C a h p a t p er 3 The Importance of State Functions: Internal Energy and Enthalpy

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1 Chapter 3 he Importance of State Functions: Internal Energy and Enthalpy Engel & Reid 1

2 Outline 3.1 he Mathematical roperties of State Functions 3.2 he Dependence of U on and 3.3 Does he Internal Energy Depend More Strongly on or? 3.4 he ariation of Enthalpy with emperature at Constant ressure 3.5 How Are C p and C v Related? 3.6 he ariation of Enthalpy with ressure at Constant emperature 3.7 he Joule-hompson Experiment 3.8 Liquefying Gases Using an Isenthalpic Expansion 2

3 Mathematical roperties of State Functions 3

4 3.1 he Mathematical roperties of State Functions artial derivatives of pressure can be written as a function of the two variables and. he change in resulting from a change in or is proportional to the following partial derivatives:,, R lim 0 2,, R lim 0 d d d 4

5 3.1 he Mathematical roperties of State Functions State function f You are on a hill and have determined your altitude above sea level. How much will the altitude (denoted z) change if you move a small distance east (denoted by x) and north (denoted by y)? he change in z as you move east is the slope of the hill in that direction, ( z/ x) y, multiplied by the distance that you move. A similar expression can be written for the change in altitude as you move north. herefore, the total change in altitude is the sum of these two changes, f, f his equation can be used to determine if function f is a state function If f is a state function df depends only on the initial and final state. In this case df is called an exact differential. f f df f f i final initial 5

6 Example roblem 3.1 x f x, y ye xyxln y f x f x x ye y ln y e x xy y y f f x 2 2 x ye x y y y x f f x y x 1 y x x 1 e 1 e 1 y y x y x f f x y y y x x x x f x y x, y is a state function. f f x x x df dx dy ye y ln ydx e x dy xy y y x 6

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13 3.1 he Mathematical roperties of State Functions Quantities that can be measured in laboratory in order to obtain numerical values for /) v and /) 1 1 ; 1 volumetric thermal expansion coefficient 1 isothermal compressibility f f d d d; Δ= d d f i ln i i f i 13

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16 Example roblem 3.2 Example roblem 3.2 You have accidentally arrived at the end of the range of an ethanol in glass thermometer so that the entire volume of the glass capillary is filled. By how much will the pressure in the capillary increase if the temperature is increased by another 10.0 C? glass = ( C) -1, ethanol = ( C) -1, and ethanol = (bar) -1 Do you think that the thermometer will survive your experiment? ethanol 1 ethanol 1 d d ln 1 i 1 ethanol glass ethanol 1 i glass ln C 1 ethanol galss i 10.0C 65.5 bar (bar) f i i 16

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18 3.2 he dependence of U on and U U du d d du dq d dq U d f i external 2 2 m, m, 1 1 f U dq d; q U i C U C d n C d nc 18

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20 Figure 3.1 Figure 3.1 Molar heat capacities C v,m are shown for a number of gases. Atoms have only translational degrees of freedom and, therefore, have comparatively low values for C v,m that are independent of temperature. Molecules with vibrational degrees of freedom have higher values of C v,m, at temperatures sufficiently high to active the vibrations. 20

21 3.2 he dependence of U on and Dependence of U on U U derived in section 5.12 du du du Cd d 3 2 Internal ressure J/m J/m / m force/area=pressure Figure 3.2 Because U is a state function, all paths connecting i, i and f, f are equally valid in calculating U. herefore, a specification of path is irrelevant. 21

22 du C d du d 22

23 Example roblem 3.3 Example roblem 3.3 Evaluate (U/) for an ideal gas and modify Equation (3.20) accordingly for the specific case of an ideal gas. Solution nr / nr 0 herefore, du C d, showing that for an ideal gas, U is a function of only. 23

24 3.3 Does the internal Energy Depend More Strongly on or? Joule Experiment Figure 3.3 Schematic depiction of the Joule experiment to determine (U/ ). wo spherical vessels, A and B, are separated by a valve. Both vessels are immersed in a water bath, the temperature of which is monitored. he initial pressure in each vessel is indicated. 24

25 3.3 Does the internal Energy Depend More Strongly on or? Joule Experiment U U du dq externald d d 0 external U U dq d d Within experimental accuracy, Joule found d d 0; dq 0 dq U d U Since d surrounding Joule s experiment was not definitive because the experimental sensitivity was limited. 25

