Constructing the Phase Diagram of a Single-Component System

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

pubs.acs.org/jchemeduc

Constructing the Phase Diagram of a Single-Component System Using Fundamental Principles of Thermodynamics and Statistical Mechanics: A Spreadsheet-Based Learning Experience for Students Arthur M. Halpern*,† and Charles J. Marzzacco‡ †

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

‡

J. Chem. Educ. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/17/18. For personal use only.

S Supporting Information *

ABSTRACT: A spreadsheet project that enables students to construct chemical potential vs temperature plots, μ(T), of liquid and gaseous mercury and all three phases of diiodine and ammonia is presented. Statistical thermodynamics is used to determine the entropies of the substances in the gas phase. For the solid and liquid phases, students start with published heat capacities, third law entropies, and other thermodynamic quantities to obtain entropies over the range of temperatures needed. By using thermodynamic cycles, they calculate the chemical potentials of the metastable phases at standard conditions. With these values as starting points, students use the Gibbs equation and stepwise integration to obtain the chemical potentials over a range of temperatures. Using this data, they calculate the melting and boiling points and transition enthalpies and entropies from the intersection points of the two phases. The calculated thermochemical quantities are in excellent agreement with literature values. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Inorganic Chemistry, Computer-Based Learning, Phases/Phase Transitions/Diagrams, Thermodynamics

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article by Castellan in this Journal.1 That article, and others that followed, nicely illustrates how one could apply the fundamental Gibbs equation

INTRODUCTION Instructors in all disciplinary areas recognize the value and effectiveness of symbols and images to help introduce students to new subject matter or ideas and to engage them in the challenges (and rewards) of learning and applying those concepts. Chemistry instructors benefit from a surfeit of such tools as instructional aids, one of the most prominent being the periodic table, which is both a symbol of the chemical sciences and a template for understanding the properties and electronic structure of the elements. Another example is the use by many instructors of physical chemistry courses of schematic chemical potential vs temperature, μ(T), diagrams to introduce students to the fundamental ideas and applications of phase equilibrium, an important gateway topic to further studies, including materials science. The doorway to the thermodynamics of phase equilibrium usually opens with the presentation, interpretation, and application of the chemical potential to material equilibrium. Subsequently, it is often used to introduce phase equilibrium in one-component systems and the derivation, use, and significance of the phase rule. The qualitative portrayal of μ(T) plots of the solid, liquid, and gas phases of a compound that so well captures the fundamental principles of phase equilibrium perhaps first appeared in the chemistry education literature in 1955 in an © XXXX American Chemical Society and Division of Chemical Education, Inc.

dμ = −Sm dT + Vm dP

(1)

(where Sm and Vm are the molar entropy and volume of the substance) and the partial derivatives ij ∂μ yz i ∂μ y jj zz = −Sm and jjj zzz = Vm ∂ T k {P k ∂P {T

(2)

to interpret and understand this simple diagram. These equations help the student recognize which phase is stable in a given temperature range, visualize phase transition temperatures Tmp and Tbp, observe how the presence of a solute changes their values, and grasp the concepts of metastable phases. Castellan subsequently used these diagrams in his physical chemistry textbook.2 The usefulness of these simple and effective portraits is indicated by the fact that many physical chemistry textbook authors use μ(T) plots in the initial, qualitative coverage of phase equilibrium.3−10 Several publications have appeared in this Journal that advantageously Received: July 15, 2018 Revised: August 30, 2018

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DOI: 10.1021/acs.jchemed.8b00560 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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for translational and internal degrees of freedom. The equations needed for this task are presented in physical chemistry texts.15,16 With guidance from the instructor, students can readily employ this information to obtain Sm,gas(T) values. This part of the exercise is a good opportunity to introduce students to the principles and applications of statistical mechanics, if they have not yet studied this topic. The total molar entropy of the gas is the sum of the components terms, viz.

utilize chemical potential schematic diagrams to illustrate the properties of phase diagrams of systems such as alloys and liquid crystals.11−13 Because these qualitative μ(T) plots are so useful in aiding students’ initial understanding of phase equilibrium, we were motivated to prepare this article to provide resources for an enhanced learning experience: the numerical calculation of μ(T) curves of the solid, liquid, and gas phases of a substance using the fundamental principles of thermodynamics and statistical mechanics. From these quantitative μ(T) results, students can determine the standard melting and boiling points, Tsmp and Tsbp, as well as the molar transition enthalpies and entropies, e.g., ΔfusHm and ΔfusSm of the substances studied. These learning opportunities are readily accessible because they use a spreadsheet application such as Microsoft Excel to perform the calculations, display the μ(T) graphs, and compute these thermochemical quantities. This article presents a detailed approach to performing the calculations, and presents three examples of their application. These examples progress in complexity, from monatomic (mercury) to diatomic (diiodine) to polyatomic (ammonia). An additional example (dibromine), and all spreadsheets, are available in the Supporting Information (SI) along with detailed guides to performing the calculations. In a companion article, we present a detailed study of water, which, in addition to the thermodynamics of phase transitions of pure water, gives students the opportunity to calculate the boiling point at higher pressure and the boiling point elevation and freezing point depression of ideal and nonideal solutions.14 Ideally, students should have had some exposure to quantum chemistry, at least energy quantization (particle in a box, and rotational and vibrational states), and elementary statistical mechanics. However, the calculations can nonetheless be a valuable learning tool for students who have not had formal or detailed instruction in these topics if the instructor revisits the general concepts of energy quantization, usually covered in first year chemistry, and discusses with them entropy in the context of the third law of thermodynamics and the important idea of the sum over states.

Sm,gas = Sm,tr + Sm,rot + Sm,vib + Sm,el

Here, the subscripts denote translational, rotational, vibrational, and electronic contributions. In the examples presented here, the electronic entropy contribution is zero (the partition function is 1) because, for Hg, I2, and NH3, one need consider only the electronic ground (singlet) state of the species to be populated at the temperatures involved. Translational Entropy

The largest contribution to Sm,gas is from translational motion and is calculated using the Sackur−Tetrode equation, which is based on the statistical mechanical treatment of the number of quantum states available to noninteracting particles in a box. This equation, adapted for convenient use in these calculations, is17 Sm,tr = R[1.5ln(M ) + 2.5ln(T / K ) − ln(P /bar) − 1.15169] (6) −1

where M is the molar mass in g mol and bar.

The rotational and vibrational entropy terms in eq 5 (for diatomic and polyatomic molecules) depend on the molecular constants of the molecule. These values are provided in the spreadsheets for each example used in this article. In the case of a diatomic molecule (other than light atom species such as H2, HD, and D2), the rotational contribution to the entropy can be accurately calculated from eq 7 (in the rigid rotor approximation)18 ÄÅ É ij T yzÑÑÑÑ ÅÅÅ zzÑ Sm,rot = RÅÅÅ1 + lnjjj j σ Θrot zzÑÑÑÑ ÅÅ (7) k {ÑÖ ÅÇ

COMPUTATIONAL OVERVIEW The construction of the μ(T) diagrams, for P = P° = 1 bar, is based on the left-hand partial derivative of eq 2, which, after rearrangement and integration, reads

where Θrot is the characteristic rotational temperature (K), which is related to the rotational constant B̃ (cm−1), as Θrot = (hc/k)B̃ = 1.4388B̃ , and σ is the symmetry number (σ = 2 for a homonuclear diatomic molecule and 3 for NH3, which has a 3fold axis of symmetry). A nonlinear polyatomic molecule has three principal moments of inertia and hence three rotational constants and the corresponding characteristic rotational temperatures. The rotational entropy is obtained from the equation19 ÄÅ ÉÑ l o o ÅÅÅ 1 ji πT 3 zy1/2 ÑÑÑ| o o o ÅÅ jj zz ÑÑÑo Sm,rot = R m 1.5 ln + Å j z ÅÅ σ j Θ Θ Θ z ÑÑ} o o o ÅÅ k A B C { ÑÑo o o (8) ÅÇ ÑÖ~ n

T

μ

(3)

and μ(T ) = μ° −

T

∫T ° Sm(T ) dT

and T and P are in K

Rotational Entropy

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∫μ° dμ = μ(T ) − μ° = −∫T ° Sm(T ) dT

(5)

(4)

In eqs 3 and 4, the lower limit T° denotes the standard temperature, 298.15 K, and μ° is the corresponding chemical potential. The value of μ° is set to zero for the stable (lowest energy) phase of the substance at 298.15 K and 1 bar, i.e., liquid for Hg, solid for I2, and gas for NH3. The values of μ° of the unstable phases are determined using methods described later in this article.

where ΘA, ΘB, and ΘC are the characteristic rotational temperatures. The use of these equations is illustrated in the examples below.

Calculation of Entropies

Vibrational Entropy

The Gas Phase. For the gas phase, S(T) can be calculated using statistical mechanical principles based on the ideal gas model and the expressions of the molecular partition functions

Students will readily anticipate that the vibrational entropy of a molecule depends on its vibrational frequency, or frequencies B



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ji T zy Sn = Sn − 1 + C Pn lnjjj n zzz for T > T • j Tn − 1 z (14) k { The index n represents a spreadsheet row number. The two propagation algorithms are needed because the absolute entropy at a specific temperature, T•, is needed, along with the corresponding CP,m value, to obtain the entropy at the other temperatures used in the calculation. Thus, since the cells are configured in order of increasing temperature, to find entropy values below T•, the integral in eq 12 is evaluated from the higher T cell to the lower T cell (eq 13). Likewise, to find Sm for a temperature above T•, one integrates from the lower T cell to the higher T cell (eq 14). Equations 13 and 14 are easily implemented using Excel and are good approximations of the entropy values at each temperature because CP,m varies slowly with T and also because of the small temperature increments used in the calculations (i.e., 1 K). The implementation of these algorithms is shown in the Supporting Information workbooks. Calculation of the Chemical Potentials The final step in calculating the μ(T) plots consists of applying eq 4 to obtain the chemical potential at a given T from the corresponding entropy. As indicated earlier, μ° is assigned a value of zero for the stable (lowest free energy) phase at 298.15 K. The method used to calculate μ values for the higher energy phase(s) is presented below. The integration of eq 4 may be carried out as the stepwise integration illustrated above for Sm. In the array of temperatures chosen to construct a μ(T) diagram, i.e., T1 to TN, the temperature corresponding to 298.15 K is denoted as T° (with corresponding value μ°), and the algorithm used to calculate μT is

for polyatomic molecules. In the latter case they should know that the number of vibrational modes of an N atom molecule is 3N − 6 for a nonlinear molecule and 3N − 5 for a linear one. The expression for the vibrational entropy of a diatomic molecule with characteristic vibrational temperature Θvib is20 ÄÅ ÉÑ Ñ 1 ÅÅÅ Θvib −Θvib / T Ñ Sm,vib = RÅÅ − ln(1 − e )ÑÑÑ ÅÅÇ T e Θvib / T − 1 ÑÑÖ (9) Although one may use a value of Θvib based on the harmonic frequency, it is preferable to include a term that accounts, at least in part, for anharmonicity; however, this is usually impractical for polyatomic molecules. In any case, because the vibrational entropy contribution is small, the use of harmonic frequencies to obtain Θvib values may be regarded here as acceptable (since the temperature is not too high). For polyatomic molecules, the vibrational entropy is the sum of the contributions from each of the 3N − 6 (or 3N − 5) vibrational modes, viz. 3N − 6(5)

Sm,vib =

∑

Sm,vib, i

(10)

i

For the examples given in this article and in the Supporting Information, the student is given the vibrational frequencies and their characteristic temperatures. The Liquid and Solid Phases. A statistical mechanical calculation for obtaining the entropy of the condensed phases is far from practical. A straightforward approach that will be familiar to the student uses the equation (for isobaric processes without composition change) CP ,m dT i dq y dH m = (dSm)P = jjj zzz = T T k T {P

(11)

μn = μn + 1 − Sn(Tn − Tn + 1) for T < T °

where CP,m is the molar isobaric heat capacity. The integrated form of eq 11 is Sm(T ) = Sm(T •) +

∫T

T •

CP ,m T

dT

and μn = μn − 1 − Sn(Tn − Tn − 1) for T > T °

(12)

(16)

Two propagation algorithms are needed for the same reasons as given above for eqs 13 and 14: To obtain μ at temperatures above T°, one integrates eq 4 from a lower to a higher T and vice versa for μ at temperatures below T°. As an example of the use of this algorithm, the top portion of the Mercury.xlsx spreadsheet is reproduced below. In this calculation, T1 = 298.15 K, TN = 750 K, and N = 453. The value of 0.0 is entered in Cell D21 for μoliq and the following command (see eq 16) is entered in Cell D22: D21-C22*(A22A21). The resulting value of μ liq, −64.69, appears. The remaining values are then propagated downward in Column D. The same calculation applies to the gas phase, Columns E and F. The calculation of μ°gas (Cell F21) is discussed below.

Unlike the case for the statistical mechanical third law entropy calculated for the gas phase, the use of eqs 11 and 12 requires the absolute entropy at the reference temperature T•, which may be 298.15 K, or some other temperature at which Sm for the substance in the particular phase is known. Students must be prepared to use the temperature dependence of CP,m in eq 12 if the temperature range is large or CP,m varies appreciably. CP,m values, obtained using power series expressions from literature sources, are provided for the substances described in this article in the Supporting Information. The technique used here to obtain Sm(T) for the solid and liquid phases from eq 12 uses a stepwise integration of CP,m/T. The CP,m values in the workbooks are given in 1 K increments in a temperature range T1 to TN, chosen to suit the specific calculation (T1 < TN). If the absolute entropy of a given phase at a temperature T• is denoted as S•m, where T1 < T• < TN, the entropy values at temperatures below and above T• are calculated using the algorithm ij T yz Sn = Sn + 1 + C Pn lnjjj n zzz for T < T • j Tn + 1 z k {

