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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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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.
the heat of the pure
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
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-
volume of (1) the mass of the sample and (2) the mass of water, together with
appropriate tolerances.
CONTENTS
Page
I. Nomenclature 526
1 1. Introduction 527
III. Calorimetry and the first law of thermodynamics 528
V. Proposed standard states for constant-volume combustion re-
actions 530
proposed standard states 531
VII. The total energy of combustion defined by the proposed standard
states 531
VIII. Definitions of some auxiliary quantities 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
solved carbon dioxide 541
525
of
Standards Journal of
Research [Voi.io
XIII.
The negligible energy quantities—Continued.
page
(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 magnitude of the correction in relation to the type of sub-
stance burned 545
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
The temperature coefficient of the heat of combustion 551
Standardizing substances 552
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
a, b, c Coefficients in the chemical formula, C a
H b
O c
.
C Concentration; molal heat capacity.
c Specific heat,
g
(g)
h
= 1.70a: (1+x).
g. f. w. Gram-formula-weight.
(1)
Liquid state.
M Molecular weight; g. f. w.
w
Number of grams of water placed in the bomb.
n Number of moles or of g. f. w. of substance burned.
Number of moles of C 2
M
Number of moles of gas in the bomb after the combustion.
o
2
Number of moles of O2 in the bomb before the combustion.
P Pressure; per cent by weight.
Pressure or partial pressure.
Pressure in the bomb
before the combustion.
2
Pressure in the bomb
after the combustion.
w
Q
R Gas constant.
S Solubility.
(s) Solid state.
T Absolute centigrade temperature.
t Centigrade temperature.
t
or intrinsic energy content.
528 Bureau of
Standards Journal of
Research [Voi.w
within the bomb does not start from
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
(2)
as it does, a
which is accompanied by a heat effect, due correction must be made
At present the bomb calorimeter appears to have no serious com-
conditions as here proposed is necessary, if one is to
tage of the highest precision attainable.
The purpose of this discussion is to
propose suitable standard states
by
which the heat of the bomb process may
be corrected so as to yield the total or
tion it is impossible to obtain
from the existent precise data of bomb
calorimetry the frequently
of the reaction at constant pressure."
This is
to-day almost univer-
sally calculated
the simple addition of a
AnRT quantity to the
result
is
when
applied to
calorimetric data of
high precision.
THERMO-
DYNAMICS
According
2
-U 1
= AU=Q-W (1)
2
u
respectively)
is the "total,"
"internal," or "intrinsic"
state
2 (or 1,
increase in
this intrinsic energy
which takes place when the system
to
state 2
Q
is the quantity of
heat
the
during the process and W is the work
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
Qv
and is commonly called the "heat at constant
for three
reasons: (1)
process is not
necessarily a zero-work process (for example, when
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
(3)
I sufficient
the heat of such a process may also be a function of the pressure, for example). Case
constant external pressure p. The heat of the process under these
Q H and is commonly called "the
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
Case 8. W= W m&x . =
energy " of the process. This case is rarely encountered in calori- metry except when it is identical with case 2. The corresponding heat
absolute temperature is the "entropy of the process."
Any one of the above heat quantities is, in principle, calculable
Qv for
= 1
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
= 1 atmosphere. This can be conveniently ac-
Q v of the bomb process in
Qp
by the empirical formula C a
b O c
weights) of this material in a thermodynamically defined physical state or states (solid and/or liquid) are placed in the bomb.
1
w
of water are also placed in the bomb, this amount being at least
sufficient to saturate the gas phase (volume —
vapor. The bomb is then closed and filled with n 02 moles of oxygen,
The above quantities will completely define the initial system and this definition will subsist, if the quantities are all increased in the
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
corrected
definite temperature, t H ,
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
at the temperature t
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
contrasted with the actual bomb process, the nature of which has
already been discussed in
proposed standard states consists solely in the reaction of unit
t
,
to produce pure carbon dioxide
pure
,
work.
This process is not experimentally realizable. The intrinsic energy
—
n
,
the decrease in
intrinsic energy for the following reaction at t
°
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-
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
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
To obtain from the heat,
—
B ,
of the bomb process, the energy
quantity,
—
which we will now proceed to discuss.
we are therefore at liberty to make use of any desired imaginary
The process employed for this purpose should obviously
contain
the actual bomb process as one of its steps. Its other steps
should be
A review of the available data for various
alternative processes
indicates that the following process
will yield trustworthy values of
the temperature t
.
of
Research [Voi.w
Step 1
.-
—
no 2
°
the bomb which contains n g. f. w. of the substance to be burned and
m w g
of liquid H 2
is pi
atm. at t
°.
AU^ AUo 2 }
1
W
02
S (3)
1
Ui
p
AUo 2 ]
1
i
takes place when it is compressed from 1 to p
x
atm. at t
°
;
W
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
S
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
—
B
is calculated for t
°.
The final pressure in the bomb
is p
2
p
w
atm. at t
°
y p
w
in the final system.
