In this article we will discuss about the Fick’s law of diffusion.
Consider a chamber in which two different gas species, B and C, at the same temperature and pressure are initially separated by a partition. The left compartment has a high concentration of gas B (dark circles) whereas the right compartment is rich in gas C (white circles). Higher concentration means more molecules per unit volume.
When the partition wall is removed there occurs a driving potential which tends to cause the concentration difference to equalize or become uniform. The molecules escape from the zone of higher concentration to travel towards the zone of lower concentration. Apparently both the species are transported by diffusion and gradually mix with each other.
ADVERTISEMENTS:
Fig. 14.1 represents the situation at a certain instant shortly after the removal of partition. The concentration of species B decreases with increasing x while the concentration of C increases with x. Mass diffusion is in the direction of decreasing concentration. Evidently there is net transfer of species B to the right and of species C to the left. After a sufficiently long period, equilibrium conditions prevail, i.e., uniform concentration of species B and C is achieved and then the mass diffusion ceases.
Experimental evidence indicates that molecular diffusion is governed by the empirical relation suggested by Fick:
The dependence of diffusion on concentration profile has been illustrated in Fig. 14.2. The concentration of species B is higher on the left side of the dotted plane and apparently the mass flow of species B occurs towards the right.
The diffusion rate for species C would be:
The -ve sign in equations 14.1 and 14.2 accounts for the fact that diffusion takes place in the direction opposite to that of increasing concentration. The diffusion coefficient Dbc or Dch is dependent upon the temperature, pressure and nature of the components of the system.
ADVERTISEMENTS:
The Fick’s law of diffusion as prescribed by equations 14.1 and 14.2 is analogous to the Fourier law of heat conduction-
Apparently the Fourier equation describes the transport of heat energy due to temperature gradient, the shear equation describes the transport of momentum due to velocity gradient and the Fick’s law describes the mass transport due to concentration gradient.
The dimensions of diffusion coefficient D as described by equation 14.1 are –
The units of mass diffusion coefficient are thus seen to be identical with those of thermal diffusivity α and kinematic viscosity v. Diffusion coefficient is thus a transport property of the fluid.
From characteristic equation applied to species B,
where pb is the partial pressure of species B and Mb is the molecular weight and G is the universal gas constant (G = 8314 Nm/kg mol K) . The mass density ρb represents the mass concentration Cb as used in the Fick’s law.
Substituting the value of Cb in equation 14.1, the Fick’s law of diffusion for component B into constituent C may be written as
Likewise, the diffusion of component C into constituent B would be
Equations 14.3 and 14.4 are essentially valid only for isothermal (constant temperature) diffusion.
Some aspects of Fick’s law of diffusion are:
(i) Fick’s law cannot be derived from first principles, i.e., it is a generalisation based on experimental evidence.
(ii) Fick’s law is valid for all matter regardless of its state; solid, liquid or gas. Mass transfer is strongly influenced by molecular spacing and as such diffusion occurs more readily in gases than in liquids and more readily in liquids than in solids.
(iii) A diffusion substance moves in the direction of decreasing concentration. The difference in concentration in a diffusing process is similar to the difference in temperature in a heat flow process.
(iv) Besides concentration gradient, the mass diffusion may occur as the result of a temperature gradient, a pressure gradient or an external force. While applying Fick’s law it is to be presumed that these additional effects are either not present or are too small.
(iv) In general, the diffusivity or diffusion coefficient D depends upon temperature, pressure and nature of the component of the system. However, for ideal gases and dilute liquids, the diffusivity coefficient can be presumed to remain practically constant for a given range of pressure and temperature.
Empirical relations for the diffusion coefficient of gases have been developed from the concept of kinetic theory of gases and the most general expression is of the form-
Where,
pt and T = total pressure in atmosphere and absolute temperature of the binary gaseous system, K
Mb and Mc = molecular weights of the gas species
Vb and Vc = molecular volumes of constituent species at normal boiling points cm3/gm-mol.
Obviously the diffusion coefficient for gases depends upon pressure, temperature and other molecular properties of the diffusing gases. These values can be extended to other pressures and temperatures by using the equations
Example:
Estimate the diffusion coefficient for ammonia in air at 25°C temperature and one atmospheric pressure.
For ammonia:
molecular weight = 17 and molecular volume = 25.81 cm3/gm mole
For air:
molecular weight = 29 and molecular volume = 29.89 cm3/gm mole
Solution:
The diffusion coefficient for binary gaseous mixtures is worked out from the relation:
