Behavior of Sulfur Dioxide Undergoing Phase
Change from Liquid to Gas
Dr. Adonios Karpetis
Team 6
__________________________________
Sualeh Khurshid
__________________________________
Ricardo Martinez
__________________________________
Matthew Osvog
___________________________________
Christopher White
Abstract
Phase changes are omnipresent in nature. Nature and human design employ phase changes in a
variety of phenomenon and cycles. Some everyday examples include evaporation of water in a water
cycle, melting of ice, condensation of water vapor on a soda can etc.
This project studies the phase transition of Sulfur Dioxide from liquid to gas based on the Van der
Waals equation of state. The emphasis has been put on the behavior inside the vapor dome below the
critical temperature, between 100 K and 300 K. Inside the vapor dome Sulfur dioxide undergoes
non-linear/retrograde behavior. Sulfur dioxide exhibits some interesting features in the given
temperature range. This behavior helps describe and explain various phenomena such as the rising of
sap in trees.
Table of Contents
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
Introduction…………………………………………………………………………2
Theory……………………………………………………………………………….2
Results……………………………………………………………………………….4
Discussion…………………………………………………………………………...4
Summary…………………………………………………………………………….5
Conclusion…………………………………………………………………………..5
Appendix………………………………………………………………………….....6
References…………………………………………………………………………...8
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I.
Introduction
Most textbooks use an idealized image of isotherms and isobars. However, as mentioned above, a
realistic isobar or isotherm follows a non-linear path inside the vapor dome. This occurs due to the
quickness of a realistic transition from phase to phase that does not allow for quasi-equilibrium. This
project aims to demonstrate this nonlinear behavior for Sulfur Dioxide using The Van der Waals
equation of state. By using Van der Waals equation and data from NIST, values of pressure and
temperature can be calculated for different isotherms and isobars respectively. These values can then
be plotted to show the behavior of pressure and temperature for realistic data inside the vapor dome.
In order to fully explain the behavior exhibited by Sulfur Dioxide we will discuss the theory behind
the Van der Waals equation of state, present our findings, and analyze the validity of the results we
obtained.
II.
Theory
Phase changes can occur due to changes in temperature, pressure or specific volume (or density) of a
substance. They are represented using a P-v, T-v, P-T diagram keeping third variable constant. The
resulting plots are called isotherms, isochores or isobars. The plots are developed through
experimental data, but can also be modeled using equations. This project employs the Van der Waals
equation of state (VDW-EOS) to describe the curves. VDW-EOS can be written explicit in pressure
and temperature as shown
The VDW-EOS is a modification on the ideal gas law, where ‘a’ is a positive constant dealing with
the size of the gas particles and ‘b’ is related to the radius of influence of the Van der Waals forces
between the particles. The values of both the constants depend on the gas under consideration. Their
values can be theoretically calculated by taking the first and second derivatives of the VDW-EOS
and solving the three equations[1]:
RTc
2a
 P 
 3 0
  
2
 v T
 vc  b  vc
 2 P 
RTc
6a
 4 0
 2 
3
 v T  vc  b  vc
a
27 Rc 2Tc 2
64 Pc
b
RTc
8Pc
vc 
8 RTc
3 Pc
Where, a = 6.883 bar(m /kmol) , b = 0.0569 m /kmol, and R = 8.314 kJ/kmol.K. Pc , Tc and vc are the
critical pressure, temperature and molar specific volume, respectively.
3
2
3
2
VDW-EOS predicts some unforeseen results, such as the existence of negative pressures and the nonlinear behavior in liquids undergoing phase change. Although the equation does not fully describe
this behavior, with some assumptions it predicts the behavior of fluids very accurately.
Figure 1: Generalized behavior of phase change, Liquid-Vapor[4]
The general idea about phase change is that it occurs at constant temperature/pressure (Figure 1).
