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and 11, to obtain μliq(T). Sm,liq(T) is found using the integrated form of eq 3 Sm(T ) = Sm(T •) +

∫T

T

CP ,m

•

T

dT

the 25 K temperature range used. Note that the overall process is isothermal. For steps 1 and 3, ΔHm values are calculated from ΔHm = Cp,m(T2 − T1), where CP,m for the appropriate phase is used and T1 and T2 are the initial and final temperatures of the step. The ΔSm values are found from ΔSm = CP,m ln(T2/T1). For step 2, ΔHm is simply −ΔfusHm, and ΔSm = −ΔfusH/273.15. The results of each step are displayed in Table 1.

(12)

•

where T is the reference temperature (298.15 K) at which the entropy of liquid water is known. For liquid water, CP,m between 240 and 410.15 K (a range that includes the supercooled and superheated liquid) is obtained from two sources11,12 and splined into an array of equal 1 K increments (see Sheet2 of the Water workbook in the Supporting Information). Sm,liq(T) is then calculated using eq 12 and the algorithms ji T zy Sn = Sn + 1 + CP ,m lnjjj n zzz for T < T • j Tn + 1 z k {

(13)

ji T zy Sn = Sn − 1 + CP ,m lnjjj n zzz for T > T • j Tn − 1 z k {

(14)

Table 1. Values of ΔH and ΔS for Steps 1−3 and the Overall Values of ΔHm and ΔSm for H2Oliq(298.15 K) → H2Osolid(298.15 K) at P = 1 bar

and

Process

ΔHm/(J mol−1)

ΔSm/(J K−1 mol−1)

Step 1 Step 2 Step 3 Overall

−1,882.80 −6,009 922.56 −6,969.24

−6.5955 −22.00 3.2317 −25.36

Using the overall values of ΔHm and ΔSm for the three process and the Gibbs equation Δμ = ΔHm − T ΔSm, one finds Δμ = μ°solid = (−6969.24) − (298.15)(−25.46) = 591.84 J mol−1. These values of μsolid ° and μgas ° are used to complete the acquisition of the data needed for the μ(T) curves of the three phases of water. They are illustrated in Figure 1.

These expressions provide good approximations of Sn because CP,m varies slowly with T and the increment in T is small (1 K). The need for the two expressions, eqs 13 and 14, and the details of their implementation are provided in the Supporting Information. Having calculated Sm,liq (T), students now proceed to obtain μliq(T) using the same method described for gas phase water and employing the analogous expressions in eqs 10 and 11. As was pointed out above, μ°liq = 0. To complete this part of the project, students find μsolid(T) utilizing the same tools they employed to calculate μliq(T). For this purpose, they will need CP,m values for the solid between 240 and 410 K and the entropy at a particular temperature. This information is provided in the Water workbook in the Supporting Information. They must also obtain a value of μsolid ° to quantitatively offset the μsolid(T) curve relative to the μliq(T) and μgas(T) curves. Calculation of μsolid °

This quantity is obtained using a series of steps that includes the equilibrium between the solid and liquid phases. These steps, all at P = 1 bar, are presented in Scheme 2.

Figure 1. Plot of μgas(T), μliq(T), and μsolid(T) for water at 1 bar. The transition temperatures are indicated by the arrows and calculated to be 272.89 and 373.29 K, respectively (see the Water.xlsx workbook in the Supporting Information for details). These results are in reasonable agreement with the respective accepted values of 273.15 and 372.75 K.

Scheme 2. Thermochemical Steps Used To Calculate μ°solid For Water

Calculation of the Standard Freezing and Boiling Points and the Enthalpies and Entropies of Fusion and Vaporization

Students can obtain approximate values of Tsmp and Tsbp from an inspection of their μ(T) plots. With the mouse curser, they probe the T values at the intersection of the curves corresponding to the two particular phases, e.g., the μliq(T) and μgas(T) curves, to find Tsbp. They can expand the chart area containing the crossing point in order to improve the sensitivity of this graphical analysis. To obtain quantitative results, students use the spreadsheet entries containing the μ(T) values of the phases. For example, to find Tsmp, they calculate in the adjacent column μliq(T) − μsolid(T), i.e., Δμ, for temperatures a few kelvins below, and above, the approximate value of Tsmp. These values of Δμ will change from positive to negative. About 4−6 Δμ values are

