Thermodynamics of Bomb Calorimetry: Determining Heat of Combustion, Slides of Thermodynamics

Download Thermodynamics of Bomb Calorimetry: Determining Heat of Combustion and more Slides Thermodynamics in PDF only on Docsity! RP546 STANDARD STATES FOR BOMB CALORIMETRY By Edward W. Washburn ABSTRACT An examination of the thermodynamics of the conditions existing in bomb calorimetry shows that the heat of combustion per unit mass of substance burned is a function of the mass of sample used, of the initial oxygen pressure, of the amount of water placed in the bomb, and of the volume of the bomb. In order to eliminate the effects of these at present unstandardized variables and to obtain a more generally useful thermal quantity which characterizes the pure chemical reaction for stated conditions, it is suggested that every bomb-calori- metric determination be first corrected (where such correction is significant) so as to give the value of A£/r, the change of ''intrinsic" energy for the pure isother- mal reaction under the pressure condition of 1 normal atmosphere for both re- actants and products. From this value the more generally useful quantity, Q v , the heat of the pure reaction at a constant pressure of 1 atmosphere is readily calculable. An equa- tion for calculating the correction is given and illustrated by examples. The magnitude of the correction varies from a few hundredths of 1 per cent up to several tenths of 1 per cent according to the nature of the substance burned and the conditions prevailing in the bomb during the combustion. It is further recommended that, in approving, for the purpose of standardizing a calorimeter, a particular value for. the heat of combustion (in the bomb) of a standardizing substance, such as benzoic acid, the value approved be accom- panied by specification of the oxygen concentration and of the ratios to the bomb volume of (1) the mass of the sample and (2) the mass of water, together with appropriate tolerances. An equation is given for correcting to any desired standard temperature the heat measured in the bomb calorimeter. CONTENTS Page I. Nomenclature 526 1 1 . Introduction 527 III. Calorimetry and the first law of thermodynamics 528 IV. The nature of the bomb process 529 V. Proposed standard states for constant-volume combustion re- actions 530 VI. Comparison of the actual bomb process with that defined by the proposed standard states 531 VII. The total energy of combustion defined by the proposed standard states 531 VIII. Definitions of some auxiliary quantities 534 1. The initial system 534 2. The final system 534 IX. Correction for dissolved carbon dioxide 535 X. The energy content of the gases as a function of the pressure 536 XI. Correction for the change in energy content of the gases 537 XII. Calculation of the change in pressure resulting from the com- bustion 538 XIII. The negligible energy quantities 540 1 . The energy content of the water 540 (a) The change in the energy content of the water vapor 540 (b) The change in the energy content of the liquid water 540 2. Combined energy corrections for water vapor and for dis- solved carbon dioxide 541 525 526 Bureau of Standards Journal of Research [Voi.io XIII. The negligible energy quantities—Continued. page 3. The energy content of the dissolved oxygen 541 4. The energy content of the substance 542 (a) The energy of compression of the substance 542 (6) The energy of vaporization of the substance 543 XIV. The total correction for reduction to the standard states 543 1. The general correction equation 543 2. An approximate correction equation 544 XV. The magnitude of the correction in relation to the type of sub- stance burned 545 XVI. Computation of the correction 545 1. General remarks 545 2. Computation for benzoic acid 547 3. Computation for a mixture 547 XVII. Corrections for iron wire and for nitrogen 548 XVIII. Reduction of bomb calorimetric data to a common temperature-. 550 XIX. The temperature coefficient of the heat of combustion 551 XX. Standardizing substances 552 XXI. Standard conditions for calorimetric standardizations 552 XXII. The heat of combustion of standard benzoic acid 553 Appendix I. Concentration of saturated water vapor in gases at various pressures 554 Appendix II. Empirical formula of a mixture 557 Appendix III. Summary of numerical data employed 557 I. NOMENCLATURE (Additional subscripts and superscripts are used in the text as further distinguishing marks) A Maximum work. a, b, c Coefficients in the chemical formula, CaHbOc . C Concentration; molal heat capacity. c Specific heat, g Gram. (g) Gaseous state. h = 1.70a: (1+x). g. f. w. Gram-formula-weight. (1) Liquid state. M Molecular weight; g. f. w. m Mass of sample burned. mw Number of grams of water placed in the bomb. n Number of moles or of g. f. w. of substance burned. nD Number of moles of C02 in solution in the water in the bomb. nM Number of moles of gas in the bomb after the combustion. no2 Number of moles of O2 in the bomb before the combustion. P Pressure; per cent by weight. p Pressure or partial pressure. p x Pressure in the bomb before the combustion. p2 Pressure in the bomb after the combustion. pw Vapor pressure of water. Q Heat absorbed. R Gas constant. S Solubility. s Heat capacity, (s) Solid state. T Absolute centigrade temperature. t Centigrade temperature. tH Standard temperature °C. U Total or intrinsic energy content. Washburn] Standard States for Bomb Calorimetry 529 into a vacuum); and (3) because constancy of volume alone is not a I sufficient characterization of an isothermal zero-work process (since the heat of such a process may also be a function of the pressure, for example). Case 2. W= fpdv^pAv, where Av is the volume increase