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

The thermodynamics of bomb calorimetry, a method used to determine the heat of combustion of substances. the importance of using pure substances, standard oxygen, and standard conditions for accurate results. It also explains the concept of standard states and the need to adopt standard states for various elements and their products of combustion.

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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 afunction 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 amore 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 1normal atmosphere for both re-
actants and products.
From this value the more generally useful quantity, Qv,the heat of the pure
reaction at aconstant pressure of 1atmosphere is readily calculable. An equa-
tion for calculating the correction is given and illustrated by examples. The
magnitude of the correction varies from afew hundredths of 1per cent up to
several tenths of 1per 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
acalorimeter, aparticular 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 afunction 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
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RP

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

  1. Combined energy corrections for water vapor and for dis-

solved carbon dioxide 541

525

Bureau

of

Standards Journal of

Research [Voi.io

XIII.

The negligible energy quantities—Continued.

page

  1. The energy content of the dissolved oxygen 541
  2. 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

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, C a

H b

O c

.

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.

m

w

Number of grams of water placed in the bomb.

n Number of moles or of g. f. w. of substance burned.

n

D

Number of moles of C 2

in solution in the water in the bomb.

n

M

Number of moles of gas in the bomb after the combustion.

n

o

2

Number of moles of O2 in the bomb before the combustion.

P Pressure; per cent by weight.

p

Pressure or partial pressure.

px

Pressure in the bomb

before the combustion.

p

2

Pressure in the bomb

after the combustion.

p

w

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.

t

H

Standard temperature °C.

U Total

or intrinsic energy content.

528 Bureau of

Standards Journal of

Research [Voi.w

obvious from the above picture that the process which takes place

within the bomb does not start from

a definite and standardized

initial condition nor end with a definite final condition, either chemi-

cally or physically.

Now the initial condition should obviously be

(1)

pure substance or

definite material in a definite phase state or states (solid

and/or

liquid); and

(2)

pure gaseous oxygen at some standard concentra-

tion; and the final condition should be pure gaseous carbon dioxide

and pure liquid water each under some standard pressure. Further-

more, if the excess oxygen undergoes,

as it does, a

change in condition

which is accompanied by a heat effect, due correction must be made

therefor, so that only the oxygen consumed will be involved in the

process to which the final heat quantity belongs.

At present the bomb calorimeter appears to have no serious com-

petitor for the precise determination of heat of combustion of organic

solids and liquids of low volatility, and the exact standardization of

conditions as here proposed is necessary, if one is to

take full advan-

tage of the highest precision attainable.

The purpose of this discussion is to

propose suitable standard states

and to describe methods

by

which the heat of the bomb process may

be corrected so as to yield the total or

intrinsic energy change for the

reaction defined by these standard states. Without such standardiza-

tion it is impossible to obtain

from the existent precise data of bomb

calorimetry the frequently

wanted

quantity ordinarily called the "heat

of the reaction at constant pressure."

This is

to-day almost univer-

sally calculated

by

the simple addition of a

AnRT quantity to the

result

obtained with the calorimetric bomb, a

procedure which

is

thermodynamically inexact

when

applied to

calorimetric data of

high precision.

III. CALORIMETRY AND THE FIRST LAW OF

THERMO-

DYNAMICS

According

to the first law of

thermodynamics

U

2

-U 1

= AU=Q-W (1)

U

2

(or U

u

respectively)

is the "total,"

"internal," or "intrinsic"

energy of any system in the

state

2 (or 1,

respectively). AU is the

increase in

this intrinsic energy

which takes place when the system

changes from state 1

to

state 2

by

any path.

Q

is the quantity of

heat

absorbed by

the

system

during the process and W is the work

done

by

the system on the surroundings. The quantity A U is inde-

pendent of the path and is completely defined by the initial and final

states. For isothermal processes three cases are, for practical or

conventional reasons of special interest, as follows:

Case 1. W=0. For this case

QW=

= AU; that is, the increase

in intrinsic energy is equal to the heat absorbed. The heat of a

process (any process whatsoever) under these conditions is usually

designated by

Qv

and is commonly called the "heat at constant

volume." This designation is somewhat unfortunate

for three

reasons: (1)

Because in general a constant-volume

process is not

necessarily a zero-work process (for example, when

accompanied by

external electrical work); (2)

because a

zero-work process is notneces-

sarily a constant-volume process (for

example, when a gas expands

Washburn] Standard States for Bomb Calorimetry

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

  1. 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

Q H 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= W m&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

Q v of the bomb process in

such a way as to obtain the quantity A U 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 C a

H

b O c

. 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

m

w

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 n 02 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 n

02 /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, t H ,

and divided

by n so as to obtain the quantity —AU B 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 t

H

  • 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 a Hb0 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.