(15)

A crucial part of calculating data for the μ(T) plots is the determination of the standard chemical potential values of the higher energy, metastable phases at 298.15 K, i.e., phases for which μ° > 0, e.g., Hg (solid and gas), I2 (liquid and gas), and NH3 (solid and liquid). Students may be concerned that the calculation of these values is challenging, but they will be

(13)

and C



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An alternative method is to carry out a regression analysis on Δμ = m(T − Tsmp), which requires an initial guess of Tsmp. This approach avoids the large correlation between m and b caused by the extended extrapolation of the data to the yintercept (T = 0). The optimized value of Tsmp and the calculated values of ΔfusSm (from −m) and ΔfusHm (from −mTsmp) are identical to those obtained from eqs 18−21. The implementation of these methods to determine the transition temperatures enthalpies and enthalpies is illustrated in the workbooks in the Supporting Information. A summary of these results for Hg, I2, and NH3 is presented later in the article in Tables 3 and 4. A comment about the uncertainties in these values is presented at the end of this article.

reassured when they discover that the task relies on the logical construction of simple thermochemical cycles and the application of basic thermodynamic principles. In this section, the methods and results of these calculations are presented for each of the three substances described in this article. Unlike the gas phase entropies, the chemical potentials are not absolute. This distinction can be seen in the equations used to obtain Sm(T) and μ(T), i.e., eqs 12 and 4. In the former case, the integration constant, Sm,T°, is the absolute entropy of the substance in the particular phase at the reference temperature T°, whereas for the chemical potential, the value of μ° is set equal to zero for the stable phase of the substance at 298.15 K. The value of μ° for a metastable phase represents the molar free energy of the phase relative to that of the stable phase, i.e., zero. The μ(T) curve of a metastable phase is therefore shifted upward relative to the curve of the stable phase by an amount equal to the value of μ° of the metastable phase. The determination of these values is an important, assessable outcome of the students’ work because they can compare their results of Tsmp and Tsbp and the respective transition enthalpies and entropies with the respective literature values.

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Mercury

To quantitatively display the plot of μgas(T) relative to μliq(T), the student must obtain the chemical potential of gaseous mercury in the standard state (298.15 K, 1 bar, ideal gas). Since the liquid phase is the stable one at this temperature and pressure, μ°liq = 0. The objective, then, is to obtain Δμ for the process

Calculation of the Transition Temperatures and Enthalpy and Entropy Changes

Hg liq(298.15 K, 1 bar) → Hggas(298.15 K, 1 bar)

As an example, we show how students can obtain the standard melting point. First, they can obtain an approximate value of Tsmp from their μsolid(T) and μliq(T) plots by using the cursor to locate the T value where they cross. To improve the sensitivity of this graphical analysis, they can expand that area of the chart. A more desirable, quantitative method is to use the spreadsheet containing the μ(T) values of the solid and liquid phases. They calculate μliq(T) − μsolid(T), i.e., Δμ, for temperatures a few kelvins below and above the approximate value of Tsmp. These differences will change from positive to negative. About 4−6 Δμ values are selected that range between similar positive and negative values. Over this narrow temperature range, Δμ can be expressed as a linear equation Δμ(T ) ≡ μ liq (T ) − μsolid (T ) = mT + b

b m

Scheme 1. Thermochemical Steps Used To Calculate μ°gas for Mercury

Students can readily verify that the sum of the three steps in Scheme 1 is equal to eq 22, and that μ°gas is the sum of the Δμ values. These three steps represent, respectively, the lowering of the pressure on liquid mercury from 1 bar to Pvap; the vaporization of liquid mercury at its equilibrium vapor pressure; and the compression of mercury vapor from Pvap to 1 bar (Pvap < 1 bar). From the integration of eq 1, the familiar result Δμ1 = ∫ Vm dP is obtained. To perform these calculations, the student will need Pvap and Vm, which are, respectively, 2.6133 × 10−6 bar21 and 1.4821 × 10−5 m3 mol−1, calculated from the atomic mass of Hg (0.20058 kg mol−1) and its liquid density (1.3534 × 104 kg m−3)22 at 298.15 K. Step 1 describes the isothermal decrease in pressure of liquid mercury. Assuming that Vm is independent of P, one obtains Δμ1 = VmΔP = Vm(2.6133 × 10−6 bar − 1 bar) = (1.4821 × 10−5 m3 mol−1)(−1 bar)(105 Pa/bar) = −1.48 J mol−1. Step 2 expresses the equilibrium condition at constant T and P, and therefore Δμ2 = 0. Step 3 denotes the isothermal compression of an ideal gas, and students will realize that for this process Δμ3 = RT ln(1/ Pvap). Thus, Δμ3 = RT ln(1/2.6133 × 10−6) = 31,867 J mol−1. The sum of steps 1−3 gives μ°gas = 31,865 J mol−1. This is the value used in the Mercury workbook (see Cell F21 above) in the Supporting Information for the calculation of the μ(T) curve for gas phase mercury.

