Lab report

Bomb CalorimetryObjectives
In this experiment, bomb calorimetry will be used to determine the heat of combustion and
standard enthalpy of formation of a pure substance (e.g. sucrose) as well as the “calorie content”
of a food product.
Introduction/Theory
A bomb calorimeter is a sealed container capable of holding several atmospheres of gas pressure.
A weighed sample of substance is placed in contact with an ignition wire inside the bomb. The
bomb is filled to about 20 atm of pressure with O2 , sealed, and placed in a known amount of
water. An electric current is passed through a wire to ignite the mixture. As the combustion
takes place, the heat evolved raises the temperature of the calorimeter and its surrounding water,
as measured by a thermometer. In order to prevent heat loss from the calorimeter system, it is
surrounded by a second water bath, whose temperature is continuously adjusted (by the
experimenter) to match that of the calorimeter. Thus, the heat transfer between the system
(calorimeter and contents) and the surroundings (the water jacket, primarily) is zero, making the
process adiabatic: q = 0. Some calorimeters use good thermal insulation instead of a second
water bath to prevent loss of heat to the surroundings.
Since the combustion takes place in a sealed container with constant volume, the work done on
or by the system is also zero, so that ∆U = q + w = 0 for the overall process shown below.
The initial temperature of the system is T1 and the final temperature is T2 after combustion of
reactants to products and heating of the calorimeter. Since U is a state function, the pathway
from initial to final state does not affect the value of ∆U. This allows us to separate the two
processes (combustion and heating), although, in reality, they occur simultaneously.
With this in mind, it is true that
∆Uoverall = ∆Ua + ∆Ub = 0
[1]
− ∆Ua = ∆Ub
[2]
or,
But, since Step b simply involves heating up the calorimeter and contents from T1 to T2 ,
∆Ub = CV ∆T = CV (T2 − T1 )
[3]
Note that CV is the heat capacity of the bomb calorimeter and its contents. From equations [2]
and [3], the change in internal energy of combustion can be readily found from the change in
internal energy of heating.
The molar enthalpy of combustion ∆Hm is given by the equation:
∆Hcomb, m = ∆Ucomb ,m + RTcomb∆ngas
[4]
where Tcomb is the temperature at which combustion occurs, ∆ngas is the change in the number of
moles of gas (in the balanced reaction representing the combustion of one mole of compound).
To ensure that units match, ∆ngas should be treated as unitless.
Determining the heat evolved during the combustion from Equation 3 requires knowledge of the
heat capacity of the calorimeter often called the calorimeter constant. By combusting a sample
with a known molar heat of combustion, we can use the measured temperature change and
Equation 3 to calculate the calorimeter constant.
Procedure
1.
Turn on the water and the hot water heater.
2.
Form and accurately weigh three pellets of special grade benzoic acid not exceeding1 gram.
If any pellet exceeds this limit, it must be shaved off to a 1 gram maximum.
3.
Place one of the pellets in the stainless steel cup and attach ~10 cm of ignition wire to the
electrodes tightly with a loop of wire just touching the top of the pellet. Make sure that the
ignition wire does not touch metal or it will short and the sample will not ignite.
4.
When closing bomb, tighten down the cap by HAND only. Close the outlet valve. Attach
the union from the oxygen tank, turning it down lightly by hand. Close the filling valve
between the bomb and gauge (turn clockwise). Open the main tank valve 1/4 turn, open the
filling valve slowly and allow the pressure to build up slowly to 25 atm. Purge the bomb
with pure oxygen twice. Then close the filling valve and the tank valve and open the relief
valve under the gauge. The excess pressure in the bomb closes the inlet valve. Unscrew the
union.
5.
Attach the electrode wires to the contacts in the bomb head.
2
6.
Place 2000 mL of water measured from a volumetric flask at 25°C in the stainless steel
water bucket. Be sure the ignition wires are towards the front of the bucket, away from the
stirrer. Close the lid and turn on the stirrer.
7.
Adjust jacket temperature to within 0.03°C of calorimeter temperature by running hot or
cold water into the jacket. The pilot light on the water heater should be off before beginning
a run.
8.
Take temperature readings every 30 seconds for two minutes BEFORE igniting the bomb.