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

t

H

,

to produce pure carbon dioxide

gas and

pure

liquid water, both

under a pressure of 1 atmosphere and

at the same temperature t

H

,

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

AU

n

,

the decrease in

intrinsic energy for the following reaction at t

H

°

C

C

a

H

b

O

c ( 3 ) or (i), i atrn.

(

a+ - )02(g), 1 atm.

— aC 2 ( 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 —AU

B ,

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,

AU

B ,

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 C

R

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 t

s

.

Bureau oj Standards Journal

of

Research [Voi.w

Step 1

.-

no 2

moles of oxygen at t

H

°

and 1 atm. are compressed into

the bomb which contains n g. f. w. of the substance to be burned and

m w g

of liquid H 2

  1. The initial pressure of the oxygen in the bomb

is pi

atm. at t

H

°.

AU^ AUo 2 }

1

+ AU

W

  • AUd

02

+ AU

S (3)

1

A

Ui

represents the value of A U for step 1 and the terms on the right

obviously have the following significance:

p

AUo 2 ]

1

is the increase in the total energy of the oxygen which

i

takes place when it is compressed from 1 to p

x

atm. at t

H

°

;

AU

W

is the increase in the total energy of the water which accom-

panies its compression and the evaporation of the amount necessary

to saturate the bomb volume, V, which is filled with oxygen at

p

x

atm.

AUd

02

is the corresponding quantity for the solution of the amounts

of oxygen which dissolve in the water and in the substance ; and

AU

S

is the increase in total energy which accompanies the com-

pression of the substance and the evaporation of whatever amount

evaporates before the ignition.

Step 2. —The combustion is carried out in the usual way and the

quantity

AU

B

is calculated for t

H

°.

The final pressure in the bomb

is p

2

p

w

atm. at t

H

°

y p

w

being the partial pressure of the water vapor

in the final system.

AU

2

= nAU B (4)

Step 3.—The aqueous solution of C 2

2

is separated from the gas

phase and

is confined

under the pressure

pz

p

w

atm.

AC/

3

=

(5)

Step 4-

—With the aid of a membrane permeable only to C 2

the

dissolved C 2

is allowed

to escape from its aqueous solution into a

space at zero pressure after which it is

compressed to 1 atmosphere.

The value of All for this process will be

called

ATJ

D

.

AU

4

= AU

D (6)

Step 5. —The pressure on the water is now reduced to 1 atmosphere

and the dissolved oxygen is removed as a gas at 1 atmosphere. At

the same time the water vapor present in the bomb at the comple-

tion of the combustion is removed in the form of pure liquid water

under a pressure of 1 atmosphere.

AU

5

= AU'd

02

  • AU' w (7)

Step 6. —The gas phase (0 2

  • C 2 )

in the bomb is now expanded to

zero pressure

AU^AUuW (8)

in which the subscript

M is used to

indicate the mixture of 2

and

CO*.

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 n

0i

moles of

oxygen

and

n g. f. w.

of the substance or

mixture having the

compo-

sition

expressed

by

the formula C a

H b

O c

and

whose

heat of combus-

tion

in the bomb is

AU

B

energy

units per g. f. w. There are

also

placed in

the bomb m w

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

n

T

= U

^^\n + 4> (20)

where is the oxygen consumed in producing nongaseous products

other than C 2

and H 2

0; for example, HN 3 ,

Fe

2 3 ,

etc.

The initial

oxygen pressure

in the bomb will be

=

n

0l

BT{l-ix

0i

'p

l )

,

x

in which

mo 3 Pi

is a small correction term calculable from the equation

of

state. At

C.

