MSE 3002 HW1 SP2024

1 MSE 3002 – HW #1 (50 pts total) Due January 23, 2023 1. Enthalpy of a Compound – Ba 3 Al 2 O 6 (12 pts) Enthalpies of Formation at 298 K & 1 atm: Ba + ½ O 2 → BaO -554 kJ/mol 2Al + 3/2 O 2 → Al 2 O 3 -1675 kJ/mol Al + ½ O 2 → Al 2 O -130 kJ/mol 2Al + O 2 → SiO 2 -435 kJ/mol Thermodynamic Information about Ba 3 Al 2 O 6 : T m = 2023 K (Melting point)  H f = 3519 kJ/mol (Heat of Fusion) C p = 327.6 J/mol-K (for the LIQUID phase of Ba 3 Al 2 O 6 ) (a) Suppose we want to describe Ba 3 Al 2 O 6 as a phase in a binary system, what two components are necessary and sufficient to describe this binary system? ( 2 pts ) (b) The reaction enthalpy to form Ba 3 Al 2 O 6 from these two components at 298 K and 1 atm is -618 kJ/mol (This value is for a stoichiometrically balanced reaction to form 1 mole of Ba 3 Al 2 O 6 , that is to say the general reaction of: A  BaO x + B  Al y O z → Ba 2 Al 2 O 6 ). What is the enthalpy of formation of Ba 2 Al 2 O 6 at 298 K? Show your work . Hint: An absolute value for the enthalpy of a compound can be defined by referencing its enthalpy to its elemental constituents ’ enthalpy at their standard state. ( 2 pts ) (c) At the bottom of this homework set is the temperature-dependent heat capacity data for the solid phase of Ba 3 Al 2 O 6 . Heat capacity is often fit to polynomials to create functional equations. Try fitting this data to 1 st , 2 nd , 3 rd , 4 th , 5 th … order polynomials and decide which is sufficient to describe the data. (If you “ over fit ” , then later calculations will be more cumbersome.) i. How did you decide which polynomial was your best choice? ( 1 pt ) ii. What is your equation for the heat capacity of the solid? ( 1 pt ) (d) Using your heat capacity function, calculate the enthalpy of solid Ba 3 Al 2 O 6 at 1040 K. Clearly show all of your work. ( 2 pts ). (e) The tabulated source for this data only provides enthalpies at 100 K increments. It lists the enthalpy of Ba 3 Al 2 O 6 as -3330.5 kJ/mol at 1000 K and -3301.1 kJ/mol at 1100 K. i. Based on these two enthalpies, calculate a linearly interpolated value for the enthalpy of Ba 3 Al 2 O 6 at 1040 K. ( 1 pt ) ii. How close is this interpolated value to the value you calculated in part (d)? (Did you need to do all of that work?) ( 1 pt ) (f) Sketch the enthalpy of Ba 3 Al 2 O 6 as a function of temperature from 298 K to 2800 K labeling important points. All relevant phase transformation information for this temperature range is given above. ( 2 pts ). *Data is from: I. Barin, Thermochemical Data of Pure Substances (VCH, New York, 1995) 2. Exploring Elemental Heats of Fusion (11 pts) (a) Using data from literature source(s), plot (using your favorite software like excel or others) the heat of fusion (i.e., melting,  H f ) (in J/mol) versus the absolute melting temperature (in Kelvin) for all elements (at least 50) that are crystalline solids above room temperature. Include a citation for your data source on the plot. Do a linear regression analysis on the data and report the slope of the curve. ( 7 pts ) Tip 1: Exclude carbon (at least for your linear regression) Tip 2: When doing the linear regression, force the line through the origin

2 Tip 3: Remember to report: (1) Your plot, (2) the slope of the regression line, & (3) a reference for the data. (b) What thermodynamic parameter does the slope of this regression line represent? Explain using words and mathematical concepts. ( 2 pts ) (c) Which elements appear to be significant outliers from the linear regression line? Give a physical explanation for why these elements may deviate from the trend. (You can also choose to leave these out of your regression analysis but still show the points on your plot.) ( 2 pts ) 3. Calculating Unary Phase Diagrams (15 pts) A somewhat incompetent graduate student has fortuitously stumbled upon a new material, Gatechium, and would like help in constructing its unary phase diagram. The student has made many measurements around the known triple points, but is having difficulty with constructing the phase boundaries. See if you can help out – you can make the simplifying assumption that all phase boundaries are linear. The following information has been gathered from the experiments: • Four phases exist: Solid 1 (S 1 ), Solid 2 (S 2 ), Liquid (L), and Gas (G) • Gatechium has a molar mass of 160 g/mol • The sublimation phase boundary definitely goes through 0 K & 0 atm. • A triple point between S 1 , L, and G occurs at 5 atm and 300 K. • At this S 1 -L-G triple point, the solid has a density of 1.7 g/cm 3 and the liquid has a density of 1.1 g/cm 3 . • A second triple point between S 1 , S 2 , and L occurs at 8 atm, but the exact temp is unknown. • The heat of fusion (  H f , S 1 → L) is measured to be 90 J/mol. • The L and G phases are in equilibrium up to 7 atm and 700 K, but the phases become indistinguishable when heated any further. This point is marked Q on the phase diagram. The grad student’s current understanding of Gatechium’s unary phase diagram is shown below. We are particularly interested in knowing more about the phase boundaries for S 1 -L and L-G. (a) What is point Q on the phase diagram? Explain. ( 2 pts ). (b) What is the heat of vaporization (  H v , L → G) for Gatechium in units of kJ/mol? Show your work. ( 3 pts )

4 5. Free Energy Diagrams (6 pts) The P vs. T unary phase diagram of SiO2 is shown below. For the pressures labeled P1 and P2 draw schematic free energy (G) versus temperature (T) diagrams. Use the following abbreviations: α - quartz (α), β - quartz (β), coesite (c), st ishovite (s), and liquid (L). (For the purposes of this problem, ignore the tridymite & cristobalite phases.)

5 Appendix : Heat Capacity Data for Ba 3 Al 2 O 6 (Question #1) Temperature (K) Heat Capacity (J/mol-K) 298 215.6 300 216.4 400 245.5 500 260.6 600 270.2 700 277.2 800 282.8 900 287.7 1000 292 1100 296 1200 299.8 1300 303.4 1400 307 1500 310.4 1600 313.8 1700 317.1 1800 320.4 1900 323.6 2000 326.9 2023 327.6