2
= nAU B (4)
Step 3.—The aqueous solution of C 2
2
is confined
pz
p
w
atm.
3
=
(5)
Step 4-
—With the aid of a membrane permeable only to C 2
the
dissolved C 2
is allowed
The value of All for this process will be
called
.
4
Step 5. —The pressure on the water is now reduced to 1 atmosphere
the same time the water vapor present in the bomb at the comple-
under a pressure of 1 atmosphere.
5
= AU'd
02
Step 6. —The gas phase (0 2
in the bomb is now expanded to
zero pressure
AU^AUuW (8)
M is used to
indicate the mixture of 2
534 Bureau of
Standards Journal of
Research [Voi.io
QUANTITIES
a bomb of
liters
0i
moles of
n g. f. w.
of the substance or
sition
expressed
the formula C a
H b
O c
tion
in the bomb is
—
B
energy
also
placed in
the bomb m w
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
T
= U
^^\n + 4> (20)
other than C 2
and H 2
0; for example, HN 3 ,
2 3 ,
etc.
The initial
in the bomb will be
=
0l
0i
'p
l )
,
x
mo 3 Pi
of
JU
=
3
5
6 p
(22)
2
narily met with in bomb calorimetry) this relation may be replaced
Mo 2
3
(23)
After the combustion, the bomb will contain (n 03
—
n
r )
moles of
2 ,
Wcoa (
= an) moles of C 2 ,
(}i
bn+ ){%
m w )
moles of water. Part of
2
and C
2
in
of the preceding section, all of the water and the dissolved C
2
are
removed from the bomb leaving a gas phase which contains
M
=
0i
-n r
nco
—
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).
M
= n 0i
-(
n -n
D -
(25)
The mole fraction of the C 2
in the gas phase will be
—
D
x
=
03
-(
—
—
D -
Washburn] StandardgStates jor Bomb Calorimetry 535
The pressure of the mixture will be
tt-gr^-MMfr)
(27)
C, ju M
8
/W/*o,=
1 + 3.21a?(l + 1.33a;) (28)
After the combustion, the bomb contains m w
(119), p.
556).
VC W
=
4
55 + 0. 3
28:r)2> 2
g
(29)
C. Since this will in general be only about
1 per cent of the total, we may substitute the average values
2
= 30
atm.,
=
curacy for our present purposes, be written
F
r
w
4
55 + 0. 3
28X 0.15)30]
= 0.02 V,
g
(30)
and the amount of liquid water will be
w
g
(31)
This amount of liquid water will be saturated with C 2
at the partial
pressure
p
2
x.
p
2
x values encountered in bomb calorimetry (1 to
correcting for dissolved C 2 ,
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
(32)
The total number, n D ,
of moles of dissolved C 2
will therefore be
=
3
2
2
w
(33)
Now the total energy of vaporization of C 2
solu-
tion at
C. to produce pure C 2
is 181 liter-atm.
per mole.
9
moles of dissolved C 2
we
have, therefore,
D
= 181
X
,
liter-atm. (34)
Since, as will appear later, this
rection, we may write with
sufficient accuracy
p
2
x
= &7iET(l
From equations (23)
(28)
we have for z
=
0.15,
/z M
=
3
for 20° C.
p
2
= 30
atm.
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.
water.
waahbum] Standard States for Bomb Calorimetry 537
XL
We are now in a position to evaluate the quantity A£/
ga
as given
equation (18)
above, using the number of moles of gas involved.
(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
A)p a
^g
-n
or putting x
=
ncoJn M
=
(n 0i
-A^ a
= 0.06637i
i
^-
(
^^
2
If we now introduce the relations
b-2c
C o=&n
—
7i jD
= 0.996an, approx. (See equation (39).)
T
/n C o
2
= 1 - 1
0/an approx. (46)
gM
= 0.0660an[
2
1
^-<i±^
-AC
ga
= 0.0660a7ir^^^-^-^-
2
X X
1.004j> r
((^2)
(^jp+*/an)] (48)
m
= 0.0660a»
\j^
~ ~
0.004p,
(49)
{
1.004 (p,-l)(b-2c + 40/w)
(Per cent CoTr.)
w ^^^^^^l(p
l
--p
2
)/x + 0.0(Ap
l
-p
2
h/x
161541—
8
538
of
Standards Journal oj Research [Vol. w
Since one unit in the first decimal place of the terms within the
B
since
2>i£
atmospheres, it will be sufficiently accurate for all purposes
to write
m , \
6.60a
fl,h
i, d
(Per cent corr.)
ga
=
_
\
&Pl
x
x)~
Pl ^
x
</)/a7iV3.47J
(51)
p
x
Ap,
—
pi),
B
in
liter-atmosphere/mole.
