# 2.5(a) Enthalpy. Chapter 2. The First Law. P.27

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1 2.5(a) Enthalpy Chapter 2. The First Law. P.27

2 Justification 2.1 The relation H = q p For a general infinitesimal change in the state of the system, U changes to U + du, p changes to p + dp, and V changes to V + dv, so H changes from U + pv to Recognize U + pv = H on the right, H changes to Now substitute du = dq + dw into this expression, If the system is in mechanical equilibrium with its surroundings at a pressure p and does only expansion work, we can write dw = pdv and obtain Now impose the condition that the heating occurs at constant pressure by writing dp = 0. Then dh=dq (at constant p, no additional work)

3 2.5(b) The measurement of an enthalpy change Calorimeter at constant pressure - isobaric calorimeter. For a combustion reaction an adiabatic flame calorimeter may be used to measure T when a given amount of substance burns in a supply of oxygen Chapter 2. The First Law. P.29

4 Example 2.2 Relating H and U The internal energy change when 1.0 mol CaCO 3 in the form of calcite converts to aragonite is kj. Calculate the difference between the enthalpy change and the change in internal energy when the pressure is 1.0 bar given that the densities of the solids are 2.71 g cm 3 and 2.93 g cm 3, respectively. Answer: The volume of 1.0 mol CaCO 3 (100 g) as aragonite is 34 cm 3, and that of 1.0 mol CaCO 3 as calcite is 37 cm 3. Therefore, (1 Pa m 3 = 1 J). Hence, only 0.1% of U. Usually ignore the difference between H and U of condensed phases, except at very high p, when pv no longer negligible Self Test 2.2 Calculate the difference between H and U when 1.0 mol Sn(s, grey) of density 5.75 g cm 3 changes to Sn(s, white) of density 7.31 g cm 3 at 10.0 bar. At 298 K, H = +2.1 kj. Correct Answer: H U = 4.4 J

5 Enthalpy of perfect gas Illustration 2.4 The relation between H and U for gas-phase reactions In the reaction: 2 H 2 (g) + O 2 (g) 2 H 2 O(l), n g = 3 mol. At 298 K, when RT = 2.5 kj mol 1, the enthalpy and internal energy changes in the system

6 Example 2.3 Calculating a change in enthalpy (p.43) Water is heated to boiling under a pressure of 1.0 atm. When an electric current of 0.50A from a 12 V supply is passed for 300s through a resistance in thermal contact with it, it is found that g of water is vaporized. Calculate the molar internal energy and enthalpy changes at the boiling point ( K).

7 Answer The enthalpy change is (1 A V s = 1 J ) g of water is (0.798 g)/(18.02 g mol 1 ) = (0.798/18.02) mol H 2 O, the enthalpy of vaporization per mole of H 2 O is H 2 O(l) H 2 O(g), n g = +1 mol,

8 Test 2.3 (p.44) The molar enthalpy of vaporization of benzene at its boiling point ( K) is 30.8 kj mol 1. What is the molar internal energy change? For how long would the same 12 V source need to supply a 0.50 A current in order to vaporize a 10 g sample? Correct Answer: kj mol 1, 660 s

9 (c) The variation of enthalpy with temperature Chapter 2. The First Law. P.35

10 2.5(c) The variation of enthalpy with temperature Chapter 2. The First Law. P.36

11 Example 2.4 Evaluating an increase in enthalpy with temperature (p.45) What is the change in molar enthalpy of N 2 when it is heated from 25 C to 100 C? (Table 2-2)

12 2.5 (c) The variation of enthalpy with temperature Test 2.4 At very low temperatures the heat capacity of a solid is proportional to T 3, and we can write C p = at 3. What is the change in enthalpy of such a substance when it is heated from 0 to a temperature T (with T close to 0)? Correct Answer H = ¼aT 4 Perfect gas (2.11):

13 2.6 Adiabatic changes Chapter 2. The First Law. P.39

14 Adiabatic processes Consider a stage in a reversible adiabatic expansion when the pressure inside and out is p. The work done when the gas expands by dv is dw = pdv; however, for a perfect gas, du = C V dt. Therefore,because for an adiabatic change (dq = 0) du = dw + dq = dw, we can equate these two expressions for du and write We are dealing with a perfect gas, so we can replace p by nrt/v and obtain

15 Adiabatic processes (2) To integrate this expression we note that T is equal to T i when V is equal to V i, and is equal to T f when V is equal to V f at the end of the expansion. Therefore, (Take C V independent of temperature.) As dx/x = ln x + constant,

16 Adiabatic processes (3) Because ln(x/y) = ln(y/x), this expression rearranges to With c = C V /nr we obtain (because ln x a = a ln x) which implies that (T f /T i ) c = (V i /V f ) and, upon rearrangement,