26 Example roblem 3.4 Example roblem 3.4 In Joule s experiment to determine (U/) the heat capacities of the gas and the water bath surroundings were related by C surroundings /C system If the precision with which the temperature of the surroundings could be measured is C, what is the minimum detectable change in the temperature of the gas? Solution q C C water bath water bath gas gas Cwater bath gas water bath C6C C gas Because the minimum detectable value of is rather large, this apparatus is clearly not suited for measuring small changes in the temperature of the gas in na expansion. 0 gas 26

27 Example roblem 3.5 Example roblem 3.5 For a gas described by van der Waals equation of state, =R/( m -b) a/ 2 m. Use this equation to complete these tasks: a. Calculate (U/) using (U/) = (/). b. Derive an expression for the change in internal energy. Solution R a b R R R a a a. b b b m, 2 m m 2 2 m m m m m b. m, f m, f U m a 1 1 Um, dm d 2 ma mi, mi, m m, i m, f U is not zero if the attractive part of the intermolecular potential is zero. 27

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31 Example roblem 3.6 Example roblem 3.6 AsampleofN 2 gas undergoes a change from an initial state described by =200 K and i =5.00 bar to a final state described by =400 K and f =20.0 bar. reat N 2 as a van der Waals gas with the parameters a=0.137 a m 6 mol -2 and b= m 3 mol -1.Weuse the path N 2 (g, =200 K, =5.00 bar)n 2 (g, =200 K, =20.0bar) N2 (g, =400 K, =20.0 bar), keeping in mind that all paths will give the same answer for U of the overall process. 31

32 Example roblem 3.6a Example roblem 3.6 m, f a. Calculate U U m using the result of m, dm mi, Example roblem 3.5. Notice that m,i = m 3 mol -1 and m,f = m 3 mol -1 at 200 K, as calculated using the van der Waals equation of state. Solution 1 1 Um, a mi, m, f a m mol J m mol m mol 32

33 roblem 3.6a Example roblem 3.6 f b. Calculate U C d using the following relationship m, m, for C,m in this temperature range: i C J K mol K K K 2 3, m Solution he ratio n /K n ensure that C v,m has the correct dimension. U C d m, m, f i d J mol 2 3 K K K K kj mol 4.17 kj mol

34 Example roblem 3.6a Example roblem 3.6 c. Compare the two contributions to U m.canu,m be neglected relative to U,m? Solution U,m is 3.1% of U,m for this case. In this example, and for most processes, U,m can be neglected relative to U,m for real gas. 34

35 3.4 he ariation of Enthalpy with emperature at Constant ressure H q H H dh d d C dq H d Figure 3.4 he initial and final states are shown for an undefined process that takes place at constant pressure. f H C d n C d, m i H C nc, m f i 35

36 Example roblem 3.7 Example roblem 3.7 A g sample of C (s) in the form of graphite is heated from 300 to 600 K at a constant pressure. Over this temperature range, C p,m has been determined to be J K C p, m -1 mol K K K K Calculate H and q p. How large is the relative error in H ifyou neglect the temperature-dependent terms in C p,m and assume that C p,m maintains its value at 300K throughout the temperature interval? 36

37 Example roblem 3.7. q H From equation (3.28), kj g mol g mol p , J K K K K K g K d K K K K mol J g d C M m H f i m p Solution 37

38 Example roblem 3.7 If we had assumed C p,m =8.617 J mol K, which is the calculated value at 300 K, =143.0 g / g mol J K mol [600K-300K]=30.81 kj he relative error is (30.81kJ kJ)/46.85 kj=-34%. In this case, it is not reasonable to assume that C p,m is independent of temperature. 38

39 3.5 How are C p and C v Related? C p and C v U C U C U C C d dq d d U d C dq d U d C d U d U d dq du ext since 39

40 3.5 How are C p and C v Related? Express abstract partial derivatives with experimentally available data and C C β κ 2 or C, m C, m m β κ 2 Because and are positive for ideal and real gases, C C > 0 40

41 3.5 How are C p and C v Related? For an ideal gas nr C C nr nr and U 0 For liquids and solids C C 41

Chapter 3 The Importance of State Functions: Internal Energy and Enthalpy. Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid

Chapter 3 The Importance of State Functions: Internal Energy and Enthalpy. Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 3 he Importance of State Functions: Internal Energ and Enthalp homas Engel, hilip Reid Objectives Epress the infinitesimal quantities du and dh as eact differentials. Derive the change of U with