This is true for temperatures above and near the critical point. For temperatures below the critical
point, the behavior is a spinodal line (Figure 2). Under this spinodal line the liquid is said to be in a
metastable state (between a-b, Figure 3). Hence there is a maximum and minimum pressure that is
achieved during phase change. Such non-linear behavior inside the vapor dome is predicted by
the VDW-EOS and further explained by the Maxwell’s correction. The area of the non-linear curve
above the constant pressure line (region a-d-b-a in Figure 3) is the negative of the area under the curve
below the constant pressure line (region b-e-c-b in Figure 3). Therefore no extra work is done in the nonlinear character as compared to the linear one[3].
Figure 2 : An example of a real Isotherm of SO2
at 300 K, pressure becomes negative during
Liquid-Vapor phase change[4]
Figure 3: Maxwell’s Correction[2]
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The isotherms were plotted using Microsoft Excel and the vapor dome was created using data from
NIST.
III.
Results
Using the Van der Waals equation of state and data from NIST, plots of P-v diagram (Figure 4, see
Appendix) and T-v diagram (Figure 5, see Appendix), with the vapor dome and several isotherms
(T=200 K, 250 K and 350 K) and isobars (P=1 bar, 10 bar and 25 bar) are shown, respectively.
Because of the large range in specific volume, a logarithmic scale is used on the horizontal axis for
both diagrams. These plots clearly show the existence of a non-linear behavior for Sulfur Dioxide
inside the vapor dome. In Figure 4 lower values of specific volume lead to negative values of
pressure. This result is unexpected but can be explained with the Van der Waals equation. This is
discussed in the next section.
IV.
Discussion
It can be seen in the P-v diagram (Figure 4) that pressure does not stay constant within the vapor
dome. In fact, Figure 5 shows the minimum pressure encountered during phase change can be
negative! This is another unforeseen result of the VDW-EOS. However, if pressure is taken to be the
normal component of stress on a surface, it can be negative. When phase change occurs between 200
K and 300 K, the minimum pressure on the isotherm becomes negative. At these temperatures when
we try to expand the liquid, the attractive Van der Waals forces between the particles resist this
expansion and attraction takes place resulting in negative pressures. The liquid surface comes under
tension. Large values of negative pressures are observed inside the xylem tubes of tall trees.
The corresponding effect can be seen in the T-v diagram (Figure 5). In our initial plot of Figure 5 the
VDW-EOS produced negative values of temperature. On an absolute temperature scale such as
Kelvin this is not possible. This occurred due to the nature of the equation itself. VDW-EOS cannot
be used to plot the behavior of a substance at all specific volumes or temperatures. As seen in Figure
6, below the triple point (T), the transition is from solid to liquid, and hence the liquid-vapor phase
change plot cannot be generated from the VDW-EOS. Also we cannot use the equation at pressures
that correspond to sublimation. Therefore the plotting of liquid-vapor phase change using the VDWEOS has a limited range.
Figure 6: A general phase diagram
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V.
Summary
It was found that pressure and temperature do not stay constant within the liquid-vapor dome and that
it actually displays non-linear behavior within the vapor dome.
VI.
Conclusions
The liquid molecules of Sulfur Dioxide are actually held together by weak, intermolecular Van der
Waals forces. Although they are weak, these forces are actually significant as they are responsible for
the non-linear behavior of the fluid shown on the diagrams in Figure 5 and 6. The Van der Waals
forces and the VDW-EOS explain the non-linear behavior of a realistic isotherm and isobar that do
not undergo a quasi-equilibrium phase change.
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VII.
Appendix
6
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VIII. References
[1] Moran, Michael J., and Howard N. Shapiro. Fundamentals of Engineering Thermodynamics. New
York: Wiley, 2000. Print.
[2] Wikipedia. Wikimedia Foundation, n.d. Web. 02 Apr.
2013. <http://en.wikipedia.org/wiki/File:MaxwellEqArea.svg>.
[3] Phase Transitions. N.p., n.d. Web.
<http://www.uam.es/personal_pdi/ciencias/gnavascu/3_Phase_Transitions_Part_A.pdf>.
[4] "8.1 Behavior of Two-Phase Systems." 8.1 Behavior of Two-Phase Systems. MIT OCW, n.d. Web. 02
Apr. 2013. <http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node61.html>
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