9.
Ignite the sample by pressing the firing button for 5 seconds AND NO LONGER! (The
indicator light may be burned out.)
10. After a delay of 10 or 15 seconds, the temperature should begin to rise in the inner bucket.
Keep the jacket and bucket temperature to within 0.03°C by running hot water slowly into
the jacket. Read the temperature of the calorimeter every minute for 5 to 8 minutes after
igniting the bomb.
11. Once the temperature reaches a plateau, remove the bomb and release the pressure. Open
the bomb and recover any fuse wire remaining. Weigh or measure the length of the
remaining fuse wire.
12. Combust at least three samples of the benzoic acid standard and at least one sucrose and one
food product sample (with a clearly labeled caloric content!).
3
Calculations
For each combustion, determine ∆T.
During ignition, part of the ignition wire combusts, thereby adding heat to the system. The extra
heat must therefore be subtracted from the total. The amount of heat liberated per gram and per
cm of fuse wire is listed on the spool.
∆Ua = −CV∆T + qwire
[5]
Heat Capacity of the Calorimeter
The balanced equation for the combustion of one mole of benzoic acid [122.1 g mol-1 ] is:
C6 H5 CO2 H(s) + 152 O2 (g) à 7 CO 2 (g) + 3H2 O(l)
The standard molar enthalpy of combustion for benzoic acid is ∆Hcomb,m = – 3227 kJ mol-1 . The
standard molar energy of combustion for benzoic acid can thus be calculated.
o
o
∆Ucomb,m
= ∆Hcomb
,m − RT∆ngas
= −3227 kJ mol –1 − (8.314 E − 3 kJ mol –1K–1 )(298K)(7 − 15 2 )
= −3227 kJ mol –1 + 1.24 kJ mol –1
= −3226 kJ mol –1
Using ∆Ucomb,m the heat capacity for the calorimeter, Ccal can be calculated using equation 5.
Molar Enthalpy of Combustion for Sucrose
Write the chemical reaction for the complete combustion of one mole of sucrose. Calculate ∆U°
for the combustion of your sample from
o
∆Ucomb
= −Ccal (T2 − T1) + qwire
[6]
Determine the molar change in internal energy in kJ mol–1 . Then calculate the molar enthalpy of
combustion from equation [4] using T=298K ~ T1 .
Compare your value with that listed in the literature.
Now calculate the standard enthalpy of formation of sucrose from your enthalpy of combustion
and the enthalpies of formation of CO2 and H2 O.
4
Molar Enthalpy of Combustion for Food Product
Since these substances are not pure, one can only calculate the heat of combustion per gram,
assuming that ∆U ~ ∆H. Find ∆H (in kJ/gram) for each substance tested. Compare to the stated
“calorie content” on the product label.
Questions
1. What are the two major assumptions implicit in the use of Equation [4] in this experiment?
2. When 2.25 mg of anthracene, C14 H10 (s), was burned in a bomb calorimeter the temperature
rose by 1.05K. Calculate the calorimeter constant. By how much will the temperature rise
when 5 mg of phenol, C6 H5 OH(s), is burned in the calorimeter under the same conditions?
3. Using the same calorimeter, by how much will the temperature rise when 5 mg of aniline,
C6 H5NH2 (l) is combusted?
5
A SAMPLE REPORT
Activation Energy: Enzyme-Catalyzed Hydrolysis of Sucrose
Abstract
The kinetics of the inversion of sucrose by the invertase enzyme was explored at several
temperatures using a polarimeter. The first order rate constants for the inversion were determined to be
0.065 +/- 0.001 min-1, 0.095 +/- 0.002 min-1, and 0.135 +/- 0.006 min-1 at 300K, 310K, and 320K
respectively. The temperature dependence of the rate constant followed Arrhenius behavior and resulted
in an activation energy of 29.2 +/- 0.2 kJ/mol in good agreement with the literature value of 31.4 kJ/mol.
Introduction
The activation energy and the rate of reaction for the invertase-catalyzed hydrolysis of sucrose will
be measured by following changes in optical rotation of an aqueous solution of sucrose and invertase.
