Phase EquilibriumPhase EquilibriumWhen a gas and a liquid phase which are not thermodynamically in equilibrium are brought into close contact, transfer of one or more components may occur from the gas phase to the liquid or, vice versa, by the mechanism of molecular diffusion. Mass transfer by molecular diffusion is the basic physical mechanism underlying many important areas of soil science, petroleum engineering, chemical engineering, biotechnology and nuclear engineering. In this experiment, a method for determining diffusion coefficients of Carbon dioxide gas in Stoddard solvent at constant volume, pressure and temperature is developed using Integral Phase Equilibria Unit.

Determine diffusion coefficient, Solubility, Henrys Constant The enthalpy of solution of carbon dioxide

in Stoddard solvent in the range of 18 - 35°C and at 1.0 atmosphere pressure.

Objective

IntroductionIntroduction

Diffusion CoefficientDiffusion Coefficient– Measures the rate of diffusionMeasures the rate of diffusion– Time-dependentTime-dependent

SolubilitySolubility– Measures maximum amount of gas dissolved in liquidMeasures maximum amount of gas dissolved in liquid– Time-independentTime-independent

Henry’s Law constantHenry’s Law constant– Dissolved gas in liquid is proportional to partial pressure in Dissolved gas in liquid is proportional to partial pressure in

vapor phasevapor phase Heat of mixingHeat of mixing

– Correlation between Henry’s Law constant and TCorrelation between Henry’s Law constant and T

Determination of diffusion coefficient Determination of diffusion coefficient from experimental datafrom experimental data

C

Gas phase

Interface

Z Z(t) Z=0

Cav

rJCutC vv +∇=∇+∂

..∂

A number of mathematical models have been proposed to determine the diffusion coefficients from experimental volume–time profiles, however all these models are developed from the equation of continuity for the solute component:

where r = Rate of reaction (kg/m3s)J = Mass transfer by the mechanism of molecular diffusion (kg/m2s)v = Molar volume (m3)

Stoddard Solvent

V=100 CC

Referring to Fig 1&2for a one-dimensional diffusion cell absence of chemical reaction, movement of the interface in the boundary conditions of the system, in which a component in the gas phase is absorbed into a liquid phase starting at time zero and continuing at longer times.

Based upon a model proposed by Higbie (penetration theory)

the liquid interface is thus always at saturation, since the molecules can diffuse in the liquid phase away from the interface only at rates which are extremely low with respect to the rate at which gaseous molecules can be added to the interface. It is also assumed that the distance between the interface It is also assumed that the distance between the interface and the bottom of the cell is semi-infinite; that is, diffusion and the bottom of the cell is semi-infinite; that is, diffusion is slow enough that the concentration at the bottom of the is slow enough that the concentration at the bottom of the cell is negligible compared to the concentration at the cell is negligible compared to the concentration at the interface.interface.

According to the film theorythe gas and the liquid phases at the interface are thermodynamically in equilibrium, i.e. the interface concentration of the solute, Ci remains unchanged as long as temperature and pressure of the system are kept constant.

Ci

C(t,Z)

Z(t)

Z

Gas V=104 CC

Thus the unsteady-state differential equation representing concentration changes with time and position is:

Solution of Fick’s 2Solution of Fick’s 2ndnd Law using the boundary conditions described is: Law using the boundary conditions described is:

Solve for the number of moles added up to a time t:Solve for the number of moles added up to a time t:

If one plots NT versus t1/2, the slope of this line is equal to

2ACi (D12/)1/2

€

D12∂ 2C∂Z2 = ∂C

∂t

whereC = Concentration of dissolved CO2 in the liquid phase at Z and t.Z = Distance in cm traveled from the liquid interface.t = timeD12 =Diffusion coefficient of species 1 in 2.

€

Ci − CCi

= erf Z2 D12t

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

The boundary conditions are:

Z = 0 C = Ci

Z ∞: C = 0

The initial condition is

C = 0at t = 0:€

NT = 2ACiD12

π ⎛ ⎝ ⎜

⎞ ⎠ ⎟12

t

SolubilityHenry’s Law constant

The solubility of a gas in a liquid solvent may be represented to good accuracy at dilute concentrations of the dissolved gas by Henry's Law:

f = H X

where f is the fugacity of the gas in the gas phase in equilibrium with the liquid phase of concentration X of dissolved gas.

