1. Phase Equilibrium  When 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.

2. 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

3. Introduction Diffusion Coefficient Diffusion Coefficient –Measures the rate of diffusion –Time-dependent Solubility Solubility –Measures maximum amount of gas dissolved in liquid –Time-independent Henry’s Law constant Henry’s Law constant –Dissolved gas in liquid is proportional to partial pressure in vapor phase Heat of mixing Heat of mixing –Correlation between Henry’s Law constant and T

4. Determination of diffusion coefficient from experimental data 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 (mole/m 3 s) J= Molar flux, transfer by the mechanism of molecular diffusion (mole/m 2 s) C Gas phase Interface ZZ(t)Z=0 CiCi Increasing time

5. Diffusion coefficient The diffusivity or diffusion coefficient D AB of constituent A in solute B which is measure of its diffusive mobility, is defined as the ratio of its flux J A to its concentration gradient. D=is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10−9 to 2x10−9 m 2 /s. For biological molecules the diffusion coefficients normally range from 10 −11 to 10 −10 m 2 /s.

6. Derivation of Fick's 1st law in one dimension Consider a collection of particles performing a random walk in one dimension with length scale ∆z and time scale ∆t. Let N(z,t) be the number of particles at position z at time t. At a given time step, half of the particles would move left and half would move right. Since half of the particles at point z move right and half of the particles at point z+∆z move left, the net movement to the right is: The flux, J, is this net movement of particles across some area a, normal to the random walk during a time interval ∆t. Hence we may write:

7. Multiplying the top and bottom of the right-hand side by (∆z) 2 But the concentration can be defined as particles per unit volum In addition, (∆z) 2 /2∆t is the definition of the diffusion coefficient D Thus the above equation can be written as: At small distance interval ∆z  zero

8. Fick's second law Fick's second law predicts how diffusion causes the concentration to change with time can be derived from Fick's First law and the mass conservation in absence of any chemical reactions: At isothermal and isobaric conditions, D the diffusion coefficient is constant & Rearrange the above equation will get Fick's second law

9. Referring to Fig 1&2 for 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 and the bottom of the cell is semi-infinite; that is, diffusion is slow enough that the concentration at the bottom of the cell is negligible compared to the concentration at the interface. It is also assumed that the distance between the interface and the bottom of the cell is semi-infinite; that is, diffusion is slow enough that the concentration at the bottom of the cell is negligible compared to the concentration at the interface. According to the film theory the gas and the liquid phases at the interface are thermodynamically in equilibrium, i.e. the interface concentration of the solute, C i remains unchanged as long as temperature and pressure of the system are kept constant. Stoddard Solvent V=100 CC Ci C(t,Z) Z(t) Z Gas V=104 CC C Gas phase Interf ace Z Z(t) Z=0 CiCi Incre asing time

10. Fick's second law) Thus the unsteady-state differential equation (Fick's second law) representing concentration changes with time and position is: Solution of Fick’s 2 nd Law using the boundary conditions described is: Solution of Fick’s 2 nd 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 N T versus t 1/2, the slope of this line is equal to 2AC i (D 12 /  ) 1/2 where C = Concentration of dissolved CO 2 in the liquid phase at Z and t. Z = Distance in cm traveled from the liquid interface. t = time D 12 =Diffusion coefficient of species 1 in 2. The boundary conditions are: Z = 0 C = C i Z  ∞: C = 0 The initial condition is C = 0 at t = 0:

11. Solubility Henry’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.

12. N=gram moles of carbon dioxide absorbed in the liquid phase PT=corrected barometer reading P o =vapor pressure of Stoddard Solvent at cell temperature P o =vapor pressure of Stoddard Solvent at cell temperature Tp=temperature at the pump Tc = temperature of the cell (bath temperature) ∆V p =total gas volume delivered from the pump to the cell ∆V p =total gas volume delivered from the pump to the cell Vcg=volume of the gas phase in the cell Zp =compressibility factor of CO2 at pump T and PT Zc =compressibility factor of CO2 at cell T and PT Vd=dead volume in the system (cc)

13. The fugacity, f, can be determined from the Lewis and Randall Rule, which gives f=fugacity of CO 2 in the gas phase f o =fugacity of pure gaseous CO 2 at P T and cell T y=mole fraction of CO 2 in gas phase Thus by definition: the fugacity coefficient for pure CO 2 in the gas phase at cell T and P T

14. Use Henry’s Law coefficients at the three experimental temperatures to obtain the heat of mixing: Use Henry’s Law coefficients at the three experimental temperatures to obtain the heat of mixing: Plotting ln(H) vs. 1/T gives a line with a slope of ΔH mix /R. Plotting ln(H) vs. 1/T gives a line with a slope of ΔH mix /R. ΔH mix is expected to be negative, which would indicate that CO 2 and Stoddard solvent are more energetically stable than apart (i.e., the interactions are favorable). ΔH mix is expected to be negative, which would indicate that CO 2 and Stoddard solvent are more energetically stable than apart (i.e., the interactions are favorable). Heat of Mixing

15. Experimental: Cell Evacuation -

16. Experimental: Filling Syringe -

17. Experimental: Reduce to Atmospheric Pressure +

18. Experimental: Fill Cell 0   between V4 and the cell is 40.5 cm and the pipe diameter is 0.15 cm

19. Penetration Model Brownian motion Brontan motion Brontan motion Brontan motion Brontan motion

24. Properties of Stoddard Solvent and Cell Dimensions Molecular Weight of Stoddard Solvent = 136 g/mole Density of Stoddard Solvent:  = 0.8022 g/cc at 20°C (The isobaric expansion coefficient is 0.00104/°C.) Weight of Stoddard Solvent in the cell = 40.11 g at room temp Volume of the Cell 150 ml Di of the cell = 50.4 mm Volume of the magnetic stirrer = 1.1 ml Length of the tube from the top of the cell to the bulkhead = 9.75” Length of the tube going through the panel to valve 4 = 7.5” Di of the tube 1/8” From these info please Calculate the Void volume (the volume of the gas above the liquid phase) From these info please Calculate the Void volume (the volume of the gas above the liquid phase) Saddawi Jan 2015

25. References Koretsky, Milo D. Engineering and Chemical Thermodynamics. John Wiley & Sons, Inc., 2004. Koretsky, Milo D. Engineering and Chemical Thermodynamics. John Wiley & Sons, Inc., 2004. Ophardt, Charles E. Virtual Chembook. Elmhurst College, 2003. [Online] Available at: http://www.elmhurst.edu/~chm/vchembook/174temppres.html Ophardt, Charles E. Virtual Chembook. Elmhurst College, 2003. [Online] Available at: http://www.elmhurst.edu/~chm/vchembook/174temppres.html http://en.wikipedia.org/wiki/Lake_Nyos http://en.wikipedia.org/wiki/Lake_Nyos