Iodine sublimation
Davíð Örn Þorsteinsson (1), Guðjón Henning Hilmarsson (2) and Saga Huld Helgadóttir (3)
1) dth21@hi.is 2) ghh9@hi.is and 3) shh14@hi.is
Abstract: A spectrophotometer is used to measure absorbance of iodine vapor in equilibrium with
solid iodine at different temperatures. From this
the vapor pressure of solid iodine is determined.
The heat of sublimation, i.e. the enthalpy change,
of solid iodine is then calculated with the ClausiusClapeyron equation. For comparison, the enthalpy
change is also calculated using statistical mechanics.
Introduction
At several temperatures, vapor pressures of solid
iodine are obtained from optical absorbtion measurements. The enthalpy change, in this case the
heat of sublimation of solid iodine, can be calculated with the Clausius-Clapeyron equation. This
requires at least two values of vapor pressures.
Application of statistical mechanics to derive thermodynamic information from spectroscopic data
is made possible by the system I2 (s) − I2 (g). The
vibrational partition function for the solid is calculated using vibrational frequencies given by the
Einstein model for a solid. The vibrational frequency of the I2 molecule in the gas phase, which
is needed to formulate the vibrational partition
function, is obtained from the absorbtion spectrum in the visible region. Also, the gas phase’s
rotational partition function can be calculated
from the known internuclear distance in the iodine
molecule. With this information at hand along
with statistical mechanical theory, one can calculate the molar energy difference ∆Ẽ00 between
the vibrational ground states of crystalline and
gaseous iodine from a measurement of vapor pressure at one temperature. Crystalline and gaseous
iodine partition functions make the entropy and
enthalpy changes attending sublimation of iodine
calculable at any temperature T . The calculated
value of ∆H̃sub from this method can then be compared with the measured value obtained from the
Clausius-Clapeyron equation.
Theory
In order to obtain ∆H̃sub , phase equilibrium is required:
I2 (s) = I2 (g)
(1)
For equilibrium it is required that the chemical
potential of I2 is the same in both the gas and
solid phase:
µs = µg
(2)
These chemical potentials need to be determined.
For a one-component system the chemical potential per mole is given by
µ=
∂A
∂n T,V
= −RT
= −kT
∂ ln Q
∂N
∂ ln Q
∂n
T,V
(3)
T,V
where n is the number of moles in the system and
N is the number of molecules in the system.
A crystal may be considered to a good approximation as an aggregate of independent harmonic
oscillators. Then from equation (3) one can get
that
µs = −3tRT ln qs
(4)
qg
(5)
N
where qs and qg are microcanonical partition functions of each oscillator.
The partition function qg is well approximated as
a product of terms arising from translational, rotational, vibrational and electronic degrees of freedom:
qg = qtrans qrot qvib qel
(6)
µg = −RT ln
The translational, rotational, vibrational and electronic partition functions are
3/2
V
qtrans = 2πmkT
h2
T
σΘrot
−Θvib /T −1 −∆0 /kT
qrot =
qvib = 1 − e
e
qel = 1
where m is the molecular mass, k is the Boltzmann constant, T is the absolute temperature, h
is Planck’s constant, V is the volume within which
the molecule is constrained to move, σ is the symmetry number of the molecule and ∆0 is the energy required to remove a molecule from the crystal at absolute zero temperature.
For the crystalline state of I2 the partition function
consists only of a vibrational part.
"M
#
Y
−Θvib /T −1
qs =
(1 − e
)
(7)
i=1
Inserting qs and qg into eqs. (4) and (5) results in


12
Y
RT 
µs =
ln
(1 − e−Θj /T ) 
(8)
2
j=1
Table 1: Molar absorption coefficients of iodine
vapor at λ = 520 nm
µg = ∆Ẽ00 − RT
"
#
kT T
−Θvib /T −1
· ln
(1 − e
)
p σΘrot
(9)
where σ = 2 is the end-for-end symmetry and
Θvib/rot is the vibrational/rotational characteristic temperature.
Equating eqs. (8) and (9) gives, after some rearranging and simplifying
7/2 Q12
T
(1−e−Θj /T )1/2
j=1
ln p − ln
(1−e−Θvib /T )
2πmkT
h2
"
= ln
3/2
2πmk 3/2
h2
!
#
k
∆Ẽ00
−
σΘrot
RT
(10)
Knowing both µs (T ) and µg (T ), one can calculate ∆S̃sub and ∆H̃sub . By contrast, the ClausiusClapeyron equation, given by
∆H̃sub
ln p = constant −
RT
(11)
requires at least two values of p at different temperatures in order to obtain a value of ∆H̃sub .
Because this is a system of only one component,
the chemical potentials for I2 in crystalline and
gaseous forms are equivalent to the molar Gibbs
free energies G̃s and G̃g , aside from an additive
constant. Entropies of the two phases can be obtained by differentiating with respect to temperature.