17 Adiabatic processes (4) The initial and final states of a perfect gas satisfy the perfect gas law regardless of how the change of state takes place, so use pv = nrt, However, we have just shown that where we use the definition of the heat capacity ratio where γ = C p,m /C V,m and the fact that, for a perfect gas, C p,m C V,m = R (Sec.2.11). Then we combine the two expressions, to obtain which rearranges to p i V i γ = p f V f γ

18 Chapter 2. The First Law. P.44

19 Illustration 2.5 Work of adiabatic expansion Consider the adiabatic, reversible expansion of mol Ar, initially at 25 C, from 0.50 dm 3 to 1.00 dm 3. The molar heat capacity of argon at constant volume is J K 1 mol 1, so c = From It follows that T = 110 K, from Note that temperature change is independent of the amount of gas but the work is not. Test 2.5 Calculate the final temperature, the work done, and the change of internal energy when ammonia is used in a reversible adiabatic expansion from 0.50 dm 3 to 2.00 dm 3, the other initial conditions being the same. Correct Answer: 195 K, 56 J, 56 J

20 Illustration 2.6 The pressure change accompanying adiabatic expansion When a sample of argon (for which γ = 5/3) at 100 kpa expands reversibly and adiabatically to twice its initial volume the final pressure will be For an isothermal doubling of volume, the final pressure would be 50 kpa.

21 Thermochemistry 2.7Standard enthalpy changes standard enthalpy change, H, the change in enthalpy for a process in which the initial and final substances are in their standard states: The standard state of a substance at a specified temperature is its pure form at 1 bar. (real gas, solution?) The standard enthalpy change for a reaction or a physical process is the difference between the products in their standard states and the reactants in their standard states, all at the same specified temperature. Example: the standard enthalpy of vaporization, vap H, is the enthalpy change per mole when a pure liquid at 1 bar vaporizes to a gas at 1 bar, as in

22 Conventional temperature for reporting thermodynamic data K (25.00 C). (a) Enthalpies of physical change The standard enthalpy change that accompanies a change of physical state - standard enthalpy of transition trs H : standard enthalpy of vaporization, vap H ; standard enthalpy of fusion, fus H, conversion of solid to liquid Chapter 2. The First Law. P.48

23 Chapter 2. The First Law. P.49

24 Chapter 2. The First Law. P.50

25 Chapter 2. The First Law. P.51

26 (b) Enthalpies of chemical change Thermochemical equation, chemical equation and change in standard enthalpy: H is the change in enthalpy when reactants in their standard states change to products in their standard states: Pure, separate reactants in their standard states pure, separate products in their standard states standard reaction enthalpy, r H

27 Standard enthalpy of combustion, c H, standard reaction enthalpy for complete oxidation of an organic compound to CO 2 (g) and H 2 O(l) if it contains C, H, and O, and to N 2 (g) if N is also present. Chapter 2. The First Law. P.53

28 Chapter 2. The First Law. P.54

29 6 (NH 2 ) 2 CO C 3 H 6 N NH CO 2 (endothermic) 1 ) urea decomposes into cyanic acid and ammonia: 6 (NH 2 ) 2 CO 6 HCNO + 6 NH 3 (endothermic) 2) cyanic acid polymerizes to form melamine and carbon dioxide: 6 HCNO C 3 H 6 N CO 2 (exothermic) Chapter 2. The First Law. P.55

30 IMPACT ON BIOLOGY I2.2 Food and energy reserves specific enthalpy, the enthalpy of combustion per gram of material. If the standard enthalpy of combustion is c H and the molar mass of the compound is M, then the specific enthalpy is c H /M Chapter 2. The First Law. P.56

31 2.7 Standard enthalpy changes (c) Hess s law Standard enthalpies of individual reactions can be combined to obtain the enthalpy of another reaction. This application of the First Law is called Hess s law: The standard enthalpy of an overall reaction is the sum of the standard enthalpies of the individual reactions into which a reaction may be divided.

32 Example 2.5 Using Hess s law The standard reaction enthalpy for the hydrogenation of propene, is 124 kj mol 1. The standard reaction enthalpy for the combustion of propane, is 2220 kj mol 1. Calculate the standard enthalpy of combustion of propene. Answer The combustion reaction we require is This reaction can be recreated from the following sum: Test 2.6 Calculate the enthalpy of hydrogenation of benzene from its enthalpy of combustion and the enthalpy of combustion of cyclohexane. Correct Answer: 205 kj mol 1

33 2.8 Standard enthalpies of formation f H of a substance is the standard reaction enthalpy for the formation of the compound from its elements in their reference states. The reference state of an element is its most stable state at the specified temperature and 1 bar. Chapter 2. The First Law. P.59