under a constant external pressure p. The heat of the process under these conditions is usually designated by QH and is commonly called "the heat at constant pressure." This designation is likewise correct and complete only when the value of the pressure is stated or implied and when all of the work done in the surroundings is accounted for by the volume change in the system. Case 8. W= Wm&x . = A, where A is the maximum work or "free energy " of the process. This case is rarely encountered in calori- metry except when it is identical with case 2. The corresponding heat quantity has received no special designation although its ratio to the absolute temperature is the "entropy of the process." Any one of the above heat quantities is, in principle, calculable from any other, but for conventional reasons the quantity Qv for p = 1 atmosphere appears to be the most wanted one. It will be advantageous therefore to standardize the initial and final states of bomb calorimetry in such a manner as to facilitate the com- putation of Qp for p = 1 atmosphere. This can be conveniently ac- complished by first correcting the quantity Qv of the bomb process in such a way as to obtain the quantity AU for the reaction standard- ized for a pressure of 1 atmosphere, from which quantity the value of Qp can be readily computed. IV. THE NATURE OF THE BOMB PROCESS Given a substance (or a material) whose composition is expressed by the empirical formula CaHbOc . m grams ( = n gram-formula- weights) of this material in a thermodynamically defined physical state or states (solid and/or liquid) are placed in the bomb. 1 mw grams of water are also placed in the bomb, this amount being at least sufficient to saturate the gas phase (volume — V liters) with water vapor. The bomb is then closed and filled with n02 moles of oxygen, this amount being at least sufficient to ensure complete combustion. The above quantities will completely define the initial system and this definition will subsist, if the quantities are all increased in the same ratio; that is, the initial state of the system is completely defined by the specification of the quantitative composition and physical state of the substance, by the temperature, and by three ratios; for example, m/V, m^/V, and n02/V. When the calorimeter fore period has been established, the charge is ignited with the aid of a known amount of electrical energy. When the after period has been established, the heat liberated is computed, corrected to some definite temperature, tH , and divided by n so as to obtain the quantity —AUB which we shall designate as the evolved heat of the bomb process per g. f. w. (gram-formula-weight) of material burned at the temperature tH - AUb^Qv 1 If the material is volatile, it must be inclosed in a suitable capsule in order to prevent evaporation, and in some cases a combustible wick or admixture with some more easily combustible material must be employed in order to ensure complete combustion. In the latter case we are dealing with a mixture of combustible materials, and the formula C aHb0 should express the empirical composition of this mixture. The heat of combustion of the added material must be separately determined and the two heats are, in principle, not additive in the bomb process. 530 Bureau of Standards Journal of Research [Vol. 10 V. PROPOSED STANDARD STATES FOR CONSTANT-VOLUME COMBUSTION REACTIONS For the purpose of recording such thermodynamic quantities as heat of formation, free energy of formation, entropy of formation, heat content, etc., for chemical substances, it is necessary to adopt some standard reference state, at least for each of the chemical elements. From some points of view a logical standard state for each element might be the state of a monatomic gas at zero degrees absolute. 2 For obvious practical reasons, however, it has not hitherto been feasi- ble to advantageously utilize a standard state denned in this way. Most of the compilations of thermodynamic properties of chemical substances define the standard state of a chemical element as the thermodynamically stable form of the element under a pressure of 1 normal atmosphere at the standard temperature (usually 18° or 25° C). The immediate requirements of bomb calorimetry will be met, if, in conformity with this general practice, we adopt the following standard states : 1. For 2 and C02 .—The pure gaseous substance under a pressure of 1 normal atmosphere 3 at the temperature tH . 2. For H20.—The pure liquid under a pressure of 1 normal atmos- phere 4 at the temperature tH . 3. For the substance or material burned.—In a thermodynamically defined state or states (solid and/or liquid 6 ) under a pressure of 1 normal atmosphere 6 at the temperature tH . The temperature tH is the temperature at which the heat of the reaction is desired. For purposes of record this is usually made either 18°, 20°, or 25° C. The utility of thermodynamic data for chemical substances and reactions would be increased, if this tem- perature could be standardized by international agreement. At all events the temperature coefficient should be stated for each new determination of the heat of a reaction which is recorded in the literature. The above definitions will make it possible to obtain from the data of bomb calorimetry a clearly defined and generally useful thermo- dynamic quantity. 7 1 A still more fundamental (but likewise inaccessible) reference state might be pure proton gas and pure electron gas at zero degrees absolute. 3 For the purposes of bomb calorimetry this pressure is at present indistinguishable from 1 bar (mega- barye) . * For a condensed phase not in the neighborhood of its critical temperature this pressure is at present calorimetrically indistinguishable from its own vapor pressure or in fact from zero pressure. 