Chapter 3 The Importance of State Functions: Internal Energy and Enthalpy. Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid
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1 Chapter 3 he Importance of State Functions: Internal Energ and Enthalp homas Engel, hilip Reid
2 Objectives Epress the infinitesimal quantities du and dh as eact differentials. Derive the change of U with and and the change in H with and to eperimentall accessible quantities. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
3 Outline 1. he Mathematical roperties of State Functions 2. he Dependence of U on and 3. Does the Internal Energ Depend More Strongl on or? 4. he ariation of Enthalp with emperature at Constant ressure 5. How Are C and C Related? 6. he ariation of Enthalp with ressure at Constant emperature 7. he Joule-hompson Eperiment 8. Liquefing Gases Using an Isenthalpic Epansion Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
4 3.1 he Mathematical roperties of State Functions Consider 1 mol of an ideal gas for which f (, ) R he change in from a change in or is proportional to the following partial derivatives: lim lim 0 0 (, ) (, ) (, ) (, ) R 2 R Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
5 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 3.1 he Mathematical roperties of State Functions 3.1 he Mathematical roperties of State Functions When changes to d, When function f is a state function, d d d ( ) ( ) f f,,
6 Eample 3.1 a. Calculate for the function f (, ) e ln b. Determine if f(,) is a state function of the variables and. c. If f(,) is a state function of the variables and, what is the total differential df? Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
7 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp Solution Solution a. e f e f f e f e f e f 1 1, 1 1,, ln e f ln ), (
8 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp Solution Solution b. Because we have shown that f(,) is a state function of the variables and. Note that an well-behaved function that can be epressed in analtical form is a state function. f f
9 Solution c. he total differential is given b df f d f d ( ) e ln d e d Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
10 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 3.1 he Mathematical roperties of State Functions 3.1 he Mathematical roperties of State Functions wo differential calculus that used frequentl: a)a function of zf(,), b)he cclic rule z z 1 1 z z z
11 3.1 he Mathematical roperties of State Functions Combination of previous equations give the coefficients. 1 1 β and κ β volumetric thermal epansion coefficient κ isothermal compressibilit Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
12 Eample 3.2 You have accidentall arrived at the end of the range of an ethanol in glass thermometer so that the entire volume of the glass capillar is filled. B how much will the pressure in the capillar increase if the temperature is increased b another 10.0 C?,, 4 5 β ( ) 1 β ( C) 1 ethanol glass C κ ethanol 5 ( ) bar Do ou think that the thermometer will survive our eperiment? Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
13 Solution β ethanol ethanol d d κ κ κ ( β β ) ( ) ethanol κ glass 1 β ln κ C 100. bar In this calculation, we have used the relations: ( ) ( )( β [ ]) he glass is unlikel to withstand such a large increase in pressure ( 1 ) if 1 ln << 4 f i Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
14 3.2 he Dependence of U on and As U is a state function, an infinitesimal change in U can be written as du U d he differential epression of the first law for constant volume can be written as U dq v /d corresponds to a constant volume path and is called the heat capacit at constant volume, C v Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd dq d U C d
15 3.2 he Dependence of U on and Atoms have onl translational degrees of freedom and low C,m independent of temperature. Molecules with vibrational degrees of freedom have higher C,m. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
16 3.2 he Dependence of U on and After C has been determined as a function of, the integral is numericall evaluated: U 2 C d 1 n 2 1 C, m d Over a limited temperature range, C,m can be simplified into q U Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
17 3.2 he Dependence of U on and he total differential of the internal energ can be written as du du du d U is a state function, all paths connecting i, i and f, f are equall valid in calculating U. C d Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
18 3.3 Does the Internal Energ Depend More Strongl on or? U is a function of alone for an ideal gas. Not true for real gases, liquids, and solids as the change in U with must be considered. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