34 Chapter 2. The First Law. P.60

35 Chapter 2. The First Law. P.61

36 Chapter 2. The First Law. P.62

37 (a) The reaction enthalpy in terms of enthalpies of formation Chapter 2. The First Law. P.63

38 Illustration 2.7 Using standard enthalpies of formation The standard reaction enthalpy of 2 HN 3 (l) + 2 NO(g) H 2 O 2 (l) + 4 N 2 (g) is calculated as follows: (b) Enthalpies of formation and molecular modelling No thermodynamically exact way of expressing enthalpies of formation in terms of contributions from individual atoms and bonds. Approximate procedures based on mean bond enthalpies, H(A B), the average enthalpy change associated with the breaking of a specific A B bond, (unreliable?)

39 2.9 The temperaturedependence of reaction enthalpies Chapter 2. The First Law. P.65

40 Kirchhoff s law Assume that no phase transition takes place in the temperature range of interest Normally good approximation to assume that r C p is independent of the temperature, at least over reasonably limited ranges

41 Example 2.6 Using Kirchhoff s law The standard enthalpy of formation of H 2 O(g) at 298 K is kj mol 1. Estimate its value at 100 C given the following values of the molar heat capacities at constant p: H 2 O(g): J K 1 mol 1 ; H 2 (g): J K 1 mol 1 ; O 2 (g): J K 1 mol 1. Assume heat capacities are independent of temp T. Answer H 2 (g) + ½O 2 (g) H 2 O(g) Test 2.7 Estimate the standard enthalpy of formation of cyclohexene at 400 K from the data in Table 2-5. Correct Answer: 163 kj mol 1

42 State functions and exact differentials 2.10 Exact and inexact differentials An exact differential is an infinitesimal quantity that, when integrated, gives a result that is independent of the path between the initial and final states. An inexact differential is an infinitesimal quantity that, when integrated, gives a result that depends on the path between the initial and final states. Chapter 2. The First Law. P.68

43 Ex.2.7 Calculating work, heat, and internal energy Perfect gas inside a cylinder fitted with a piston. Initial state T, Vi and final state T, V f. The change of state can be brought about in many ways, of which the two simplest are: Path 1, free expansion against zero external p; Path 2, reversible, isothermal expansion. Calculate w, q, and U for each process. Method U of a perfect gas depends only on T and is independent of V those molecules occupy, so for any isothermal change, U = 0. In general U = q + w. The question depends on being able to combine the two expressions. Answer Because U = 0 for both paths and U = q + w, in each case q = w. The work of free expansion is zero; so in Path 1, w = 0 and q = 0. For Path 2, work w = nrt ln(v f /V i ) and consequently q = nrt ln(v f /V i ). These results are consequences of the path independence of U, a state function, and the path dependence of q and w, which are path functions. Test 2.8 Calculate the values of q, w, and U for an irreversible isothermal expansion of a perfect gas against a constant nonzero external p. Correct Answer: q = p ex V, w = p ex V, U = 0

44 2.11 Changes in internal energy (a) General considerations Chapter 2. The First Law. P.70

45 Chapter 2. The First Law. P.71

46 Chapter 2. The First Law. P.72

47 internal pressure Chapter 2. The First Law. P.73

48 (b) The Joule experiment Chapter 2. The First Law. P.74

49 Chapter 2. The First Law. P.75

50 (c)changes in internal energy at constant pressure expansion coefficient isothermal compressibility Chapter 2. The First Law. P.76

51 Chapter 2. The First Law. P.77

52 Ex 2.8 Calculating the expansion coefficient of a gas Derive an expression for the expansion coefficient of a perfect gas. Method Substitute the expression for V in terms of T obtained from the equation of state for the gas. Answer Because pv = nrt, we can write The higher the temperature, the less responsive is the volume of a perfect gas to a change in temperature. Test 2.9 Derive an expression for the isothermal compressibility of a perfect gas. Correct Answer κt. = 1/p

53 For a perfect gas, T = 0, so

54 Further Information 2.2 The relation between heat capacities Section 3-8

55 2.12 The Joule Thomson effect for a closed system of constant composition Joule Thomson coefficient, µ

56 Liquefaction of gases. Gas expands through a porous barrier from one const. p to another, monitor the diff. of T from adiabatic expansion. Observed lower T on low p side, diff. in T proportional to p diff. The cooling by isenthalpic expansion is called Joule Thomson effect. Chapter 2. The First Law. P.82

57 The Joule Thomson effect Chapter 2. The First Law. P.83

58 The modern method of measuring µ is indirect, and involves measuring the isothermal Joule Thomson coefficient, the quantity Chapter 2. The First Law. P.84