Because steps 1 and 3 are not isothermal, one cannot calculate their Δμ values; instead, it is necessary to determine ΔHm and ΔSm for each step. For this calculation, one needs the isobaric heat capacities of the liquid and solid and the enthalpy of fusion at 273.15 K. These values are CP,m,liq(298.15 K) = 75.312 J mol−1 K−1, CP,m,solid (273.15 K) = 36.902 J mol−1 K−1, and ΔfusH (273.15 K) = 6,009 J mol−1.11−14 For the purpose of this calculation, the CP,m values are considered to be constant over C



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ln 2 RT. The values of μ°gas and μ°liq at a given temperature can be adjusted to 2 bar by applying eq 1

selected that range between similar positive and negative values. Linear regression is is performed on these Δμ,T values to find the temperature for which Δμ = 0 (i.e., the phase equilibrium condition). The regression equation of Δμ vs T is Δμ ≡ μ liq − μsolid = mT + b

dμα = Vm, α dP

which, after integration between 1 and 2 bar, becomes

(15)

μα (298.15 K, 2 bars) = μα° +

where m and b are the slope and intercept. From eq 1, one can see that m = −ΔfusSm

2

Vm, α dP

(20)

For the liquid phase, Vm is constant (1.8064 × 10 m mol−1), and thus, μliq(298.15, 2 bar) = μ°liq + (1.8064 × 10−5 m3 mol−1)(2 bar − 1 bar)(105 Pa/bar) = 0 + 1.8 J/mol. For the gas phase, Vm = RT/P, and μgas(298.15, 2 bar) = μ°gas + RT ln (2 bar/1 bar) = 8,554.38 J/mol + 1,718.28 J/mol = 10,272.66 J/ mol. The calculations used to obtain the μgas(T) and μliq(T) curves at 2 bar are contained in Sheet5 of the Water workbook. These plots are shown in Figure 2 along with μgas(T) for P = 1 bar. The

and using eq 15, one obtains (17)

Students can then readily obtain ΔfusHm at Tsmp by realizing that, at this temperature, Δfusμ = 0 and therefore ΔfusHm = (Tsmp)(ΔfusSm). Thus ΔfusHm = ( −b/m)( −m) = b

∫1

−5

(16)

Tsmp = −b/m

(19)

3

(18)

In a similar fashion, by analyzing the μliq(T) and μgas(T) data in the vicinity of Tsbp, they can determine values of Tsbp, ΔvapSm, and ΔvapHm. Thus, from their μ(T) data they can obtain the three fundamental thermochemical properties of phase transition. Each calculated quantity can be compared with the values reported in the literature. The use of this technique to determine the transition temperature entropies and enthalpies is illustrated in the Water workbook in the Supporting Information. The results of this study are displayed in Table 2 along with the respective literature values. Table 2. Standard Melting and Boiling Points and Transition Enthalpies and Entropies Calcd Lit.

Tsmp/K

ΔfusHma

ΔfusSmb

Tsbp/K

ΔvapHma

ΔvapSmb

272.89c 273.15

5.941d 6.007f

21.77e 22.00f

373.30c 372.78g

40.80d 40.67g

109.29e 109.10g

Figure 2. Graphs of μliq and μgas vs T for water. Gas phase at 1 bar (red ) and 2 bar (red ---), and liquid phase (blue ). The difference between the liquid at 1 bar and at 2 bar is negligible.

a

kJ/mol. bJ/(K mol). cEquation 17. dEquation 18. eEquation 16. f Reference 14. gReference 15.

boiling points are indicated; their calculated values are 373.30 and 394.30 K. The boiling point is thus elevated by ca. 21 K and shows that if, for example, a rate constant approximately doubles for a 10 K increase in temperature, foods prepared in a pressure cooker may require about 1/4 the usual cooking time.

An alternative method is to perform a regression analysis on the function Δμ = m(T − Tsmp) in which Tsmp is a parameter. In this case the approximate value of Tsmp is used as an initial guess. This approach has the advantage of avoiding the large correlation between m and b caused by the large extrapolation of the data to the y-intercept. Students will find that the optimized value of Tsmp as well as the values of ΔfusSm (from −m) and ΔfusHm (from −mTsmp) are identical to those obtained using eqs 16−18. Further details are given in Sheet4 of the Water workbook in the Supporting Information. The uncertainties in the calculated quantities are discussed at the end of this article.