4 For pure substances only one state, solid or liquid, is involved, but for mixtures more than one state or Ehase may be present. The gaseous state is excluded from the present discussion for two reasons: First, ecause the heat content of the gas and of its mixtures with oxygen must be evaluated as a function of the pressure; and second, because the bomb calorimeter is not the best instrument for the determination of heats of combustion of gases (or of volatile liquids). The flame calorimeter is more accurate and convenient for such cases. (See Rossini, B. S. Jour. Research, vol. 6, pp. 1, 37, 1931; vol. 7, p. 329, 1931; and vol. 8, p. 119, 1932.) * See footnote 4. T For purposes of computing heats of formation from heats of combustion it is further necessary to adopt standard states for hydrogen and for carbon, and it would be desirable to agree upon values for the heats of formation of water and of carbon dioxide, such international values being subject to revision when desirable, in the same manner as the atomic weight table. Discussion of these questions is, however, outside the scope of this paper. For the combustion of substances which contain elements other than carbon, hydro- gen, and oxygen it is likewise necessary to adopt standard states for these elements and for their products of combustion. These cases will not, however, be discussed in the present paper, which will be confined to materials the composition of which can be expressed by the formula C»HbO«. Washburn] Standard States for Bomb Calorimetry 531 VI. COMPARISON OF THE ACTUAL BOMB PROCESS WITH THAT DEFINED BY THE PROPOSED STANDARD STATES As contrasted with the actual bomb process, the nature of which has already been discussed in detail, the analogous process defined by the proposed standard states consists solely in the reaction of unit quantity of the substance with an equivalent amount of pure oxygen gas, both under a pressure of 1 atmosphere and at the temperature tH , to produce pure carbon dioxide gas and pure liquid water, both under a pressure of 1 atmosphere and at the same temperature tH , the reaction taking place without the production of any external work. This process is not experimentally realizable. The intrinsic energy change associated with this process is, however, a definite and useful thermod3mamic quantity and is equal to — AUn , the decrease in intrinsic energy for the following reaction at tH° C : C aHbOc (3 ) or (i), i atrn. + ( a+ - )02(g), 1 atm. — aC02 (g), i atm. + f)ll2^(l)) 1 atm. (2) From this quantity, the heat, Qp , of the isobaric reaction at 1 atmos- phere can be readily calculated by adding the appropriate work quantity. The quantity, — AU-r, of course, differs but slightly from —AUB , the heat of the actual bomb process, and for many purposes the difference is of no importance. Indeed, only a few years ago the two were calorimetrically indistinguishable. To-day, however, this is not always the case. The difference, while small, may be many times the uncertainty in determining the heat of the bomb process and may amount to from a few hundredths of 1 per cent up to several tenths of a per cent of this value, depending upon the particular substance and the experimental conditions of the measurement. To obtain from the heat, — AUB , of the bomb process, the energy quantity, — AUr, for the process defined by the standard states requires the computation of certain " corrections" the nature of which we will now proceed to discuss. VII. THE TOTAL ENERGY OF COMBUSTION DEFINED BY THE PROPOSED STANDARD STATES Since the quantity ATJ for any process is completely defined by the initial and final states of the system, the proposed standard states do not define any particular path. In order to arrive at the value of A C7R we are therefore at liberty to make use of any desired imaginary process as long as it does nor violate the first law of thermodynamics. The process employed for this purpose should obviously contain the actual bomb process as one of its steps. Its other steps should be selected on the basis of the availability of the necessary data for com- puting the AU terms for these steps. A review of the available data for various alternative processes indicates that the following process will yield trustworthy values of the AU terms. The process is isothermal at the temperature ts . 534 Bureau of Standards Journal of Research [Voi.io VIII. DEFINITIONS OF SOME AUXILIARY QUANTITIES 1. THE INITIAL SYSTEM Given a bomb of volume V liters in which are placed n0i moles of oxygen and n g. f. w. of the substance or mixture having the compo- sition expressed by the formula CaHbOc and whose heat of combus- tion in the bomb is —AUB energy units per g. f. w. There are also placed in the bomb mw grams of water, a quantity sufficient to saturate the oxygen. The gram-formula-weight (g. f. w.) of the substance is evidently 12a+1.0078b + 16c (19) The number of moles of oxygen required for the combustion is nT=U+^^\n + 4> (20) where <j> is the oxygen consumed in producing nongaseous products other than C02 and H20; for example, HN03 , Fe2 3 , etc. The initial oxygen pressure in the bomb will be = n0lBT{l-ix0i'p l ) , 21 x in which mo 3Pi is a small correction term calculable from the equation of state. At 20° C. JU02 = 0.03732 - 0.052256p (22) For 2 pressures between 20 and 40 atmospheres (the range ordi- narily met with in bomb calorimetry) this relation may be replaced by the following approximate but sufficiently exact equation: Mo 2 =0.03664 (23) 2. THE FINAL SYSTEM After the combustion, the bomb will contain (n03— n r ) moles of 2 , Wcoa ( = an) moles of C02 , and (}i bn+ ){% mw ) moles of water. Part of the water will be in the gaseous state and part of the 2 and C02 in the dissolved state in the liquid water. As explained in steps 3 and 5 of the preceding section, all of the water and the dissolved C02 are removed from the bomb leaving a gas phase which contains nM = n0i-nr + nco2— nD (24) moles of gas, where nD is the number of moles of dissolved C02 re- moved with the liquid water. The dissolved 2 may be neglected in equation (24). nM = n 0i -( ± J n-nD -<f> (25) The mole fraction of the C02 in the gas phase will be an— nD x = n03-(—j--fn— nD -<t> (26) Washburn] StandardgStates jor Bomb Calorimetry 535 The pressure of the mixture will be y.