(17)

(18)

For the solid−liquid phase transition, eq 2 can be written as dΔμfus dT

= −Δm,fusS

(19)

and by comparing eq 19 with eq 17, it is evident that ΔfusSm = −m

(20)

Since the two phases are in equilibrium, Δfusμ = 0, and therefore i by ΔfusHm = TsmpΔfusSm = jjj− zzz( −m) = b k m{

(22)

which is equal to μ°gas (since μ°liq = 0). The process in eq 22 can be represented as the sum of three isothermal steps shown in Scheme 1.

where m and b are the slope and intercept. When Δμ = 0, T = Tsmp, and from eq 17, we have Tsmp = −

APPLICATION TO MERCURY, DIIODINE, AND AMMONIA

(21)

Students can readily obtain Tsmp, ΔfusHm, and ΔfusSm (at Tsmp) from their μliq(T)and μsolid(T) results. In the same way, they can find Tsbp, ΔvapHm, and ΔvapSm (at Tsbp) from the μgas(T) and μliq(T) data. D



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Plots of μgas(T) and μliq(T) are shown in Figure 1. The location of the transition temperature, Tsbp, is indicated by the

Scheme 3. Thermochemical Steps Used To Calculate μ°liq for Diiodine

individually for steps 1−3, add them to find ΔH1−3 and ΔS1−3, and then use the Gibbs equation Δμ1−3 = ΔH1−3 − TΔS1−3 to obtain Δμ1−3. The heat capacities of liquid and solid diiodine and the heat of fusion are needed to perform this calculation. For liquid iodine, CP,m,liq = 80.669 J K−1 mol−1 and may be considered constant over the temperature range considered.27,28 The heat capacity of solid iodine, however, increases linearly with temperature, i.e., CP,m,solid = 0.1545T + 5.0904.22 The heat of fusion of iodine at the melting point is 15,517 J mol−1.29 Students should demonstrate that, for step 1, ΔH = 0.1542(T22 − T12)/2 + 5.0904(T2 − T1) and ΔS = 0.1545(T2 − T1) + 5.0904 ln(T2/T1). For step 2, ΔH = 15,517 J mol−1 and ΔS = 15,517 J mol−1/386.75 K. For step 3, ΔH = 80.699(T1 − T2) and ΔS = 80.699 ln(T1/T2). The results of the calculation are summarized in Table 1.

Figure 1. Plot of μgas(T) and μliq(T) at 1 bar. The transition temperature, Tsbp, calculated using eq 18, is 629.71 K.

arrow, and its value, calculated using eq 18, is 629.71 K, which is in good agreement with the reported value of 629.839 K.23 Students can recognize that the accuracy of Tsbp (and any other transition temperature) relies on the accurate vertical displacement of the μgas(T) curve, which depends on the value of μ°gas. All the data used to calculate the μ(T) curves and Tsbp are in the Mercury workbook.

Table 1. Values of ΔHm and ΔSm for Steps 1−3 in Scheme 3 and the Overall Values of ΔHm and ΔSm for I2 solid (298.15 K) → I2 liquid (298.15 K) at P = 1 bar

Diiodine

Diiodine presents a slightly more complicated system than mercury because one must consider rotational and vibrational degrees of freedom. Since solid diiodine is the stable phase at 298.15 K and 1 bar, its chemical potential, μ°solid, is zero. Although the chemical potential of gaseous diiodine is available in the NIST JANAF webbook, it is instructive for students to determine μ°gas in a manner similar to that used for mercury. The vapor pressure and molar volume of solid iodine at 298.15 K are required to carry out the calculation. These values are 4.07 × 10−4 bar24 and 5.145 × 10−5 m3 mol−1,25 respectively. The steps are shown in Scheme 2.