19 Eample 3.3 Evaluate ( U / ) for an ideal gas and modif accordingl 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 onl. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
20 3.3 Does the Internal Energ Depend More Strongl on or? U is a function of alone for an ideal gas. Ideal gas molecules do not attract or repel one another, no energ is required to change their average distance of separation (increase or decrease ). Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
21 Eample 3.4 ( ) In Joule s eperiment to determine U /, the heat capacities of the gas and the water bath surroundings were related b C surroundin / C 1000 g sstem 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? Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
22 Solution iew the eperimental apparatus as two interacting sstems in a rigid adiabatic enclosure. he first is the volume within vessels A and B, and the second is the water bath and the vessels. Because the two interacting sstems are isolated from the rest of the universe, q C gas water bath C C water bath water bath gas C gas water bath gas ( ± C) 6 C Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
23 3.4 he ariation of Enthalp with emperature at Constant ressure he initial and final states for an undefined process that takes place at constant pressure. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
24 3.4 he ariation of Enthalp with emperature at Constant ressure As f i we have H q Since H is a state function, dh is an eact differential. dh H d H d Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
25 3.4 he ariation of Enthalp with emperature at Constant ressure he heat capacit at constant pressure, C, is defined as C p dq d p H In general, a constant pressure process with no change in the phase of the sstem and no chemical reactions, f ( ) d n C m ( ) H C, i f i d Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
26 Eample 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,m has been determined to be Calculate H and q. How large is the relative error in H if ou neglect the temperature-dependent terms in C,m and assume that C, maintains its value at 300 K throughout the temperature interval? Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
27 Solution Answer: H m M f i C, m ( ) d kJ K K K K K K K K K d K Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
28 Solution (cont d) If we had assumed C,m J mol -1 K -1, which is the calculated value at 300 K, [ ] 30. kj H / ( ) % he relative error is / In this case, it is not reasonable to assume that C,m is independent of temperature. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
29 3.5 How Are C and C Related? he relationship between C p and C v is defined as 2 β C C or C, m C, m κ m 2 β κ When C C for a liquid or a solid, C >> U Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
30 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 3.6 he ariation of Enthalp with ressure at Constant 3.6 he ariation of Enthalp with ressure at Constant emperature emperature When all sstems containing pure substances or mitures at a fied composition, provided that no phase changes or chemical reactions take place, we appl H
31 Eample 3.8 Evaluate H for an ideal gas. Solution: ( ) ( ) ( [ ] 2 nr / / ) nr / and ( ) nr / for an ideal gas. herefore, H nr nr nr nr 0 2 nr Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
32 3.7 he Joule-hompson Eperiment he Joule-hompson eperiment allows to be measured with a much higher sensitivit than in the Joule eperiment. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
33 3.7 he Joule-hompson Eperiment he eperimentall determined limiting ratio of to at constant enthalp is known as the Joule-hompson coefficient: µ J lim 0 H H Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
34 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 3.7 he Joule 3.7 he Joule-hompson Eperiment hompson Eperiment From an isenthalpic process, alues of µ J- for selected gases are shown. J p H C H H C µ giving 0
35 2010 earson Education South Asia te Ltd Chapter 3: he Importance of State Functions: Internal Energ and Enthalp Eample 3.11 Eample 3.11 Show that for an ideal gas. Solution: U H 0 µ J [ ] 0 1 / nr C nr C C U C H C µ J
36 3.8 Liquefing Gases Using an Isenthalpic Epansion Heat is etracted from the gas eiting from the compressor in Joule-hompson epansion It is further cooled in the countercurrent heat echanger before epanding through a nozzle. Because its temperature is sufficientl low, liquefaction occurs. Chapter 3: he Importance of State Functions: Internal Energ and Enthalp 2010 earson Education South Asia te Ltd
C a h p a t p er 3 The Importance of State Functions: Internal Energy and Enthalpy
Chapter 3 he Importance of State Functions: Internal Energy and Enthalpy Engel & Reid 1 Outline 3.1 he Mathematical roperties of State Functions 3.2 he Dependence of U on and 3.3 Does he Internal Energy
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