- tt-gr^-MMfr) (27) At 20° C, juM is given by the equation 8 /W/*o,= 1 + 3.21a?(l + 1.33a;) (28) IX. CORRECTION FOR DISSOLVED CARBON DIOXIDE After the combustion, the bomb contains mw + 9bn g of water. Of this, the amount (see Appendix I, equation (119), p. 556). VCW = 0M73V+ (0.0455 + 0.0328:r)2>2 V, g (29) is in the vapor state at 20° C. Since this will in general be only about 1 per cent of the total, we may substitute the average values p2 = 30 atm., and a; = 0.15, and the above equation may, with sufficient ac- curacy for our present purposes, be written F6r w,= F[0.0173+(0.0455 + 0.0328X 0.15)30] = 0.02 V, g (30) and the amount of liquid water will be mw + 9bn- 0.02V, g (31) This amount of liquid water will be saturated with C02 at the partial pressure p2x. For the range of p2x values encountered in bomb calorimetry (1 to 8 atmospheres) it will be sufficiently accurate, for the purpose of correcting for dissolved C02 , to assume that the solubility is propor- tional to the partial pressure, using a proportionality constant com- puted from the solubility of C02 at about 7 atmospheres. (See fig. 1.) With this assumption the number, SC 02, of moles of dissolved C02 per cm3 of liquid water will be SC o2= 0.0^82p2x, M/cm3 at 20° C. (32) The total number, nD , of moles of dissolved C02 will therefore be nD = 0.030382p2x(mw + 9bn - 0.02 V) (33) Now the total energy of vaporization of C02 from its aqueous solu- tion at 20° C. to produce pure C02 gas at 1 atmosphere is 181 liter-atm. per mole. 9 For the nD moles of dissolved C02 we have, therefore, AUD = 181 X nD , liter-atm. (34) Since, as will appear later, this term is a small part of the total cor- rection, we may write with sufficient accuracy p2x = &7iET(l - hmP2)/V (35) From equations (23) and (28) we have for z = 0.15, /zM = 10" 3 . Hence for 20° C. and p2 = 30 atm. p2x = an X 24.05 (1 - 10"3 X30)/F (36) « From unpublished measurements in this laboratory by the method described by Washburn, B. S. Jour. Research, vol. 9, p. 271, 1932. • Computed from the temperature coefficient of the solubility of COi in water. 536 Bureau of Standards Journal oj Research [Vol. 10 and equation (33) becomes Wd = 0.089 a,n(ma + 9bn- 0.02V)/V or putting ra«, = l g, 9bn = 0.55 g, and V= 1/3 liter nD = (approx.), 0.039a7i(l + 0.55-0)/0.33 = 0.004a^ (approx.) for average calorimetric conditions. The discussion of this correction is continued on page 541. (37) (38) (39) Figure 1. — Solubility of C02 in H2 as a function of the pressure at 20° C. The points indicated are taken from International Critical Tables, vol. 3, p. 260. The lineA is determined by the values at and 1 atmosphere. The locus of the curve B is estimated. The line C is drawn so as to pass through and the point for 7 atmospheres on the curve B. The equation of this line is that shown. X. THE ENERGY CONTENT OF THE GASES AS A FUNCTION OF THE PRESSURE Direct calorimetric measurements of AC/] p i for oxygen and for mixtures of oxygen with carbon dioxide have been made by Rossini and Frandsen. 10 Their results reduced to 20° C give and AU0i] p o = 0.0663p, liter-atm./mole at 20° C. (40) AUM ] P = 0.0663 (l + h)p = 0M6S[l + 1.70x(l+x)]p, liter- uu atm./mole at 20° C. ^ 1; Both relations are valid for pressures up to 45 atmospheres and for values of x up to 0.4. For C02 between and 1 atmosphere, we shall employ the relation - A£/co2] po = 0.287^, liter-atm./mole at 20° C. (42) which is derived from the Beattie-Bridgeman equation of state. 11 >o Rossini, F. D., and Frandsen, M., B. S. Jour. Research, vol. 9, p. 745, 1932. 11 See Washburn, B. S. Jour. Research, vol. 9, p. 522, 1932. It is of interest tonote that equation (41) above, when extrapolated to z=»l gives— A(7co»] p =0.291p. Washburn] Standard States for Bomb Calorimetry 539 and the above equation becomes 10- 3^3.21 (1 + 1.33a;) X 0.664Ap_ x Pi 1+x 10-3^3.21 (1 + 1.33a;) X 0.664 + (l-6.64X 10-^)(^-C +^); + (1 - 6.64 X 10-VO Ckz2?+«£±*V (58) or (59) — Aj) _ IT Pi 1 + 7T For — we have (see equation (37)), for 20° C. ^=0 ' (m. +OTm-OW an V This gives 7r = JlO- 3 pi2.07(l + 1.33a;) + (1- 6.64 X 10-%)^^+ 4>/&n + 0.089(m„+.9bw-0.02F)/Fn (61) If <£ = 0, the second term in the expression for ir is positive except when 2c>b. r and hence also Ap may therefore be either positive or negative according to circumstances. The term containing (mw + 9bn — Q.02V)/V is usually very small, almost negligible in fact. For V=% liter, and mw =lg, it ordinarily lies within the range (42 ± 7) X 10~4 , and in many cases it will suffice to assume this value for it. In certain particular cases, however, this term is the principal one in the expression for t. b — 2c Thus for the case 12 x = 0.149, ^i = 22 atm., = 0, and —r— = — 1/18, we have w = x[0 + 0.039(mw + 9bn-0.02V)/V} = 0.149 X (42 ± 7) X 10" 4 = 0.0363 ± 0.03 1 (62) and - Ap/pi = tt/(1 + w) = 0.0363 ± 0.03 1 (63) Aj) This is neghgibly small. In fact any value of— within the Umits ±0.001, may be taken as zero. For most cases an approximate form of equiation (61) will be sufficiently exact. This can be obtained by substituting the average values 0.15 for x and 42 X 10~4 for the last term, giving 7r = aJ~2.07X 10-^(1 + 1.33 X 0.15) +-^p + 0/an + 42 X 10~ 41 ^ = ^2.5X10-^1 +^—+ 0/an + 42XlO- 4l (64) 12 For example, myristicinic acid, CjHjOs, or dimethoxydihydroxybenzoic acid, CiHioO*. 