H is the Henry’s law constant, which is a function of temperature. Thus, by measuring the solubility one can obtain an estimate of the Henry's law

constant.

€

X = nn + nsolvent

nn == gram moles of carbon dioxide absorbed in the liquid phasegram moles of carbon dioxide absorbed in the liquid phase

PTPT == corrected barometer readingcorrected barometer reading == vapor pressure of Stoddard Solvent at cell temperaturevapor pressure of Stoddard Solvent at cell temperature

TpTp == temperature at the pumptemperature at the pump

Tc Tc = = temperature of the cell (bath temperature)temperature of the cell (bath temperature) == total gas volume delivered from the pump to the celltotal gas volume delivered from the pump to the cell

VcgVcg == volume of the gas phase in the cellvolume of the gas phase in the cell

Zp Zp == compressibility factor of CO2 at pump T and PTcompressibility factor of CO2 at pump T and PT

Zc Zc == compressibility factor of CO2 at cell T and PTcompressibility factor of CO2 at cell T and PT

VdVd == dead volume in the system (cc) dead volume in the system (cc)

€

n = PT (ΔVP − Vd )ZpRTP

−Vcg(PT − P1

o)ZcRTc

The fugacity, f, can be determined from the Lewis and Randall Rule, which gives

f = fugacity of CO2 in the gas phase

fo = fugacity of pure gaseous CO2 at PT and cell T y = mole fraction of CO2 in gas phase

Thus

by definition:

the fugacity coefficient for pure CO2 in the gas phase at cell T and P T

€

f = f ° y

€

y = PT − P1°

PT

€

f = f °

PT

⎛ ⎝ ⎜

⎞ ⎠ ⎟ PT −P1

°( )

€

f °

PT

= φ

Use Henry’s Law coefficients at the three experimental temperatures to Use Henry’s Law coefficients at the three experimental temperatures to obtain the heat of mixing:obtain the heat of mixing:

Plotting ln(H) vs. 1/T gives a line with a slope of Plotting ln(H) vs. 1/T gives a line with a slope of ΔΔHHmixmix/R./R.

ΔΔHHmixmix is expected to be negative, which would indicate that CO is expected to be negative, which would indicate that CO22 and and Stoddard solvent are more energetically stable than apart (i.e., the Stoddard solvent are more energetically stable than apart (i.e., the interactions are favorable).interactions are favorable).

Heat of Mixing

€

∂ln Hi( )∂ 1/T( )

⎛

⎝ ⎜

⎞

⎠ ⎟P ,xCO2

= ΔHmix

R

Experimental: Cell Evacuation

-

Experimental: Filling Syringe

-

Experimental: Reduce to Atmospheric Pressure

+

Experimental: Fill Cell

0 between V4 and the cell is 40.5 cm and the pipe diameter is 0.15 cm

Penetration ModelPenetration Model

0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time [=] Seconds

Concentration [=] Molarity

Concentration as a function of time for a given height

Concentration at bottom of tankConcentration at middle of tankEnding time of experiment

ReferencesReferences Koretsky, Milo D. Koretsky, Milo D. Engineering and Chemical Engineering and Chemical

ThermodynamicsThermodynamics. John Wiley & Sons, Inc., 2004.. John Wiley & Sons, Inc., 2004. Ophardt, Charles E. Ophardt, Charles E. Virtual ChembookVirtual Chembook. Elmhurst College, . Elmhurst College,

2003. [Online] Available at: 2003. [Online] Available at: http://www.elmhurst.edu/~chm/vchembook/174temppres.http://www.elmhurst.edu/~chm/vchembook/174temppres.htmlhtml

http://en.wikipedia.org/wiki/Lake_Nyoshttp://en.wikipedia.org/wiki/Lake_Nyos

Cell informationCell information the dimensionthe dimension Diameter = 51.43 mmDiameter = 51.43 mm Height of the lid = 21.7 mmHeight of the lid = 21.7 mm Diameter to the lower section = 50.4 mmDiameter to the lower section = 50.4 mm Depth of the lower section averaged = 70.5 Depth of the lower section averaged = 70.5

mmmm Volume of the Stoddard liquid 100mlVolume of the Stoddard liquid 100ml Volume of the space (Gas) 104 mlVolume of the space (Gas) 104 ml