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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
- The initial system 534
- 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
- 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
- The energy content of the dissolved oxygen 541
- 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
- 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
General remarks 545
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
- 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
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.
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
- The initial pressure of the oxygen in the bomb
is pi
atm. at t
H
°.
AU^ AUo 2 }
1
+ AU
W
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
Step 6. —The gas phase (0 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
- 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)
- 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
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
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
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)
= ft .X 0.0663 (pi-l)+n M X 0.0663(1+
A)
(0-p 2 )
7icoiX0.287(l-0)
,-w r
)0.0663(l-0),liter-atm.
This may also be written
-A^
= 0.0663n oo ,r^Pi-^—
-(WwooO(l
^g
-n
r )lncoA
or putting x
=
ncoJn M
=
ncoj
(n 0i
-A^ a
= 0.06637i
coLi
i
^-
(
^^
2
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+^[^
-AC
ga
= 0.0660a7ir^^^-^-^-
2
X X
1.004j> r
((^2)
*/a»)-1.
(^jp+*/an)] (48)
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
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
- 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
- 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
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
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
is equal to AC7«,
vap
as given by equation
(66)
above.
- 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.
- 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,
(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-
- 7X10-
19 X10-
19 X10-
11 X10-
14 X10-
12 X10-
- 7X10-*
5.4X10-
liter-
atm./g
jj-Heptane.... „ -
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
- 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
(m
Uf
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.
- 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.