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FREEZING POINT DEPRESSION AND BOILING POINT ELEVATION OF AQUEOUS SOLUTIONS

Ideal Solution

Students can use their μ(T) data and calculations to explore the properties of an ideal solution by finding Tsmp and Tsbp for such a solution. To accomplish this, they will employ the fundamental equation that expresses the chemical potential of the solvent (water) in an ideal solution in terms of the temperature and solution composition, i.e.16

Tsbp at P = 2 bar

To extend their study, students can observe the effect of an increase in pressure on the boiling point of water. They should realize that, in the statistical thermodynamic calculation of Sgas, only the translational term (eq 4) is pressure dependent. Since μ°gas and μ°liq pertain to a pressure of 1 bar, those values must also be adjusted to the higher pressure. As an example, students can determine the boiling point of water in a typical homekitchen pressure cooker, which can achieve a pressure of about 1 bar above ambient pressure, or an absolute pressure of 2 bar. First, Strans is readily recalculated using P = 2 bar in eq 4, resulting in a uniform decrease in Strans of R ln 2 and an increase in μgas of

μw = μw* + RT ln x w

(21)

where μw* is the chemical potential of pure liquid water at temperature T; xw is the mole fraction of water. To find Tsmp and Tsbp for the ideal solution, the chemical potential of liquid phase water is now expressed using eq 21 as μw (T ) = μ liq (T ) + RT ln x w D

(22) DOI: 10.1021/acs.jchemed.8b00561 J. Chem. Educ. XXXX, XXX, XXX−XXX


Article

where μliq(T), which the students have already calculated, replaces μ*w . Thus, the quantity RT ln xw is added to each value of the μliq(T) array. The composition terms in eqs 21 and 22 are temperature independent. It is assumed that the presence of the solute has no effect on the chemical potentials of the solid and gas phases. Values of Tsmp and Tsbp are calculated as described previously (eqs 15 and 17). Students can choose a value of xw for these calculations, which are shown in the water workbook in the Supporting Information. If they prepare their spreadsheet so that the relevant calculations are dynamically linked, they can readily change the value of xw and immediately observe the changes in Tsmp and Tsbp. To demonstrate the application of eq 22, the freezing point depression, T*smp − Tsmp (ΔTf), and the boiling point elevation, Tsbp − Tsbp * (ΔTb), are calculated for an aqueous solution with 1 mol % impurity, i.e., xw = 0.99. This composition corresponds to a molality, m, of 0.5607.17 The results are shown in Table 3 and displayed in Figure 3.

Table 4. Values of kf and kb

a

T*smp

T smp

ΔTf

T*sbp

Tsbp

ΔTb

272.885

271.840

1.045

373.295

373.581

0.286

Table 3

Equation 23b

Literaturec

kf kb

1.864 0.510

1.871 0.511

1.86 0.512

K kg/mol. bUsing results from this work. cReference 19.

remarkable and indicates the validity and the internal consistency of the methods students used to obtain them. To further test the robustness of the calculations, students can obtain ΔTf for a 0.3241 m solution and compare their result with the experimental value reported for a 0.3241 m aqueous of urea.20 The calculated value of ΔTf is 0.6069 K, which compares well with the experimental result of 0.5953 K.20 The corresponding kf values are 1.873 and 1.837 K kg/mol. This outcome supports the soundness of the calculations and implies that the urea solution at this molality exhibits nearly ideal behavior. Nonideal Solution

Table 3. Freezing and Boiling Points of a 0.01 Mole Fraction (0.5607 Molal) Solute in an Ideal Aqueous Solutiona Value

Constanta

The chemical potential of water in a nonideal solution is21 μw (T ) = μw*(T ) + RT ln a w

a

where aw is the activity of water in the solution. As an example of a nonideal solution, we use a 0.10 m aqueous solution of an electrolyte, MX, which completely dissociates into unassociated M+ and X− ions. ΔTf and ΔTb are calculated from the transition temperatures, as before, but students must now take into account the activity coefficient of the solute, MX. The Debye−Hückel (DH) equation, which students have likely encountered before, provides a means for expressing the activity coefficient of the solute, γMX, viz.22