When sucrose is hydrolyzed to a mixture of glucose and fructose, the optical rotation changes from
clockwise (+) to counter clockwise (-).
O3
H12
+C
H12
O
C
224
11 + H2 O invertase C
62
1124H2
164
4O
4
4
464
44
36
(+) rotation
(1)
(−) rotation
The rotation will be monitored with an optical polarimeter and then used to calculate the amount of
sucrose left unhydrolyzed. The angle of rotation is determined at the beginning of the experiment (α0)
and at equilibrium (α∞). The algebraic difference (α0 – α∞) is a measure of the original sucrose
concentration. The concentration of H2O during the reaction remains essentially constant, since H2O is
present in large excess.
The reaction is known to be first order in sucrose,1-6 thus the concentration of sucrose as a function
of time should follow the relationship
C 
ln t  = −kt
 C0 
(2)
where C0 is the initial concentration of sucrose when the reaction mixture is first placed in the
polarimeter and Ct is the sucrose concentration at time t after the addition of invertase. Since Ct is
proportional to (αt – α∞) and C0 (a constant) is proportional to (α0 – α∞), the rate constant k can be
determined from a plot of ln(αt – α∞) versus time.4-6
The rate constant k depends on the absolute temperature T via the Arrhenius Equation
E
− a
k = Ae RT
(3)
where Ea is the activation energy. Thus if the rate constant is determined at several different
temperatures, the activation energy can be determined via a modification of Equation (3) by plotting lnk
versus 1/T.
Ea
RT
Once the plot is constructed the slope can be used to calculate the activation energy as follows.
ln k = lnA −
1
(4)
Ea = -R•slope
(5)
The data treatment consists of determining the rate constants for the invertase-catalyzed hydrolysis of
sucrose at three different temperatures followed by determining the activation energy from an Arrhenius
Plot (lnk versus 1/T).
Materials and Methods
Optical rotation was measured at a wavelength of 589 nm using a digital polarimeter (Jasco, DIP360) and a quartz cell with a 10 cm path length. Sucrose (Sigma-Aldrich, #47289) and invertase (from
baker’s yeast, Sigma-Aldrich, #I9274) were used as received and all solutions were made with distilled
water. The concentration of the sucrose solution used in each experiment was 50 g/L. The invertase
solution was prepared in an acetate buffer with a pH of 5.0 at a concentration of 0.04 g/L. Temperature
was controlled with a refrigerated circulating water bath (VWR, 1140) connected to the polarimeter cell.
All solutions were placed in the temperature bath prior to mixing for thermal equilibration.
Experiments were conducted at 300K, 310K, and 320K using the same procedure. First the
polarimeter cell was filled with sucrose solution (~10 mL) and the optical rotation was measured
producing α0. Next 5 mL of sucrose solution was removed and 5 mL of the invertase solution was
added. After the addition of invertase the optical rotation was measured every 5 minutes until the
measured value approached the equilibrium value (a∞). The equilibrium optical rotation (a∞) was
measured at t = 60 minutes for each temperature.
Data
Tables 1 through 3 give the values of αt (the measured optical rotation), (αt – α∞), and ln(αt – α∞)
measured as a function of reaction time.