59 Chapter 2. The First Law. P.85

60 Chapter 2. The First Law. P.86

61 Chapter 2. The First Law. P.87

62 Chapter 2. The First Law. P.88

63 Chapter 2. The First Law. P.89

64 Chapter 2. The First Law. P.90

65 Chapter 2 Checklist of Key Ideas 1. Thermodynamics ( 熱力學 ) is the study of the transformations of energy. 2. The system ( 系統 ) is the part of the world in which we have a special interest. The surroundings ( 環境 ) is the region outside the system where we make our measurements. 3. An open ( 開放 ) system has a boundary through which matter can be transferred. A closed ( 封閉 ) system has a boundary through which matter cannot be transferred. An isolated ( 孤立 ) system has a boundary through which neither matter nor energy can be transferred. 4. Energy ( 能量 ) is the capacity to do work. The internal energy ( 内能 ) U is the total energy of a system. 5. Work ( 功 ) is the transfer of energy by motion against an opposing force, dw = Fdz. Heat ( 熱 ) is the transfer of energy as a result of a temperature difference between the system and the surroundings.

66 Chapter 2 Checklist of Key Ideas 6. An exothermic 放熱 process releases energy as heat to the surroundings. An endothermic 吸熱 process absorbs energy as heat from the surroundings 7. A state function 状態函數 is a property that depends only on the current state of the system and is independent of how that state has been prepared 8. The First Law of thermodynamics states that the internal energy of an isolated system is constant, U = q + w 9. Expansion work is the work of expansion (or compression) of a system, dw = p ex dv. The work of free expansion is w = 0. The work of expansion against a constant external pressure is w = p ex V. The work of isothermal reversible expansion of a perfect gas is w = nrt ln(v f /V i ) 10. A reversible 可逆 change is a change that can be reversed by an infinitesimal modification of a variable

67 Chapter 2 Checklist of Key Ideas 11. Maximum work is achieved in a reversible change 12. Calorimetry is the study of heat transfers during physical and chemical processes 13. Heat capacity 熱容量 at constant volum: C V = ( U/ T) V. The heat capacity at constant pressure is C p = ( H/ T) p. For a perfect gas, the heat capacities are related by C p C V = nr 14. Enthalpy 焓 : H = U + pv. The enthalpy change is the energy transferred as heat at constant pressure, H = q p 15. During a reversible adiabatic 絕熱 change, the temperature of a perfect gas varies according to T f = T i (V i /V f ) 1/c, c = C V,m /R. The pressure and volume are related by pv γ = constant, with γ = C p,m /C V,m

68 Chapter 2 Checklist of Key Ideas 16. The standard enthalpy change is the change in enthalpy for a process in which the initial and final substances are in their standard states. The standard state is the pure substance at 1 bar. 17. Enthalpy changes are additive, as in sub H = fus H + vap H 18. The enthalpy change for a process and its reverse are related by forward H = reverse H 19. The standard enthalpy of combustion is the standard reaction enthalpy for the complete oxidation of an organic compound to CO 2 gas and liquid H 2 O if the compound contains C, H, and O, and to N 2 gas if N is also present. 20. Hess s law states that the standard enthalpy of an overall reaction is the sum of the standard enthalpies of the individual reactions into which a reaction may be divided.

69 Chapter 2 Checklist of Key Ideas 21. The standard enthalpy of formation ( f H ) is the standard reaction enthalpy for the formation of the compound from its elements in their reference states. The reference state is the most stable state of an element at the specified temperature and 1 bar. 22. The standard reaction enthalpy may be estimated by combining enthalpies of formation, r H = Product sν f H Reactants ν f H 23. The temperature dependence of the reaction enthalpy is given by Kirchhoff s law, 24. An exact differential is an infinitesimal quantity that, when integrated, gives a result that is independent of the path between the initial and final states. An inexact differential is an infinitesimal quantity that, when integrated, gives a result that depends on the path between the initial and final states

70 Chapter 2 Checklist of Key Ideas 25. Internal pressure: π T = ( U/ V) T For a perfect gas, π T = The Joule Thomson effect is the cooling of a gas by isenthalpic expansion. 27. Joule Thomson coefficient: µ = ( T/ p) H Isothermal Joule Thomson coefficient: µ T = ( H/ p) T = C p µ 28. The inversion temperature is the temperature at which the Joule Thomson coefficient changes sign.

71 Problem A sample consisting of 1 mol of perfect gas atoms (for which C V,m = 3/2R) is taken through the cycle shown in Fig (a) Determine the temperature at the points 1,2, and 3. (b) Calculate q, w, U, and H for each step and for the overall cycle. If a numerical answer cannot be obtained from the information given, then write in +,, 0, or? as appropriate. Chapter 2. The First Law. P.97

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