JU

=

3

5

6 p

(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

3

(23)

  1. THE FINAL SYSTEM

After the combustion, the bomb will contain (n 03

n

r )

moles of

2 ,

Wcoa (

= an) moles of C 2 ,

and

(}i

bn+ ){%

m w )

moles of water. Part of

the water will be in the gaseous state and part of the

2

and C

2

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 C

2

are

removed from the bomb leaving a gas phase which contains

n

M

=

n

0i

-n r

nco

n

D (24)

moles of gas, where n D

is the number of moles of dissolved C 2

re-

moved with the liquid water. The dissolved 2

may be neglected in

equation (24).

n

M

= n 0i

-(

J

n -n

D -

(25)

The mole fraction of the C 2

in the gas phase will be

an

n

D

x

=

n

03

-(

j-

-fn

n

D - (26)

Washburn] StandardgStates jor Bomb Calorimetry 535

The pressure of the mixture will be

y.-

tt-gr^-MMfr)

(27)

At

C, ju M

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 m w

  • 9bn g

of water. Of

this, the amount (see Appendix I, equation

(119), p.

556).

VC W

=

0M73V+ (0.

4

55 + 0. 3

28:r)2> 2

V,

g

(29)

is in the vapor state at

C. Since this will in general be only about

1 per cent of the total, we may substitute the average values

p

2

= 30

atm.,

and a;

=

0.15, and the above equation may, with sufficient ac-

curacy for our present purposes, be written

F

r

w

F[0.0173+(0.

4

55 + 0. 3

28X 0.15)30]

= 0.02 V,

g

(30)

and the amount of liquid water will be

m

w

  • 9bn- 0.02V,

g

(31)

This amount of liquid water will be saturated with C 2

at the partial

pressure

p

2

x.

For the range of

p

2

x values encountered in bomb calorimetry (1 to

8 atmospheres) it will be sufficiently accurate, for the purpose of

correcting for dissolved C 2 ,

to assume that the solubility is propor-

tional to the partial pressure, using a proportionality constant com-

puted from the solubility of C 2

at about 7 atmospheres. (See fig. 1.)

With this assumption the number, S C 02,

of moles of dissolved C 2

per cm

3

of liquid water will be

S C o

= 0.0^ 2 p

2

x, M/cm

3

at

C.

(32)

The total number, n D ,

of moles of dissolved C 2

will therefore be

n

D

=

3

2

p

2

x(m

w

  • 9bn

0.02 V)

(33)

Now the total energy of vaporization of C 2

from its aqueous

solu-

tion at

C. to produce pure C 2

gas at 1 atmosphere

is 181 liter-atm.

per mole.

9

For the n

D

moles of dissolved C 2

we

have, therefore,

AU

D

= 181

X

n

D

,

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

p

2

x

= &7iET(l

hmP2)/V (35)

From equations (23)

and

(28)

we have for z

=

0.15,

/z M

=

3

. Hence

for 20° C.

and

p

2

= 30

atm.

p

2

x

= an X 24. (

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.

waahbum] Standard States for Bomb Calorimetry 537

XL

CORRECTION FOR THE CHANGE IN ENERGY CONTENT

OF THE GASES

We are now in a position to evaluate the quantity A£/

ga

as given

by

equation (18)

above, using the number of moles of gas involved.

Performing the summation indicated by that equation between

1 and

p

atmosphere gives us

(43)

  • AE/gM

= ft .X 0.0663 (pi-l)+n M X 0.0663(1+

A)

(0-p 2 )

7icoiX0.287(l-0)

  • (w o

,-w r

)0.0663(l-0),liter-atm.

This may also be written

-A^

= 0.0663n oo ,r^Pi-^—

-(WwooO(l

  • A)p a

^g

  • (n 0i

-n

r )lncoA

or putting x

=

ncoJn M

=

ncoj

(n 0i

  • r + nco*)

-A^ a

= 0.06637i

coLi

i

^-

(

^^

2

  • -^(^-l)l (45)

If we now introduce the relations

n

and

b-2c

\ ,

n

C o=&n

7i jD

= 0.996an, approx. (See equation (39).)

we have

n

T

/n C o

2

= 1 - 1

0/an approx. (46)

and

Atf

gM

= 0.0660an[

2

1

^-<i±^

+ 4.33 + 1. OO4pi(l+^^ + 0/ar^-l. 004 (l+^[^

  • 0/araY| (47)

-AC

ga

= 0.0660a7ir^^^-^-^-

2

  • 3.33 + 1.004^

X X

1.004j> r

((^2)

*/a»)-1.

(^jp+*/an)] (48)

  • A

U

m

= 0.0660a»

\j^

~ ~

0.004p,

(49)

{

1.004 (p,-l)(b-2c + 40/w)

|C

4a

and

(Per cent CoTr.)

w ^^^^^^l(p

l

--p

2

)/x + 0.0(Ap

l

-p

2

h/x

  • 1.004(^-1) ((b-2c)/4a + 0/an) + 3.33] (50)

161541—

8

538

Bureau

of

Standards Journal oj Research [Vol. w

Since one unit in the first decimal place of the terms within the

brackets corresponds to less than 0.001 per cent of AU

B

and

since

2>i£

atmospheres, it will be sufficiently accurate for all purposes

to write

m , \

6.60a

f A

fl,h
i, d

(Per cent corr.)

ga

=

_

AU

\

&Pl

x

x)~

Pl ^

x

Cpi-l)((b-2c)/4a

</)/a7iV3.47J

(51)

which is valid for

p

x

and

Ap,

(=P

pi),

in atmospheres and AU

B

in

liter-atmosphere/mole.

XII.

CALCULATION OF

THE CHANGE IN

PRESSURE

RESULTING FROM THE COMBUSTION

For computing small pressure changes in the neighborhood of any

pressure between 25 and 45 atmospheres, at room temperatures,

the pressure of a gas can be taken as given by the relation

p

= nRT(l-np)/V (52)

in which /x is a constant characteristic of the gas, and the temperature.

The drop in pressure following the combustion will therefore be

-Ap=p

1

-p

2

=p

1

UM

y

(

fi M

p

2 )

(53)

If we put (see equation (25))

n

M

=

n

0i

—-

;

—n D

— <j)=(tin-n D

)/x

= ein/x (Approx.) (54)

and

n

orVl

VIRT{- »o>Vi)

(55)

this

becomes

-A^ =

?

,

pM-Mo

2

MMAWyi

(l +

AWyi)^

b-2c n D

  • \l

F

^ L l-Mo

2

Pi

Pi \

4a &n

/J

and

Pi^m/mo," 1)mo, + (

~

^^

X

~T^

^nr)j

~

AP

=

7I~7~i n ~Ji

,

/b-2c, n D

4>\

(57)

1 +^i(mm/mo

2

1)mo,+

(

Mo.Pi)*

(^

-4^-

+—

—J

At

C,

Mm/mo,-

l

= 3.21a;(l + 1.33a?)

(see equation (28).)

M

Ol

=

0.G4X

4

(see equation (23).)

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 t

H

°

is greater than that present in the ini-

tial

system at this temperature. The increase, AC

W

,

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 VAC W

X 22.

liter-atm. of heat energy, 22.82 liter-atm. being the total energy of

vaporization of 1 g

of H 2

  1. We have, therefore,

AC7„,

V8P

=22.82F^ 2

a:r0.0334-0. 5

('l

-^f)j,

liter-atm. (66)

An extreme case would be the following: V=}i liter, ^

= 45 atm.,

x

=

0.3 and

Ap

=

2 atm., and this

would give

A

UJ**-

= 0.043 liter-atm.

=

4

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 A U

D

, for the dissolved C 2 ,

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

C.and for the pressure range encountered in

bomb

calorimetry this increase is expressed with sufficient accuracy

by

the equation

:

A

U

w

U(l

  • = - 54 X

6

P, liter-atm./g (68)

In the process defined by the proposed standard states,

m w

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

=

p

2

Pa

=

)'i (pi

P

2pv>)

=

P. and the net change in the

energy content of the liquid water becomes

-AC u,

M<

  • =

9b7iX54Xl0-

6

P, liter-atm. (69)

An extreme

case would berep resented

by 9bn

=

2g

and P

= 45

atmospheres for which

case

-AEV

lq

  • = 0.

liter-atm. =

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 AU

W

  • AU' W

is equal to AC7«,

vap

as given by equation

(66)

above.

  1. COMBINED ENERGY CORRECTIONS FOR WATER VAPOR AND FOR

DISSOLVED CARBON DIOXIDE

The

correction for

dissolved C 2

is (see

equations (33)

and

(34))

-AU

D

= -181X0.

3

2

^

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 Vp 2

x f"o. 3

3

5

(l

r^Yl

liter-atm. (72)

For

p

2

x we have at

C. (see equation (36))

p

2

x

= 2S.

3

SinlV f

approx. (73)

The sum of equations (71)

and

(72)

combined with

(73)

and

divided by

O.Oln ATJ

B

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 .