Table 1: Optical Rotation as a Function of Time for the Hydrolysis of Sucrose at 300K
Time (min) error αt (deg)
0.00
0.07
22.52
5.00
0.07
14.54
10.00
0.07
10.51
15.00
0.07
6.52
20.00
0.07
3.41
25.00
0.07
1.53
30.00
0.07
0.51
35.00
0.07
-0.99
40.00
0.07
-1.72
45.00
0.07
-2.03
∞
-3.5
error αt – α∞ (deg) error ln(αt – α∞) error
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
26.02
18.04
14.01
10.02
6.91
5.03
4.01
2.51
1.78
1.47
0.00
2
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
3.259
2.893
2.640
2.305
1.933
1.615
1.389
0.920
0.577
0.385
0.001
0.002
0.002
0.003
0.004
0.006
0.007
0.012
0.017
0.020
Table 2: Optical Rotation as a Function of Time for the Hydrolysis of Sucrose at 310K
Time (min) error αt (deg)
0.00
0.07
22.00
5.00
0.07
11.97
10.00
0.07
5.95
15.00
0.07
3.00
20.00
0.07
0.25
25.00
0.07
-1.04
30.00
0.07
-2.12
∞
-3.50
error αt – α∞ (deg) error ln(αt – α∞)
0.02
25.50
0.03
3.2387
0.02
15.47
0.03
2.7389
0.02
9.45
0.03
2.2460
0.02
6.50
0.03
1.8718
0.02
3.75
0.03
1.3218
0.02
2.46
0.03
0.9002
0.02
1.38
0.03
0.3221
0.02
0.00
error
0.0012
0.0019
0.0032
0.0046
0.0080
0.0122
0.0217
Table 3: Optical Rotation as a Function of Time for the Hydrolysis of Sucrose at 320K
Time (min) error αt (deg) error αt – α∞ (deg) error ln(αt – α∞)
0.00
0.07
22.20 0.02
25.70
0.03
3.2465
5.00
0.07
8.90
0.02
12.40
0.03
2.5177
10.00
0.07
3.48
0.02
6.98
0.03
1.9430
15.00
0.07
0.32
0.02
3.82
0.03
1.3403
20.00
0.07
-1.92
0.02
1.58
0.03
0.4574
∞
-3.50
0.02
0.00
error
0.0012
0.0024
0.0043
0.0079
0.0190
The uncertainty in reading α is estimated to be about ± 0.02˚ and is based on the fluctuations in the
readout of the digital polarimeter. The uncertainty in the temperature at which reaction rates were
measured is about ±1˚C and is based on the observed temperature drift during each experiment.
Uncertainties in time measurements were 4 seconds and were estimated as the amount of time required
to record the rotation and time simultaneously. The errors in (αt – α∞) and ln(αt – α∞) were calculated
by propagating the error from the optical rotation using the equations shown in Appendix A.
Results
Reaction Kinetics
Figures 1 through 3 show the change in ln(αt – α∞) as a function of time for 300K, 310K, and 320K
respectively. As described above by Equation 2, the slope of the graph of ln(αt – α∞) versus time will
be equal to the negative of the rate constant. The rate constants for each temperature are indicated in
each figure and will be used to determine the activation energy for the catalyzed reaction via an
Arrhenius
Plot. Errors in the rate constants were determined by a linear regression using Excel. The linearity of
each graph is quite good, R2 values greater than 0.99 in each case, which confirms the reaction is first
order with respect to the sucrose concentration. Further, as expected the rate constants increase with
increasing temperature.
3
Hydrolysis of Sucrose @ 300K
3.500
ln( αt – α ) = -0.0651• t + 3.2571
R2 = 0.9973
-1 -1
k = 0.065 +/- 0.001 min
3.000
))
ln(ααtt-α
– α∞
2.500
2.000
1.500
1.000
0.500
0.000
0
10
20
30
40
50
Time (min)
Figure 1: ln(αt – α∞) versus time for the hydrolysis of sucrose at 300K
Hydrolysis of Sucrose @ 310K
3.5000
ln( αt – α ) = -0.0954• t + 3.2361
R2 = 0.9982
-1
-1
k = 0.095 +/- 0.002 min
3.0000
ln(ααt t- -α∞α) )
2.5000
2.0000
1.5000
1.0000
0.5000
0.0000
0
5
10
15
20
25
30
Time (min)
Figure 2: ln(αt – α∞) versus time for the hydrolysis of sucrose at 310K
4
35
Hydrolysis of Sucrose @ 320K
3.5000
ln( αt – α ) = -0.1351• t + 3.2521
R2 = 0.9941
k = 0.135 +/- 0.006 min -1 -1
3.0000
ln(αtα-t α- ∞α) )
2.5000
2.0000
1.5000
1.0000
0.5000
0.0000
0
5
10
15
20
25
Time (min)
Figure 3: ln(αt – α∞) versus time for the hydrolysis of sucrose at 320K
Activation Energy
The data obtained from Figures 1 through 3 is shown in Table 4 along with the calculated values
lnk and 1/T. Errors in the calculated values were determined by using the propagation equations shown
in Appendix A. This data is used to construct an Arrhenius Plot (lnk versus 1/T) and determine the
activation energy for the reaction.