CALCULATION OF
PRESSURE
pressure between 25 and 45 atmospheres, at room temperatures,
p
= nRT(l-np)/V (52)
The drop in pressure following the combustion will therefore be
-Ap=p
1
-p
2
=p
1
UM
(
fi M
2 )
(53)
If we put (see equation (25))
M
=
0i
—-
—
;
—n D
— <j)=(tin-n D
—
= ein/x (Approx.) (54)
orVl
VIRT{- »o>Vi)
(55)
this
-A^ =
?
,
2
MMAWyi
(l +
b-2c n D
^ L l-Mo
2
Pi
Pi \
/J
Pi^m/mo," 1)mo, + (
~
^^
X
~
=
7I~7~i n ~Ji
,
/b-2c, n D
4>\
(57)
1 +^i(mm/mo
2
1)mo,+
(
Mo.Pi)*
(^
-4^-
+—
—J
C,
l
= 3.21a;(l + 1.33a?)
(see equation (28).)
Ol
=
4
(see equation (23).)
540 Bureau of
Standards Journal oj Research [Voi.w
result of the bomb reaction the amount of water present as
°
tial
system at this temperature. The increase, AC
W
,
in the concen-
C,
is given by the equation (see
equation (122),
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
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
=
—
Ap
=
= 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
negligible. Since it is opposite
in sign to that
arising
, for the dissolved C 2 ,
a
partial
occur
tions may advantageously be
into a single expression repre-
senting the
algebraic sum of
section 2, below.
The
increase, A Z7«,
l,q
,
in the total energy content of liquid water as
a function
13
bomb
calorimetry this increase is expressed with sufficient accuracy
the equation
:
A
w
U(l
6
P, liter-atm./g (68)
m w
(pi+pj)
atmos-
phere and (m„, + 9b7i) grams are decompressed from (jp%
)
to 1
atmosphere.
Since we are dealing with a very small energy quantity, we will
write pi
=
2
=
)'i (pi
P
2pv>)
=
-AC u,
M<
6
P, liter-atm. (69)
An extreme
=
2g
and P
= 45
case
lq
liter-atm. =
cal. (70)
" Bridgeman, P. W., rroc. Am. Acad., vol. 48, p. 348, 1912.
Washburn] Standard States for
Bomb Calorimetry 541
W
is equal to AC7«,
vap
(66)
above.
DISSOLVED CARBON DIOXIDE
The
correction for
dissolved C 2
is (see
equations (33)
(34))
-AU
D
= -181X0.
3
2
2
w
(71)
(equation
(66))
= 22.82 Vp 2
x f"o. 3
3
5
(l
r^Yl
liter-atm. (72)
2
x we have at
C. (see equation (36))
p
2
x
= 2S.
3
SinlV f
approx. (73)
The sum of equations (71)
(72)
(73)
—
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 .
D +
U„ +
w )
=
q^/
I
" +
9M
5 ^
=
'j=gfL
r_
.96 +
^-±^-^Z^2l
(74)
-AU b
/8l
L
J
v
'
The quantity
—
b
/b>
}
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
w
=l g)
about
—
per cent. In the majority of
cases it will be
found
per
cent.
the solubility of
2
in water is approximately
u
2
=
X
6
O2 ,
3
(75)
The amount of
2
dissolved in the initial water will therefore be
ft£o
2
=
!- 2x 10-*m ttpi,
moles (76)
,
jDO
=1.2Xl0-
6
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
I/]
^=aP, liter-atm./g
[Data taken from
International Critical Tables. M=molecular weight]
Substance
t 10««
-AI7]
«
=45a
-AUbIM
Per cent
corr.
MAC/
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-
19 X10-
19 X10-
11 X10-
14 X10-
12 X10-
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).
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.
vaporization is negligible or he must make the necessary correction.
16
By
adding together equations (74)
(51)
units,
(13)
b-2c
(Per cent corr.) Tot.=
0.160^
-AU
B
/ei
L
Vi
\x
h/x+(l-l
Uf
1
'
l
/
EiTl
(81)
the volume of the bomb in liters
; p
x
is the initial
2
pressure at 20° C,
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 ,
w
2
initially placed in the bomb; is the number of moles of
2
—
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
—
B ,
in order to obtain
—
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
of
Standards Journal of
Research [Voi.io
B ,
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
B
at any room temperature, except possibly in
certain extreme cases.
S
,
the energy of
compression of the substance plus the energy content of its vapor.
(81).
accuracy of the determination of —AU B
1.)
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
or
absorb heat. The amount of water
with the water vapor. Presumably the amount thus adsorbed has
substantially constant
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
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
con-
ditions of the
experiment, it
can not be provided
for in
equation (81).
hygroscopic
difficult
to
therefore be
inclosed in a suitable capsule.
The
calculation
of equation (81)
is somewhat time consuming and it is desirable for many purposes
to
simpler equation for rapid
generality,
introducing certain approximations into equation (81)
typical calori-
metric conditions. In this way the
following approximate equation
may be obtained, for pi
in atmosphere, for
—
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
1 + 1.1 -j—
approx.
This approximate equation will in general give a value for
the
(per cent corr.) Total
existing data of bomb calorimetry.