540 Bureau of Standards Journal oj Research [Voi.w XIII. THE NEGLIGIBLE ENERGY QUANTITIES 1. THE ENERGY CONTENT OF THE WATER (a) THE CHANGE IN THE ENERGY CONTENT OF THE WATER VAPOR As a result of the bomb reaction the amount of water present as vapor in the final system at tH° is greater than that present in the ini- tial system at this temperature. The increase, ACW , in the concen- tration of water vapor for ##=20° C, is given by the equation (see equation (122), Appendix I) AC r „ = w[o.O,34-0.0,05,(l-^)] > g/liter (65) This increase is accompanied by the absorption of VACW X 22.82 liter-atm. of heat energy, 22.82 liter-atm. being the total energy of vaporization of 1 g of H20. We have, therefore, AC7„,V8P =22.82F^2a:r0.0334-0.03055 ('l -^f)j, liter-atm. (66) An extreme case would be the following: V=}i liter, ^2 = 45 atm., x= 0.3 and— Ap = 2 atm., and this would give A UJ**- = 0.043 liter-atm. = 1 .04 cal. (67) This will rarely amount to more than 0.01 per cent of the heat of the bomb process and will usually be negligible. Since it is opposite in sign to that arising from the quantity AUD , for the dissolved C02 , a partial compensation will occur and the expressions for the two correc- tions may advantageously be combined into a single expression repre- senting the algebraic sum of the two effects. This will be done in section 2, below. (b) THE CHANGE IN THE ENERGY CONTENT OF THE LIQUID WATER The increase, A Z7«, l,q , in the total energy content of liquid water as a function of the pressure upon it is displayed in graphic form by Bridgeman. 13 At 20° C.and for the pressure range encountered in bomb calorimetry this increase is expressed with sufficient accuracy by the equation : AUw U(l - = - 54 X 10" 6P, liter-atm./g (68) In the process defined by the proposed standard states, mw grams of liquid water are compressed from 1 atmosphere to (pi+pj) atmos- phere and (m„, + 9b7i) grams are decompressed from (jp% + pw ) to 1 atmosphere. Since we are dealing with a very small energy quantity, we will write pi + pw = p2 + Pa = )'i (pi + P2 + 2pv>) = P. and the net change in the energy content of the liquid water becomes -AC7u, M<1 - = 9b7iX54Xl0- 6P, liter-atm. (69) An extreme case would berep resented by 9bn = 2g and P = 45 atmospheres for which case -AEV lq - = 0.005 liter-atm. = 0.1 cal. (70) " Bridgeman, P. W., rroc. Am. Acad., vol. 48, p. 348, 1912. Washburn] Standard States for Bomb Calorimetry 541 a wholly negligible quantity. Hence for all practical purposes the quantity AUW + AU'W is equal to AC7«, vap - as given by equation (66) above. 2. COMBINED ENERGY CORRECTIONS FOR WATER VAPOR AND FOR DISSOLVED CARBON DIOXIDE The correction for dissolved C02 is (see equations (33) and (34)) -AUD = -181X0.030382^2^(^w + 9bn-0.02F), liter-atm. (71) and that for the excess water vapor in the final system is (equation (66)) - A Uj">- = 22.82 Vp2x f"o.0334 - 0.03055 (l - r^Yl liter-atm. (72) For p2x we have at 20° C. (see equation (36)) p2x= 2S. 3SinlVf approx. (73) The sum of equations (71) and (72) combined with (73) and divided by — O.Oln ATJB will give us the net correction in per cent arising from the two effects in question. We thus obtain the follow- ing relation : Per cent corr . (AUD + A U„ + A U'w ) = q^/ I[-• + 0.0069 (m " +9M -0.0012 5^1 = 'j=gfL r_ .96 +^-±^-^Z^2l (74)-AUb/8l L V an J v ' The quantity — AUb/b> } the heat of combustion per gram-atom of carbon, will have its minimum value in the case of oxalic acid, say 1,200 liter-atm., and its maximum value for hydrocarbons, say 7,500 liter-atm. The net correction given by equation (74) will therefore vary in practice (for V=l/3 liter and mw=l g) only between about — 0.06 and —0.008 per cent. In the majority of cases it will be found to lie between —0.01 and —0.03 per cent. 3. THE ENERGY CONTENT OF THE DISSOLVED OXYGEN At 20° C. the solubility of 2 in water is approximately u So2 = 1.2 X 10~ 6 pO2 , mole/cm3 (75) The amount of 2 dissolved in the initial water will therefore be ft£o2 = !-2x 10-*m ttpi, moles (76) and that in solution in the final water will be n , jDO2 =1.2Xl0- 6 (m u, + 9bn)2?2 (l-a:) (77) " Int. Crit. Tables, vol. 3, p. 257 and Frolich, Tauch, Hogan and Peer, Ind. Eng. Chem., vol. 23, p. 549, 1931. 544 Bureau of Standards Journal of Research [Voi.io Because of the small magnitude of this correction and the small temperature coefficient of AUB , the value of AUB used in equation (81) may be the value for any room temperature and the correction given by the equation may for the same reasons be directly applied to the value of ATJB at any room temperature, except possibly in certain extreme cases. In deriving this relation we have neglected AUS , the energy of compression of the substance plus the energy content of its vapor. This energy quantity is at present negligible and is in any case specific for each substance and is not therefore included in equation (81). It may, however, become significant for some substances, if the accuracy of the determination of —AUB is increased. (See Table 1.) If the sample is inclosed in a sealed glass capsule which it does not fill, the quantity; AUs becomes zero but in its place we would have the probably negligible energy of compression of the capsule. We have also neglected the heat of adsorption of water by the sample. When the perfectly dry weighed sample is placed in the bomb and the latter closed, the sample is in contact