Table 4: Rate constant data for different temperatures
Temperature (K)
300
310
320
error k (min-1)
1
0.065
1
0.095
1
0.135
error
0.001
0.002
0.006
5
lnk
-2.73
-2.35
-2.00
error
0.02
0.02
0.04
1/T (K-1) •10-3
3.33
3.22
3.13
error
0.01
0.01
0.01
Figure 4 shows the Arrhenius Plot obtained from the data in Table 4. As described in the introduction
the activation energy can be determined from the slope of a plot of lnk versus 1/T via Equations 4 and 5.
As shown in Figure 4 the activation energy for the invertase-catalyzed hydrolysis of sucrose is 29.2 +/0.2 kJ/mol. This value is a measure of the energy barrier that the reactants (sucrose, water, and
Arrhenius Plot
-1.5000
lnk = -3508.5• (1/T) + 8.9623
R2 = 1
Ea = 29.2 +/- 0.2 kJ/mol
-1.7000
-1.9000
-2.1000
-2.3000
-2.5000
-2.7000
-2.9000
3.100E-03
3.150E-03
3.200E-03
3.250E-03
3.300E-03
3.350E-03
1/T (K-1)
Figure 4: Arrhenius Plot for the Hydrolysis of Sucrose
invertase) must overcome before products can be formed. The uncertainty in the activation energy was
determined by a linear regression using Excel. It is important to note that the R2 value for the Arrhenius
Plot is 1, which indicates excellent agreement with the Arrhenius equation. In addition, since a linear
regression of the data was used to determine the uncertainty in the activation energy the value obtained
is quite small and may under estimate the error in the experiment.
6
Conclusions
The data and results presented here show that the invertase catalyzed hydrolysis of sucrose
C
O3
H12
+4
C64
H12
O
224
11 + H2 O invertase C
62
1124H2
164
4O
4
4
44
36
(+) rotation
(−) rotation
is indeed a first order process under the conditions studied. At higher sucrose concentrations the
kinetics may deviate from first order as the assumption that the water concentration is in large excess
may not be valid. The rate constants for the hydrolysis of sucrose are summarized in Table 5.
Table 5: Rate Constants
Temperature (K)
Rate Constant (min-1)
300
0.065 ± 0.001
310
0.095 ± 0.002
320
0.135 ± 0.006
The activation energy as determined from the slope of the plot in Figure 4 is 29.2 +/- 0.2 kJ/mol and
agrees well with the literature value of 31.4 kJ/mole4. Comparing our value to the literature value
results in a 7% deviation, [(31.4 – 29.2)/31.4]x100% = 7 %. Seven percent error is fairly good however,
the actual value of 31.4 kJ/mol does not fall within the uncertainty of our reported value, 29.2 +/- 0.2
kJ/mol. The under estimation of the error in the experiment is due to the fortuitous good fit exhibited in
the Arrhenius Plot and repetition of the experiment may lead to a better estimate of the error.
7
Appendix A: Error propagation equations.
(A1)
σ (α −α ) = σα • 2
t
∞
(A2)
σ ln(α −α ) =
(A3)
Errors in the rate constants were determined from a linear regression performed in Excel
using ln(αt – α∞) versus time data.
(A4)
σ ln k =
σk
k
(A5)
σ1 / T =
σT
2
T
(A6)
The error in the slope obtained from the Arrhenius plot was determined from a linear
regression performed in Excel using lnk versus 1/T data.
(A7)
σ Ea = R • σ slope
t
∞
σ (α −α )
t
∞
αt − α∞
8
References
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2.
3.
4.
5.
Chem-354 Lab Manual, Cal Poly State University, 2002
F. J. Kezdy and M. L. Bender, Biochem., 1, 1097 (1962)
M. l. Bender, F. J. Kezdy, and F. C. Wedler, J. Chem. Educ., 44, 84 (1967)
H. B. Dunford, J. Chem. Educ., 61, 129 (1984)
D. P. Shoemaker, C. W. Garland, and J. W. Nibler, Experiments in Physical Chemistry, pp.
264-275, McGraw-Hill, Boston (1996)
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9