This work. Values in K.

ln γMX = −

In order to assess the accuracy of the results, the cryoscopic and ebullioscopic constants, kf and kb, are calculated from the ΔTf and ΔTb values, respectively. These constants are defined as kf = ΔTf/m and kb = ΔTb/m. From ideal solution theory, kf and kb are expressed approximately as18 kf ≡

ΔfusHm

and

kb ≡

d ln γMX dm

yz zz z {

(25)

ij −1 yzji d ln a w zy 1 zzj z− = jjj j M w υm zzjk dm z{ m k {

(26)

Mw is the molar mass of water (kg/mol), and υ is the stoichiometric number of MX, i.e., 2. Equation 26, which is an application of the Gibbs−Duhem equation, is rearranged to provide the needed expression ÄÅ ÉÑ ÅÅ i d ln γ y ÑÑ d ln a w j MX z Å Ñ j z Å = −BÅÅmjj zz + 1ÑÑÑ ÅÅ k dm { ÑÑ dm (27) Ç Ö

* )2 MR(Tsbp Δ vapHm

A ij m j j 3/2 j T k1 + m

in which A is 6036.78 K3/2 and m is the MX molality, which has here been divided by m°, the standard molality. In eq 25, and the ones following, m is a dimensionless quantity. It should be noted that, in the theoretical treatment leading to the DH equation, the solution composition is expressed in terms of the ionic strength I. For the electrolyte MX, however, I is equal to m. The challenge now is to express the activity of water (solvent) in terms of the activity coefficient of MX (solute). Fortunately, such an expression is available in a textbook by Raff23 and is adapted here as

Figure 3. Graphs of μliq and μsolid vs T for water. Solid (black ), pure liquid (blue ), and ideal solution with 1.0 mol percent solute (yellow ) at 1 bar. The locations of the normal and depressed melting points are indicated. The values are 272.885 and 271.837 K, respectively.

* )2 MR(Tsmp

(24)

(23)

M is the molar mass of water in kg/mol. Table 4 presents values of kf and kb, obtained from the data in Table 3, calculated from eq 23 in which the transition temperatures and enthalpies are those from this study, as well as the literature values. The agreement among the values obtained from ΔTf and ΔTb, those calculated from eq 26, and the literature values is

in which B = Mwυ = 0.036030 kg/mol. To apply eq 27 we first obtain d ln γMX/dm, which, from eq 25, is E


Journal of Chemical Education ÄÅ d ln γMX A ÅÅÅ 1 = − 3/2 ÅÅÅ Å dm T ÅÇ 2 m (1 +

ÉÑ ÑÑ 1 ÑÑ − Ñ m) 2(1 + m )2 ÑÑÑÖ

Article

After completing these exercises, students should be encouraged to compile a list of the information they needed to obtain the results, the origin of each, and the purpos(es) for which it is used. Such a table is presented in the Supporting Information.

(28)

and substitute it into eq 27. After rearranging and expanding the derivative, one finds ÉÑ ÄÅ ÄÅ ÉÑ AB ÅÅÅ m dm ÑÑÑ AB ÅÅÅ m dm ÑÑÑ Ñ Å Å Ñ d ln a w = Å Å Ñ− 2Ñ 2T 3/2 ÅÅÅÇ (1 + m ) ÑÑÑÖ 2T 3/2 ÅÅÅÇ (1 + m ) ÑÑÑÖ

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UNCERTAINTIES IN CALCULATED RESULTS Because the calculated results are based on a variety of externally obtained data, such as molecular constants, vapor pressure, density, and heat capacities, as well as on a number of computational simplifications, such as the algorithms used to calculate the entropies and the chemical potentials, a complete uncertainty analysis in the results is not practical. One can, however, calculate the uncertainties in the enthalpy and entropy of fusion and vaporization and the transition temperatures since they are obtained from plots of Δμ vs T (eqs 15−18). These calculations are presented in Sheet4 and Sheet6 of the Water workbook (Supporting Information). Examples of these uncertainties are displayed in Table 6 for pure water and for a 1.0 mol % ideal solution. The uncertainties of the transition temperatures for the other solutions studied are similar to those in Table 6, e.g., 0.003 for the freezing point and 0.0004 for the boiling point. For the freezing point depression and boiling point elevation, the uncertainties are somewhat larger, ca. 0.004 and 0.0005 K, respectively (see Sheet7 and Sheet8 in the Water workbook in Supporting Information). Students will be impressed by the accuracy and precision of the results obtained in this study, but they should be reminded that these uncertainties apply to the results obtained using eqs 18−21 and not to the overall calculation.