with saturated water vapor. It immediately proceeds to adsorb water and to evolve or absorb heat. The amount of water adsorbed depends upon the nature of the sample, the surface exposed and the time of contact with the water vapor. Presumably the amount thus adsorbed has become substantially constant by the time the calorimeter fore period has been determined. The charge is now ignited and the adsorbed water appears as liquid water in the final system. The observed heat of combustion will therefore be less than the true value by the amount of heat required to convert this adsorbed water into liquid water. Since the correction here involved is specific for the substance burned and varies with the surface exposed and the con- ditions of the experiment, it can not be provided for in equation (81). For hygroscopic substances it might be very appreciable and difficult to determine and correct for. Such substances should therefore be inclosed in a suitable capsule. 2. AN APPROXIMATE CORRECTION EQUATION The calculation of the total correction by means of equation (81) is somewhat time consuming and it is desirable for many purposes to have available a simpler equation for rapid calculation. Such an equation can be obtained, with some loss of accuracy and generality, by introducing certain approximations into equation (81) and taking advantage of certain fortuitous compensations for typical calori- metric conditions. In this way the following approximate equation may be obtained, for pi in atmosphere, for — AUB in kg-cal.w per g. f. w., for a bomb volume of % liter, for = 0, and for mw = 1 g. (82)(Per cent corr.) Total --^jL-^- 1 + 1.1 -j— -_J approx. This approximate equation will in general give a value for the (per cent corr.) Total which is accurate within 15 per cent of itself, a degree of accuracy which is sufficient for correcting most of the now- existing data of bomb calorimetry. Washburn] Standard States for Bomb Calorimetry 545 XV. THE MAGNITUDE OF THE CORRECTION IN RELATION TO THE TYPE OF SUBSTANCE BURNED It will be noted that of the terms in the parenthesis of equation (82), only the second depends upon the nature of the substance burned. The extreme values possible for this term are: (1), 1.1X0.5 for all hydrocarbons and mixtures of hydrocarbons whose net composition is approximately expressed by the empirical formula CaH2a ; and (2), 1.1 ( — 0.75) for oxalic acid; with the value zero for all carbohydrates, carbon itself, and certain aldelrydes. The magnitude of the correction for these three types of substances is illustrated in Table 2 for three different values of the initial 2 - pressure. Table 2. — Illustrating the magnitude of the correction for reducing the data of bomb calorimetry to the -proposed standard states Percent corr. Approx.-, -A(7fl)/aL - 1+ 1 - 1-ir Pi J b-2c 4a (-AUb) a Per cent correction for Type of substance pi=20 pi= 30 pi=40 atm. Hydrocarbons of the types CaH „ , - «._ 0.5 .0 -.75 kg-cal.is 150 110 30 -0. 022 - .06o - .38 -0. 03i - .08: - .5fl -0.04o Carbohydrates, certain aldehydes, etc - - .Hi Oxalic acid - _. - .75 XVI. COMPUTATION OF THE CORRECTION 1. GENERAL REMARKS In computing the correction for reduction to the standard states, it is necessary to reduce each experimental value separately; that is, before averaging, unless the experimental conditions (values of plf m, and mw ) are substantially the same for all experiments. By making these conditions the same (see Table 5), the values of —AUB may be averaged first and the correction applied to the average, thus greatly reducing the amount of computation required. Before averaging, each observed value of —AUB must first be cor- rected to the same temperature, tH . The method of making this correction is discussed below. (See Sec. XVIII.) A convenient computation form is illustrated by the chart of Table 3. The first horizontal section of the chart provides for the entry of the necessary numerical values of the initial conditions. The second section provides similarly for the final conditions. The third section contains the steps in the calculation and the final result to which they lead. In case the sample is a pure substance, the coefficients a, b, and c in the empirical formula, CaHbOc , will be known. If the sample is a mixture, the exact composition of which is unknown, this composition must first be determined. This may be accomplished in all cases by making a combustion analysis, from the results of which an em- pirical formula for the mixture can be computed. If, as is frequently the case, the mixture is made up of known amounts of two constit- uents, of known compositions, then the composition of the mixture and its empirical formula are calculable. (See Appendix II.) 546 Bureau of Standards Journal oj Research [Vol. 10 d "7; p, oo 5; 'w <i 8 £ *w Ol g o c eo + <W ft, •« O S? £ O 5, 5" 3^ + o £ "*-> 51 g cs h _ - O ^ L* >*s -al "f-, <^j ft. fi s o 1 a + 3 ei ^ 5, 1 £ /" s -tSIH O + u r-l|H 3 ^—"'^ PU s» 1 5 B •-., 3 I 09 5j <^ 9 rH --1 N«i II 1 u OT> o W u d w Q < >-i H a O* S| ooc —ie^o6 ^ -h o (OHfll I- OlOh©NO CN CM MHOOO .-I OOOi d i ' i" \i in to © i- io io m ^j'HCOtOOOMNMiiCWHOOOOOO '+ I II I I MCOHONONrt I I v» oo i_^-< On ' ' ' w "jg ' O «* K-l 'O rH t-H l~. (Joi ih ' ' VJ JJtOo Tlco Ooi • o o eoo2 ra-r ll?2 3 8 03 a> [s o II i^' fill Ss^ IS S,k hNCO^u) suo!)i|)uo.) |ii|!i'| 05 1—i -* a ®ooo S8S §883 oo co NH O O Ol HHI >0 O O O O O ( 00 oo oo to OOO <M O O i-H CN +^ I T3 ~ I 9 OCO •& <N t^ l^ h- >o o o o) hhu:OOO o oooo CO IN SiOi*<00 3 33333 W WHWWW L3 i 7 5S e R X || V «oS r £i^ll i Q° II II l5 -?