Process

ΔHm/J mol−1

ΔSm/J K−1 mol−1

Step 1 Step 2 Step 3 Overall

5,138.7 15,517 −7,149.9 13,506

15.01 40.12 −20.99 34.14

Using Δμ = ΔHm − TΔSm for the overall process, one finds Δμ = μ°liq = 3,327 J mol−1, which compares well with the NIST-JANAF value of 3,322 J mol−1.27 The assembly of this information and the preparation of the desired temperature dependence of μgas, μliq, and μsolid are available in the diiodine workbook in the Supporting Information. The results are displayed in Figure 2, which

Scheme 2. Thermochemical Steps Used To Calculate μ°gas for Diiodine

The three free energy components Δμ1−3, calculated in the way shown above for mercury, are −5.14, 0, and 19,352 J mol−1, which yields a value of 19,347 J mol−1 for μ°gas. This result compares well with that reported by NIST-JANAF of 19,325 J mol−1.26 To construct the μliq(T) curve, the value of μ°liq at 298.15 K is determined by using the following three-step isobaric thermochemical cycle shown in Scheme 3. The melting point of iodine is 386.75 K, and thus at this temperature the solid and liquid phases are in equilibrium, and the free energy change step 2 is zero. Steps 1 and 3 are not isothermal, and therefore, Δμ cannot be calculated for them. To obtain μ°liq, one must instead calculate ΔH and ΔS

Figure 2. Plot of μgas(T), μliquid(T), and μsolid(T) for diiodine at 1 bar. The transition temperatures are indicated by the arrows and are calculated to be 386.87 and 458.20 K, respectively. E



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shows the locations of the transition temperatures Tsmp and Tsbp. These values are 386.97 and 458.18 K, which compare remarkably well with the literature values of 386.75 and 457.666 K, respectively.29

Step 1 Δμ1 = ΔHm1 − 298.15ΔSm1 = μm° liq = 5, 673.34 J/mol

Step 2

Ammonia

195.36

With ammonia, students encounter a polyatomic molecule, which adds complexity to the calculation of the gas phase entropy. As with Hg and I2, the Sackur−Tetrode equation (eq 6) is used to calculate Sm,trans. However, Sm,rot and Sm,vib must account for the additional internal degrees of freedom compared with those for diiodine. Ammonia has a 3-fold rotational axis (σ = 3) and three principal moments of inertia and, therefore, three characteristic rotational temperatures, ΘA, ΘB, and ΘC. Equation 8 is used to obtain Sm,rot. There are 3N − 6, or 6, vibrational modes (2 degenerate and 2 nondegenerate), and eqs 9 and 10 are employed to calculate Sm,vib. In the latter case, the two degenerate modes are counted twice in eq 10. Values of the rotational and vibrational molecular constants and their characteristic temperatures, including the details of calculating of S m,tot and μgas(T), are provided in the Supporting Information documentation and spreadsheets. In the case of ammonia, the gaseous state is the stable phase under standard conditions, and thus μ°gas = 0. Students must therefore obtain μ°liq and μ°solid. They find μ°liq using a technique similar to that used to obtain μ°gas for Hg (see Scheme 1), as shown in Scheme 4.

ΔHm2 =

∫298.15

CP ,m liq(T ) dT and

195.36

ΔSm2 =

∫298.15

CP ,m liq(T )/T dT

Step 3 ΔHm3 = −Δfus,m H and ΔSm3 = −Δfus,m H /Tnmp

Step 4 298.15

ΔHm4 =

∫195.36

CP ,m solid(T ) dT and

289.15

ΔSm4 =

∫195.36

CP ,m solid(T )/T dT

Overall Δμoverall = ΔHm1 + ΔHm2 + ΔHm3 + ΔHm4 − 298.15(ΔSm1 + ΔSm2 + ΔSm3 + ΔSm4)

Since ΔHm1 − (298.15)(ΔSm1) = μ°liq, we conclude that Δμoverall = ΔHm2 + ΔHm3 + ΔHm4

Scheme 4. Thermochemical Steps Used To Calculate μ°liq for Ammonia

° ° − 298.15(ΔSm2 + ΔSm3 + ΔSm4) + μ liq = μsolid

The values of ΔHm2 + ΔHm3 + ΔHm4 and ΔSm2 + ΔSm3 + ΔSm4 are shown in Table 2 along with their sums. Table 2. Values of ΔHm and ΔSm for Steps 2−4 in Scheme 5 and Their Suma Using values of Pvap (298.15 K) and V m,liq of 9.9785 bar30 and 2.3279 m3/mol,31 respectively, students will find that μ°liq = 5,681.79 J/mol. To determine μ°solid, a series of four isobaric steps is constructed that represents the process NH3,gas → NH3,solid at 298.15 K and 1 bar, as illustrated in Scheme 5.

a

Scheme 5. Thermochemical Steps Used To Calculate μ°solid for Ammonia

Process

ΔHm/J mol−1

ΔSm/J K−1 mol−1

Step 2 Step 3 Step 4 Sum

−7,749.15 −5,655.10 6,935.97 −6,468.28

−31.68 −28.95 27.97 −32.66

The details of each step are provided in the Supporting Information.