(29)

− B dm

Next, eq 29 is integrated between the limits m = 0 and m = m. The corresponding limits of ln aw are 0 (when m = 0, aw = 1) and ln aw. Equation 29 becomes ln a w =

m AB m dm 3/2 0 (1 + m ) 2T m AB m dm − −B 3/2 0 (1 + m )2 2T

∫

∫

∫0

m

dm

(30)

The first two integrals on the RHS of eq 30 may present a challenge to students. These integrals are not trivial but may be solved using techniques usually presented in undergraduate calculus, e.g., substitution, integration by parts, and using standard forms. The details of evaluating these integrals are given in the Water documentation in the Supporting Information. The integration of eq 30 yields the needed expression for ln aw as a function of m, which is ÄÅ ÉÑ | l o o A ÅÅÅij m + 2 m yz ÑÑÑ o o j z Å Ñ ln a w = Bm − 2 ln(1 + m ) − m } j z Å Ñ o o j z 3/2 Å Ñ o T ÅÅÇk 1 + m { o Ñ ÑÖ n ~

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(31)

CONCLUSIONS By applying the fundamental principles of classical and statistical thermodynamics, students will learn how to calculate the freezing and boiling points of water, in addition to the enthalpy and entropy of the phase transitions. In addition, they determine the freezing and boiling point changes for ideal and nonideal solutions. These activities, along with the uniformly good agreement with experimental results, reinforce their understanding (and appreciation) of chemical thermodynamics.

Equation 31 is used with eq 24 to calculate μliq(T) by adding RT ln (aw) to the previously calculated values for pure water (μw* in eq 24). Using due care, students can successfully code the expression in eq 31 into their spreadsheet to complete this calculation to find Tsmp and Tsbp of an MX solution. Finally, ΔTf and ΔTb are calculated for the 0.10 m solution of MX. These results are listed in Table 5, along with experimental data for a 0.10 m aqueous solution of NaCl.24,25

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Table 5. Freezing Point Depression and Boiling Point Elevation Calculated for a 0.10 m Solution of MX and Corresponding Experimental Results for a NaCl Solution Solute

ΔTf/K

ΔTb/K

MX (calcd)a NaCl (exptl)

0.3438 0.3470b

0.0964 0.0946c

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00561. Details and background information for the calculations (PDF, DOCX) Excel workbook containing multiple spreadsheets of the calculations (ZIP)

a

Using eqs 24 and 31 and the methods described earlier. bReference 24. cReference 25.

■

The close agreement of the calculated and experimental values is very gratifying. Students should feel amply rewarded and justifiably proud.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Table 6. Values of the Phase Transition Properties of Pure Water and a 1.0 mol % Ideal Solution and the Respective Uncertainties Value Uncertainty

Tsmp * /K

ΔfusHma

ΔfusSmb

Tsbp * /K

ΔvapHma

ΔvapSmb

Tsmp/K

Tsbp/K

272.885 0.0032

5.94 0.013

21.77 0.049

373.297 0.0004

40.80 0.013

109.29 0.036

271.837 0.0032

373.581 0.0004

a

kJ/mol. bJ/(K mol). F



Article

ORCID

(24) Scatchard, G.; Prentiss, S. S. The Freezing Points of Aqueous Solutions. IV. Potassium, Sodium and Lithium Chlorides and Bromides. J. Am. Chem. Soc. 1933, 55, 4355−4362. (25) Smith, R. P. The Boiling Point Elevation. II. Sodium Chloride 0.05 to 1.0 M and 60° to 100°. J. Am. Chem. Soc. 1939, 61, 500−503.

Arthur M. Halpern: 0000-0002-2211-2826 Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The authors acknowledge J. Tellinghuisen for helpful discussions. REFERENCES

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Using the Principles of Classical and Statistical Thermodynamics To

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