^ ^ es co »»« io < suoi-jipuoo lumj MOHIOONOH I I I I I rH«O00CSlO<NOO^H e>jo<-H<-io«oO'-i'-i roa^MMOmMo C-)OTt<r-<0'-lOOO I I I I io •>*< »-i oo ooHOJIOONICOJ) *) fflCrtiOOU K5 00 OShO»-(Ot-OOO ° ' \ ' 1*7 ' \ r JT51 -S; +7^2 H I 9 a? S ea r 4--5.V >. a) oo aq || .O (5,T>| M |_ t;'- 7 o o R.C I ° ° H ^ J-.' I ' N oooH --1 oj ai uOJiO^INWJ'Wtfl uoiiwjndaioo Washburn] Standard States for Bomb Calorimeiry 549 The effect on x.—Since the amount of N 2 in the bomb before ignition is almost never recorded by the investigator and is usually not known to him, we shall make use of the approximate equivalence of the molecular weights and energy contents of N 2 and 2 and shall assume- that the N2 , except so far as it combines with the 2 , can be treated as so much excess 2 . Its only effect upon the quantity A£7gas will therefore reside in its effect upon x. A corresponding effect will also be produced by the combustion of any iron wire. These two effects will combine and equation (26) will read as follows: an — nD /b-2c\ n°2 ~ \ — ^r~ ) n ~ riD ~ (t>N ~ <t)] (85) in which <j>N is given by and #Fe 7 0n = t (per cent corr. N)( — A?7R)n/l,455 (86) 3 4>Fe = 2 (Per cent corr-Fe) (~ AZ7R) n/19,000 (87) Example 0n = t (per cent corr. N) ( ) an/1,455 (88) Put -^—^=100 and (per cent corr.N) =0.2 Hence <£N = 0.024an 3 0Fe = 2 (Per cent eorr.pe) X 100an/l 9,000 = 0.0079 (per cent corr. Fe ) an (89) Put (per cent corr. Fe ) =0.05. Hence <£Fe = 0.0 34a7i. Total 0N + 0Fe = O.O244an (90) which may be compared with 7i£> = 0.004an and n02 <3an, in equation It is obvious that the combustion of the iron wire in the amount ordinarily employed is wholly negligible as regards its effect upon x. The effect of the nitrogen on x is appreciable, but also probably negligible for all practical purposes. Since, however, its effect can be readily computed all doubt can be eliminated by computing the I quantity x by means of equation (85). _ Effect on AUW V&P \—In the final system in the bomb the concentra- tion of water vapor is (see equation (119), Appendix I). C'w = 0.0173 + (0.0 455 + 0.0328x)p2 = 0.0173 + 0.0455 X 30 + 0.0328p2z = 0.019 + 0.0328X24an/F - (0.019 + 0.0069an/V), g/liter (91) 550 Bureau of Standards Journal of Research i w. w If now the water contains HN03 at such concentration (NHNOs mole/liter) that its partial vapor pressure is reduced by the fractional amount —— , its concentration in the vapor will be reduced by the amount -AC^_W = -££_„= 0>03iyHNOa (92) and - AC'„ = 0.03AWo 3 (0.019 + 0.0069an/TO = [0-03 (per cent corr. N)(- AE7R)?i/l,455] (0.019 + 0.0069an/F) (mw + 9bn)XlQ- s 0.0206(per cent corr. N) (- AUn)n(0M9 + 0.00Q9an/V) „.' , _. -~ (m„ + 9bn) ' g/llter (93) The energy required to evaporate this water is therefore 22.82VAC'w , liter-atm. or vanN , 0.0114F(per cent corr. N)(- AUR)n(0.019 + 0M69i\n/V) A[AUw *)_ mw + 9b7i (94) or 100A(A^vap >) _ 1.14 V (per cent corr. N ) (0.019 + 0.0069ayi/F) (q . (-AUn)n mw + 9bn { } In practice a,n/V usually varies between 0.1 and 0.3, whence 100A(AUwvap ') ^ 1.1 4 V (per cent corr. N ) (0.0203 ± O.O38) ( — AUR)n mw + 9bn = (0.023 ± 0.039) V (per cent corr. N) mw + 9bn For V— 1/3 liter, (per cent corr. N ) = 0.2 and mw + 9bn= 1.5 this gives 0.001 per cent, a wholly negligible quantity. Effect on the amount of dissolved carbon dioxide.—There are no data on the solubility of carbon dioxide in aqueous solutions of nitric acid for C02-pressures of the magnitudes encountered in bomb calorimetry. For pressures in the neighborhood of 1 atmosphere the data in International Critical Tables (vol. 3, p. 279) indicate that the solubility of C02 in 1/4 iVHN03 at 25°C. is about 0.9 per cent greater than in pure water. Such a difference would of course be negligible, since the total C02 correction will never exceed 0.06 per cent. In view, however, of the various corrections and uncertainties introduced by the presence of nitrogen in the bomb it is obvious that in bomb calorimetry of the highest accuracy nitrogen-free oxygen should be employed and the air should be swept out of the bomb. XVIII. REDUCTION OF BOMB CALORIMETRIC DATA TO A COMMON TEMPERATURE In the actual calorimetric determination, the calorimeter and its contents undergo a rise in temperature AtB = t2 -t l (96) Washburn) Standard States jor Bomb Calorimetry 551 as a result of the bomb reaction. In order to obtain, from this ob- served temperature rise, the heat of the bomb process at some con- stant known temperature, tH , it is necessary to know the effective heat capacity, sf , of the initial system and/or the effective heat capacity, sF , of the final system. The effective heat capacity of any system, substance or material is the quantity of heat which must be added thereto in order to raise its temperature from t to t', divided by If sB is the effective heat capacity of the calorimetric system itself, then the heat evolved by the bomb process at the constant tempera- ture, tH°, will be -AUBn = sB (t2 -t 1 )+s 1 (tH -t l )+sF (t2 -tH) (97) The temperature, tH, may be any desired value whatsoever. If, as is usually the case (although not at all necessary), it is made equal to *i or to t2) one of the terms in the above expression reduces to zero. For use in equation (97) above, the following values (average for 25° C.) for sT and sF are sufficiently accurate for any value of AtB within the region of room temperatures. s 7 =5.0l7io2 +0.995ma , + 0.7V r +Smcp + 0.108mPe , cal.i5 deg.-'C (98) sF =5.01no2 + 0.995m tt , + 0.7V r +0.158mpo + n[(1.77i + 0.0112y2)a + 7.74b + 2.5c] - 347iHNo3 , cal. 15 deg.^C (99) in which no2 = g-moles of 2 in the bomb initially. mw = g of H2 in the bomb initially. V— volume of bomb in liter. 