(A U

D +

A

U„ +

A U'

w )

=

q^/

I

[-•

(m

" +

9M

5 ^

=

'j=gfL

r_

.96 +

^-±^-^Z^2l

(74)

-AU b

/8l

L

V an

J

v

'

The quantity

AU

b

/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,

liter-atm. The net correction given by equation (74) will therefore

vary in practice (for V=l/3 liter and m

w

=l g)

only between

about

0.06 and

per cent. In the majority of

cases it will be

found

to lie between

and

per

cent.

  1. THE ENERGY CONTENT OF THE DISSOLVED OXYGEN

At

C.

the solubility of

2

in water is approximately

u

So

2

=

X

6

p

O2 ,

mole/cm

3

(75)

The amount of

2

dissolved in the initial water will therefore be

ft£o

2

=

!- 2x 10-*m ttpi,

moles (76)

and that in solution in the final water will be

n

,

jDO

=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,

Washburn] Standard States jor Bomb Calorimetry

Table 1. Energy of

isothermal compression

-A

I/]

^=aP, liter-atm./g

[Data taken from

International Critical Tables. M=molecular weight]

Substance

t 10««

-AI7]

«

=45a

-AUbIM

Per cent

corr.

MAC/

-0.01 A U

B

HjO

.. __.

°C.

20

20

25

25

20

25

25

20

25

20

20

liter/g

59

540

325

82

420

417

248

305

270

127

121

liter-

aim. \g

2.6X10-

24 X10-

15 XI

0-

  1. 7X10-

19 X10-

19 X10-

11 X10-

14 X10-

12 X10-

  1. 7X10-*

5.4X10-

liter-

atm./g

jj-Heptane.... „ -

  • 475

467

403

220

304

142

348

260

.

Naphthalene...

. .

. -.-.

.

Acetone.

Formic acid

......

Acetic acid

. .

..

.

.

Benzoic acid

J

1 Thermal expansion determined by E. R. Smith (B. S. Jour. Research, vol. 7,

p. 903, 1931).

(b) THE ENERGY OF VAPORIZATION OF THE SUBSTANCE

The energy of vaporization of the substance must either be made

negligibly small, by suitable inclosure of the sample when necessary,

or must be computed and corrected for. A safe rule to follow is to

inclose every substance whose vapor pressure is more than 1 mm Hg.

If it is not so inclosed, the investigator must show that the energy of

vaporization is negligible or he must make the necessary correction.

16

XIV. THE TOTAL CORRECTION FOR REDUCTION TO THE

STANDARD STATES

  1. GENERAL CORRECTION EQUATION

By

adding together equations (74)

and

(51)

and converting to cal.

units,

we obtain equation

(13)

in the form

b-2c

(Per cent corr.) Tot.=

0.160^

-AU

B

/ei

L

Vi

\x

*J

h/x+(l-l

  • ft>/&n^-2A i

(m

Uf

  • 9bn)/p 1

V+5.S/p

1

'

019VAp,Pl

l

/

EiTl

4a

(81)

in which V is

the volume of the bomb in liters

; p

x

is the initial

2

pressure at 20° C,

p

2

is the final pressure in the bomb at

C, both

in atmospheres; Ap=p 2

pu

x is the mole fraction of C

2

in the

final system; h=1.70x(l + x); n is the number of grain-formula-

weights of the material, C a

H b

O c ,

burned; m

w

is the mass of H

2

initially placed in the bomb; is the number of moles of

2

consumed

by auxiliary reactions;

and

AU

B

is the evolved heat of the bomb

process in kg-cal.

15

per gram-formula-weight.

By evaluating this relation for a given combustion we obtain the

total correction in per

cent which must be applied to the value of

AU

B ,

in order to obtain

AU

n ,

the decrease in intrinsic energy for

the chemical reaction, for the standard conditions at

C, (equa-

tion (2)).

1 See footnote 5, p. 530.

544

Bureau

of

Standards Journal of

Research [Voi.io

Because of the small magnitude of this correction and the small

temperature coefficient of AU

B ,

the value of AU

B

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 ATJ

B

at any room temperature, except possibly in

certain extreme cases.

In deriving this relation we have neglected AU

S

,

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 —AU B

is increased. (See Table

1.)

If the sample is inclosed in a sealed glass capsule which it does not

fill, the quantity; A U

s

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.

  1. 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

AU

B

in kg-cal.w

per

g.

f. w., for a bomb volume of % liter,

for = 0,

and for m

w

= 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.