Thus, Δμ1−3 = −6,468.28−298.15(−32.66) = 3,269.30 J/ mol and μ°solid = 3,269.30 + 5,673.34 = 8,942.64 J/mol. Having obtained values of μ°liq and μ°solid, students can now calculate the sought-after curves μgas(T), μliq(T), and μsolid(T). These graphs are shown in Figure 3. After students have completed the calculations, produced the μ(T) graphs and found Tsmp, Tsbp, ΔvapHm, and ΔvapSm, they should be encouraged to reinforce their learning experience by preparing a list of all the information and data they used to carry out these calculations and the origin of this information. Such a list is available in the Supporting Information for the NH3 calculations.

Step 1 shows the conversion of gaseous NH3 at 298.15 to the liquid at 298.15 K. Note that, since μ°gas = 0, the molar free energy change of step 1 is equal to μoliq, the value of which is 5,681.79 J/mol (as shown above). In step 2, liquid NH3 is cooled from 298.15 K to its melting point, 195.36 K. In step 3, the liquid freezes to the solid under equilibrium conditions, and step 4 is the warming of the solid from 195.36 to 298.15 K. The thermochemical data is available in the literature. The results are as follows:

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RESULTS OF THE CALCULATIONS Students may now calculate the phase transition properties of the three substances, such as the standard melting and boiling points and the entropies and enthalpies of fusion and vaporization at the respective melting and boiling points, as F



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Figure 3. Plot of μgas(T), μliquid(T), and μsolid(T) for ammonia at 1 bar. The transition temperatures are indicated by the arrows and are calculated to be 197.36 and 247.81 K, respectively.

Table 3. Values of the Transition Temperatures and Enthalpies of Mercury, Diiodine, and Ammonia Tsmp/K Substance

a

Calc

Mercury Diiodine Ammonia

Lit.

Calc

Lit.

629.71 458.18 237.73

d

386.97 197.65

ΔfusHm/kJ mol−1

Tsbp/K a

386.75 195.36e

Calc 629.84c 457.67d 239.68e

b

15.50 5.869

ΔvapHm/kJ mol−1

Lit.

Calc d

15.52 5.655e

b

59.19 41.89 23.52

Lit. 59.21c 41.96d 23.35e

a

Equation 18. bEquation 21. cReference 23. dReference 29. eReference 30.

calculations rely on molecular constants. Although uncertainties in some of the individual values may be known, they cannot be analytically propagated to express the uncertainties in the thermochemical results. Furthermore, the algorithms used to obtain the chemical potentials from the entropies (eqs 15 and 16) and, for liquid and solid, the entropies from the heat capacities (eqs 13 and 14) are approximations, which impose uncertainties. The standard combined uncertainties of the thermochemical quantities, however, calculated from eqs 18−21, are small, e.g., several millikelvins for transition Tsmp and Tsbp (complete information is provided in the Supporting Information). Although more rigorous methods may be used, the objective of this project is to give students the opportunity to use firstprinciples to create the μ(T) curves that give quantitative meaning to the qualitative curves they see in their textbooks. The good results they obtain validate the soundness and usefulness of the fundamental thermodynamic tools they used to acquire them.

described above in eqs 18−21. These results, along with the respective literature values, are summarized in Tables 3 and 4. Table 4. Values of the Transition Entropies of Mercury, Diiodine, and Ammonia ΔfusSma Substance Mercury Diiodine Ammonia

Calcb 40.06 29.69

ΔvapSma Calcb

Lit. d

40.12 28.95e

94.00 91.42 98.94

Lit. 93.84c 91.68d 97.44e

J K−1 mol−1. bEquation 20. cReference 23. dReference 29. eReference 30.

a

The agreement with the literature values is remarkably good and is a gratifying outcome of the students’ work. It will also indicate the soundness of the approach they used in the calculations and of the power of fundamental classical and statistical thermodynamics to accurately calculate measurable quantities from first principles.

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CONCLUSIONS Students will derive many benefits by working on these projects, whether alone or in groups, because by doing so they will solidify their knowledge, application, and understanding of chemical and statistical thermodynamics. An important learning outcome is their ability to calculate how the chemical potential of a substance, in different phases, varies with temperature and pressure through the application of the fundamental equation dμ = Vm dP − Sm dT and to quantitatively assess the quality of their results by determining the standard melting and boiling points of the substances studied. Students also gain experience doing spreadsheet calculations and constructing graphs. Performing these calculations and

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UNCERTAINTIES Astute students, though impressed with the good agreement between the calculated thermochemical data for the three substances and the literature values, will ask about the uncertainties in these calculations. We do not present a complete uncertainty analysis of the calculations here because the literature data and the methods used to obtain the results are subject to unknown systematic and methodological errors. For example, the μ(T) data for the liquid and solid phases depend on published sources of the liquid and solid heat capacities over a large temperature range, and their uncertainties cannot be explicitly ascertained. The gas phase G



