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| Fig.1 Concentration gradient. |
Diffusion happens due to a gradient in a medium, for example if we add a drop of ink in water the ink will diffuse in the water to ultimately make an homogeneous mixture.
The Physical Process
When a inhomogeneity is set in a mix of two liquids the process of diffusion is set. Molecules of the more concentrated fluid diffuses until the concentration is uniform. Note that the process is not set by any kind of force but simply by a random process known as Brownian motion.
Brownian Motion
A particle in a fluid is surrounded by many other particles which will interact with it though multiple collisions (see figure 2). Due to this constant bombardment the fluid particle will undergo an erratic movement throughout the space available. The size of every step is given by the mean free path (l) which is a measure of the distance covered between successive collisions, while the root-mean-squere speed is given by:
where:
k: 1.3806503 × 10-23 m2 kg s-2 K-1 (Boltzmann constant)
T: Absolute temperature (K)
M: Mass of the particle
The proportionality constant will depend of the inner degrees of freedom of the molecule (vibrational, rotational, etc).
Diffusion
Now lets consider a vessel filled with two fluids A and B where fluid B is concentrated in a small region of the overall volume and fluid A is uniformly distributed in the vessel. You can think of fluid B as a drop of ink in a glass of water (fluid A).
If c is the concentration of B at x (see Fig. 3) then, the concentrations at x - dx and x + dx will be given by:
c(+) = c(x) + dx*dc/dx < c(x) (2)
The amount of particles in the volume ΔV(-) contained between x - dx and x will be approximately N(-) = c(-) ΔV(-). Similarly, N(+) = c(+)ΔV(+) with N(-) > N(+). When a particle moves in ΔV(-) it can either move towards x or move away from x. In average, as many particles will be moving towards x as moving away from it in ΔV(-). The same arguments can be applied to ΔV(+). Therefore as many particles in ΔV(+) are moving away from x as moving towards it. But because ΔV(-) is more populated than ΔV(+) as is inferred from the premise that c(-) > c(x), then more B particles are moving to the right of x than to the left of x, i.e., there is a net flux of B particles moving towards the regions where the concentration of B is lower.
If the time between collisions is τ, and dx ~ l (mean-free-path), then the flux towards the right would be approximately:
Where Δc = dx*dc/dx, ΔV = ΔV(-) = ΔV(+) and equations (1) and (2) have been used.
Conclusion
Diffusion is the result of Brownian motion of the molecules of a fluid and not of any kind of forces appearing because of a concentration gradient. The speed of the diffusion process is given by the concentration difference between to regions, the amount of particles participating in the motion and the time between collisions. Note that higher collision rate (i.e., smaller τ) increases the speed of diffusion as collisions are at the core of the random nature of the movement of particles in a fluid.
The Physical Process
When a inhomogeneity is set in a mix of two liquids the process of diffusion is set. Molecules of the more concentrated fluid diffuses until the concentration is uniform. Note that the process is not set by any kind of force but simply by a random process known as Brownian motion.
Brownian Motion
 |
| Fig.2 Brownian motion |
v(rms) ~ sqrt(kT/M)
where:
k: 1.3806503 × 10-23 m2 kg s-2 K-1 (Boltzmann constant)
T: Absolute temperature (K)
M: Mass of the particle
The proportionality constant will depend of the inner degrees of freedom of the molecule (vibrational, rotational, etc).
Diffusion
 | |
| Fig. 3 Global flow to the right |
If c is the concentration of B at x (see Fig. 3) then, the concentrations at x - dx and x + dx will be given by:
c(-) = c(x) + dx*dc/dx > c(x) (1)
c(+) = c(x) + dx*dc/dx < c(x) (2)
The amount of particles in the volume ΔV(-) contained between x - dx and x will be approximately N(-) = c(-) ΔV(-). Similarly, N(+) = c(+)ΔV(+) with N(-) > N(+). When a particle moves in ΔV(-) it can either move towards x or move away from x. In average, as many particles will be moving towards x as moving away from it in ΔV(-). The same arguments can be applied to ΔV(+). Therefore as many particles in ΔV(+) are moving away from x as moving towards it. But because ΔV(-) is more populated than ΔV(+) as is inferred from the premise that c(-) > c(x), then more B particles are moving to the right of x than to the left of x, i.e., there is a net flux of B particles moving towards the regions where the concentration of B is lower.
If the time between collisions is τ, and dx ~ l (mean-free-path), then the flux towards the right would be approximately:
Φ ~ ΔcΔV/τ (3)
Where Δc = dx*dc/dx, ΔV = ΔV(-) = ΔV(+) and equations (1) and (2) have been used.
Conclusion
Diffusion is the result of Brownian motion of the molecules of a fluid and not of any kind of forces appearing because of a concentration gradient. The speed of the diffusion process is given by the concentration difference between to regions, the amount of particles participating in the motion and the time between collisions. Note that higher collision rate (i.e., smaller τ) increases the speed of diffusion as collisions are at the core of the random nature of the movement of particles in a fluid.