G̃s
s
S̃s = − ∂∂T
= − ∂µ
∂T
p
=
∂ G̃g
∂T
=−
p
(m2 mol−1 )
69.10
68.65
68.20
67.65
67.20
66.75
66.30
65.85
65.40
65.00
64.60
64.20
63.80
Instruments used to conduct this experiment were:
A UV/Visible Spectrophotometer, which measures absorbance of samples at certain wavelengths; water thermostat with adjustable temperature regulator and circulating pump; plastic
tubing to connect circulating pump with water
jacket of absorbtion cell; Voltmeter and a type
J thermocouple(iron-constantan); Water-jacketed
absorption cells containing solid I2 and air. The
schematic for the setup can be found in fig. 1.
No effective temperature difference was measured
between the water thermostat and samples.[2]
p
12 RX
Θj /T
−Θj /T
−
ln(1
−
e
)
2 j=1 eΘj /T − 1
S̃g = −
T ( ◦ C)
20
25
30
35
40
45
50
55
60
65
70
75
80
∂µg
∂T
(12)
p
Ẽ00 − µg
7
Θvib /T
(13)
+ R + R Θ /T
T
2
e vib − 1
Then the heat sublimation at temperature T is
=
∆H̃sub = T ∆S̃sub = T (S̃g − S̃s )
Fig. 1: Schematic setup of experiment. a) spectrophotometer; b) absorbtion cell; c) water thermostat; d) plastic tubing; e) Voltmeter and thermocouple.
(14)
Experimental
Spectrophotometric measurements of the absorbance A of the iodine vapor in equilibrium with
the solid at the absorbtion maximum, 520 nm, determines the vapor pressure of solid iodine. It is
also measured at 700 nm, where the molar absorption coefficient of iodine vapor is so small as to
be negligible. Measurements are performed from
20◦ C to 80◦ C in steps of 5◦ C. It is necessary to
wait 10 − 15 min after each temperature change
before a measurement is made, for the system to
reach stability.
Calculations
The net absorbance is determined for each temperature as the difference between the absorbance
for the iodine absorption cell and the empty absorption cell at 520 nm.
A = Aiodine − Aempty
(15)
The net absorbance is related to c, the concentration of I2 vapor, and p, the pressure of I2 vapor
(assuming the perfect-gas law):
A = dc =
d
p
RT
(16)
where d is the optical path length inside the inner
absorption cell, d = 1 cm.
The value of the molar absorption coefficient can
be found in Table 1. Then p is calculated with eq.
(16).
∆H̃sub is determined from both experiment and
theory. The experimental value is determined from
the approximate Clausius-Clapeyron equation, eq.
(11), where ln (p) is plotted against 1/T and the
slope of the best straight-line fit to the data is
∆H̃sub /R, see fig. 2.
The values of p and T from the experiment are
used for the statistical mechanical calculations.
∆Ẽ00 is calculated with eq. (10) at each temperature. Then it is possible to calculate the molar
entropies S̃g and S̃s of the crystalline and vapor
forms of I2 at every measured temperature, with
eqs. (12) and (13), and obtain the molar heat of
sublimation ∆H̃sub with eq. (14) by taking the
mean value of calculated values for each temperature.
Fig. 3: Purple iodine vapor.
of the crystal by ∼ 10% results in a change of
only ∼ 2% in ∆H̃sub . Therefore small variations
in ν̃j result in miniscule changes in ∆H̃sub . Only
substantial changes in ν̃j can result in effective difference in ∆H̃sub .
Examination of the iodine vapor in the absorbtion
cell after the measurements shows a purple color.
That is because light with wavelength 520 nm has
been absorbed, leaving a mix of blue (∼ 480 nm)
and red (∼ 700 nm), as seen in fig. 3.
References
[1] David P. Shoemaker et al, Experiments in
physical chemistry, 6th ed., p. 514-528, 1997.
[2] pyromation.com, Type J Thermocouple.
[3] en.wikipedia.org, Standard enthalpy change of
formation(data table)
Fig. 2: ln (p) as a function of 1/T
Conclusion
The two values for heat of sublimation acquired
from measurements and statistical mechanical calculations are
∆H̃sub, measured = 64 ± 4 kJ/mol
∆H̃sub, calculated = 69 ± 4 kJ/mol
The measured value of ∆H̃sub is obtained by
the best fit line through the measurement points.
However, the first two points have such great uncertainty values that they are omitted. By doing
this, a more accurate value of ∆H̃sub is acquired.
The correct value of ∆H̃sub for gaseous idodine, I2 ,
is 62.438 kJ/mol. The measured value of ∆H̃sub
is consistent with the correct value.
The cause of difference in the calculated value
from the correct one can stem from approximations made in calculations. By changing the representative values of ν̃j for the 12 vibrational modes