Article

(18) Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill: Boston, p 848. (19) Adapted from Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; pp 81−854. (20) Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill: Boston, p 849. (21) Huber, M. L.; Laesecke, A.; Friend, D. G. The Vapor Pressure of Mercury NISTIR 6643. https://nvlpubs.nist.gov/nistpubs/Legacy/IR/ nistir6643.pdf (accessed Jul 2018). (22) Herrington, E. F. G.; Brown, I.; Lane, J. E. Recommended Reference Materials for Realization of Physicochemical Properties: Density. Pure Appl. Chem. 1976, 45, 1−9. (23) Busey, R. H.; Giauque, W. F. The Heat Capacity of Mercury from 15 to 330° K. Thermodynamic Properties of Solid Liquid and Gas. Heat of Fusion and Vaporization. J. Am. Chem. Soc. 1953, 75, 806−809. (24) Shirley, D. A.; Giauque, W. F. The Entropy of Iodine. Heat Capacity from 13 to 321 K. Heat of Sublimation. J. Am. Chem. Soc. 1959, 81, 4778−4779. (25) The Royal Society. Periodic Table. http://www.rsc.org/ periodic-table/element/53/iodine (accessed Jul 2018). (26) Chase, M. W., Jr. NIST-JANAF Thermochemical Tables, 4th ed. https://janaf.nist.gov/tables/I-027.html (accessed Jul 2018). (27) Chase, M. W., Jr. NIST-JANAF Thermochemical Tables, 4th ed., https://janaf.nist.gov/tables/I-025.html (accessed Jul 2018). (28) Chase, M. W., Jr. NIST-JANAF Thermochemical Tables, 4th ed. https://janaf.nist.gov/tables/I-024.html (accessed Jul 2018). (29) Chase, M. W., Jr. NIST-JANAF Thermochemical Tables, 4th ed. https://janaf.nist.gov/tables/I-023.html (accessed Jul 2018). (30) Overstreet, R.; Giauque, W. F. Ammonia. The Heat Capacity and Vapor Pressure of Solid and Liquid. Heat of Vaporization. The Entropy Values from Thermal and Spectroscopic Data. J. Am. Chem. Soc. 1937, 59, 254−259. (31) Lange’s Handbook of Chemistry, 10th ed.; Dean, J. A., Ed.; McGraw-Hill: New York, 1967; pp 1451, 1468.

related tasks will enhance their understanding of the fundamentals of phase equilibrium.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00560. Detailed information about performing the calculations for mercury, diiodine, ammonia, and also for dibromine (PDF, DOCX) Individual Microsoft Excel workbooks (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

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

The authors declare no competing financial interest.

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

(1) Castellan, G. W. Phase Equilibria and the Chemical Potential. J. Chem. Educ. 1955, 32 (8), 424−427. (2) Castellan, G. W. Physical Chemistry, 3rd ed.; Addison-Wesley Publishing Company: Reading, MA, 1983; pp 259−263. (3) Daniels, F.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley: New York,1966; pp 119−120. (4) Moore, W. J. Physical Chemistry, 4th ed.; Prentice Hall: Englewood Cliffs, NJ, 1972; pp 216−217. (5) Mortimer, R. G. Physical Chemistry; Benjamin/Cummings Publishing, Inc.: Redwood City, CA, 1993; pp 182−183. (6) Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; John Wiley & Sons, Inc.: New York, 1997; pp 171−173. (7) Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; pp 252−256. (8) Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley & Sons, Inc.: New York, 2001; p 180. (9) Engel, T.; Reid, P. Physical Chemistry; Pearson Benjamin Cummings: San Francisco, 2006; pp 167−170. (10) Atkins, P.; de Paula, J. Physical Chemistry, 9th ed.; W. H. Freeman and Company: New York, 2010; pp 143−145. (11) Van Hecke, G. R. Thermotropic Liquid Crystals. A use of chemical potential-temperature phase diagrams. J. Chem. Educ. 1976, 53 (3), 161−165. (12) Kildahl, N. K. Journey around a Phase Diagram. J. Chem. Educ. 1994, 71 (12), 1052−1055. (13) Tomasini, P. Deconstructing Phase Diagram Calculations. J. Chem. Educ. 2014, 91 (6), 934−936. (14) Halpern, A. M.; Marzzacco, C. J. J. Chem. Educ. 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 in Ideal and Nonideal Solutions. J. Chem. Educ. 2018, DOI: 10.1021/acs.jchemed.8b00561. (15) Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill: Boston, pp 820−854. (16) Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; pp 1035−1068. (17) Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; pp 1043−1045. H


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