2racp = the total heat capacity, at constant pressure, of the car- bonaceous material or materials burned, cal.i5 deg." 1 C. n = the number of gram-formula-weights of carbonaceous material burned, the total composition of which is expressed by the empirical formula CaHbO c . P2 = Pi + Ap = final pressure in the bomb. raFe = g of iron wire burned to Fe2 3 . %no3 = gram-formula-weights of HN0 3 formed. = (per cent corr.) Hno3 ( — AZ7sWl,455. (Per cent corr.)HN03 = per cent correction applied to —AUB for the heat of formation of a small amount of HN03 . In deriving equations (98) and (99) account has been taken of the variation of the heat capacity of the gases and of the liquid water with pressure. The term 0.7V takes care of the latent heat of vaporization of the water and the term 34tiHN03 of the heat capacity of the dissolved HN03 . XIX. THE TEMPERATURE COEFFICIENT OF THE HEAT OF COMBUSTION Using the specific heat data given in International Critical Tables, the following equation, valid strictly over the interval between 20° and 30° C, but sufficiently accurate for all practical purposes over any temperature interval in the region of room temperatures, can be readily derived 554 Bureau of Standards Journal oj Research [Vol. 10 conditions pi = 30 atm., m/V=S g/liter, and m tD/V=3 g/liter (see Table 5.) So corrected 18 the values become Int. kj. per g Fischer and Wrede 26. 440 Dickinson 26. 439 Jaeger-v. Steinwehr 26. 427 Roth, Doepke, Banse 26.430 The agreement is now not so good, but the average of all four is still 26.434 with an apparent uncertainty of not more than 7 joules. There is, therefore, no reason at the present time for changing the value adopted by the International Chemical Union, provided the standard conditions assumed above are also adopted. There is, however, need for new determinations of this important quantity under more exactly controlled conditions than have prevailed here- tofore. If now we assume the value 26.434 kj/g for — AUB when p x = 30 atm., m/V=3 g/liter, and mw/V=S g/liter, then the value of — AC7R for the pure reaction C7H6 2 (8), 1 atm. + 7}2 O2 (g), 1 atm. - 7CQ2 (g), 1 atm. + 3H2 (i), i atm. will be 0.08 per cent less (see Table 3) or 26.413 X 122.05 kj. per gram- formula-weight . It is, of course, this latter value (or rather the corresponding one for — QP} the heat of the pure reaction under a constant pressure of 1 atmosphere) which would appear for benzoic acid in tables of heats of combustion of chemical substances. The former value —AUB is applicable only to the use of benzoic acid as a standardizing substance for bomb calorimetry. APPENDIX I. CONCENTRATION OF SATURATED WATER VAPOR IN GASES AT VARIOUS PRESSURES Within the pressure range involved in bomb calorimetry the con- centration of saturated water vapor varies with the temperature according to the equation logio<7„ =y + / (101) A is a constant characteristic of water and independent of the com- position and pressure of the gas phase. (See fig. 2.) / is a function of the composition and pressure of the gas phase. For water at 25° C. and under its own vapor pressure, we have <7„=^= 0.02302 g/Hter (102) and for 70° (7* = 0.1967 g/liter (103) Hence for equation (101) we find log10a.(g/iiter) = ~ 2 ^ 17 + (/= 5.465) (104) 18 An additional correction for temperature has been applied to the Dickinson value since the writer is advised by Doctor Dickinson that this value is for 25° C. (approx.) instead of 20° C. Washburn] Standard States jor Bomb Calorimetry 555 Within the range of pressures and gas compositions met with in bomb calorimetry the concentration of saturated water vapor varies with the pressure according to the equation CtD = C + aP (105) Co is a temperature function only and a varies with the nature of the gas phase. Combining this equation with the preceding one we have log10 «7 +aP) = 2,117 + 1 (106) For N2 at 50° C. Bartlett 19 found (7 + aP= 0.095 g/liter for P = 50 atmospheres I^?r2r <z Figure 2. — Temperature variation of the concentration of saturated water vapor in the presence of various gases Hence 7=5.530 and at 20° C. and 50 atmospheres log10 ((7o+aP) =-2|^+ 5.530 log10C = ^|^p4-5.465 ft+«P= 1.161CS, C = 0.01728 a = 0.0456 (107) (108) (109) (HO) (HI) (112) w F. P. Bartlett, J. Am. Chem. Soc, vol. 49, p. 66, 1927. 556 Bureau of Standards Journal of Research [Voi.io and Cw = (0.017284-0.0456P), g/liter for P in atm. (113) For air at 49.9° C. Pollitzer and Strebel 20 found <7 + aP = 0.099, g/liter for P = 72.5 atm. accuracy about 1 per cent. These data yield Cw = (0.01728 + 0.0 450 P), g/liter at 20° C. (114) For C02 at 49.9° C, Pollitzer and Strebel found C + aP = 0.1443, g/liter for P = 38.72 atm. Hence at 20° C„= (0.01728 + 0.0334P), g/liter (115) Bartlett considers that the experimental data of Pollitzer and Strebel are somewhat too high on account of certain errors in experimental technic. We notice that the a from Bartlett's results for N2 is 0.0456 while the data of Pollitzer and Strebel yield 0.0450 for air. For our present purposes we require the value of a for 2 , for which no experimental data are available. We shall, therefore, write for 2 at 20° Cw = 0.0173 + 0.0455P (116) and for C02 at 20° Cw = 0.0173 + 0.0834P (117) Bartlett found that Cw for mixtures of H2 and N2 could be calcu- lated from the values for the pure gases by the law of mixtures. We shall, therefore, write for mixtures of 2 and C02 at 20°. Cw = C +laOi(l-x) + aCox]P (118) = 0.0173 4- (0.0455 + 0.0328a:)P (119) The quantity CQ can be obtained from the expression logioC = ~ 2 j! 13 -f 5.453 (120) which is valid for room temperatures, 16° to 30°. If Cw is the value for 2 at the pressure p x and C'w the value for an 2 — C02 mixture at the pressure 2>2 = Pi + Ap, then AC„ = C'to- Cw = acoiPtX- <*o,(- Ap + xp2) (121) = ^[o.0334-0.03056(^l-^], g/liter (122) 20 F. Pollitzer and E. Strebel, Z. physik. Chem., vol. 110, p. 708, 1924.