Keith Mack
Fick's laws of diffusion, first posited by Adolf Fick in 1855, describe the movement of particles from high to low concentration. Fick's first law can be used to derive the second law, which predicts changes in concentration gradient over time due to diffusion. The first law applies to systems where conditions remain constant, while the second law is more versatile, applicable to physical science and changing systems. Fick's second law is a special case of the convection-diffusion equation, and while it can be solved numerically, it lacks an analytical solution. These two laws are fundamental to understanding diffusion and its coefficient, D.
| Characteristics | Values |
|---|---|
| Fick's First Law | Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient |
| Fick's Second Law | Prediction of change in concentration gradient with time due to diffusion; a special case of the convection-diffusion equation in which there is no advective flux and no net volumetric source |
| Applicability | Fick's First Law can be applied to systems in which the conditions remain the same; Fick's Second Law is more applicable to physical science and other systems that are changing |
| Solving the Equation | The integral can only be solved numerically with a computer, so erf tables are used to solve the diffusion equation where necessary |
| Relation to Diffusion | Fick's laws relate to the diffusion of solute in a solvent |
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What You'll Learn
- Fick's first law: movement of particles from high to low concentration
- Fick's second law: a special case of the convection-diffusion equation
- Applicability: Fick's first law for static systems, second law for dynamic systems
- Solving Fick's second law: erf tables and numerical analysis
- Fick's laws: describe diffusion and can be used to solve for the diffusion coefficient (D)
Fick's first law: movement of particles from high to low concentration
Fick's laws of diffusion describe diffusion and were first introduced by Adolf Fick in 1855, based on experimental results. Fick's first law describes the movement of particles from high to low concentration, also known as diffusive flux. This movement is directly proportional to the concentration gradient of the particles. In other words, the rate at which particles move from an area of high concentration to an area of low concentration is influenced by the difference in concentration between the two areas.
Fick's first law can be applied to various fields, including chemistry, biology, and medicine, to understand the relationship between mass transfer fluxes and concentration gradients in molecular diffusion processes. For example, it can explain the diffusion of molecules such as ethylene, which promotes plant growth and ripening, or the diffusion of water molecules in dehydration. It is also useful for everyday applications in laboratories.
Fick's first law assumes that factors such as temperature, pressure, and external forces are either negligible or absent. The diffusion coefficient, which is used to quantify the ease of diffusion, depends on temperature, pressure, and the substances in the system. In ideal gases and dilute liquids, the diffusion coefficient is typically assumed to remain constant within a certain range of pressure and temperature.
Fick's first law can be used to derive his second law, which predicts how the concentration gradient changes over time due to diffusion. Fick's second law is a special case of the convection-diffusion equation, where there is no advective flux and no net volumetric source. When the conditions within a system are constant, Fick's first law can be accurately applied, with the flux going in being equal to the flux going out. However, it's important to note that Fick's law does not consider the energy or direction of the particles, focusing only on the spatial gradient of the flux.
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Fick's second law: a special case of the convection-diffusion equation
Fick's laws of diffusion describe diffusion and were first introduced by Adolf Fick in 1855, based on experimental results. Fick's second law is concerned with concentration gradient changes over time. It is a linear equation with the dependent variable being the concentration of the chemical species under consideration.
Fick's second law can be derived from Fick's first law, which states that the molar flux due to diffusion is proportional to the concentration gradient. Fick's second law can be used to solve for the diffusion coefficient, D. Dimensional analysis of Fick's second law reveals that, in diffusive processes, there is a fundamental relation between the elapsed time and the square of the length over which diffusion takes place.
Fick's second law is a special case of the convection-diffusion equation, which is a parabolic partial differential equation combining the diffusion equation and the advection equation. It describes physical phenomena where particles or energy are transferred inside a physical system due to two processes: diffusion and convection. Diffusion results in the mixing and transport of chemical species without requiring bulk motion, while convection involves the collective movement of ensembles of molecules, typically in fluids, using bulk motion to move particles from one place to another.
The convection-diffusion equation is applicable to a range of processes, including heat diffusion and the conduction of electricity. Simplifications can be applied to the Maxwell-Stefan equations in order to employ the equivalent Fick's law diffusivity, particularly for systems involving concentrated mixtures that require convection and momentum conservation (fluid flow) to be solved with diffusion.
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Applicability: Fick's first law for static systems, second law for dynamic systems
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 based on experimental results. Fick's first law can be used to derive the second law, which is identical to the diffusion equation.
Fick's First Law
Fick's first law applies when the conditions within the system are constant, and the flux going in is equal to the flux going out. It relates the diffusive flux to the gradient of the concentration. It states that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. This law is applicable in one, two, or three dimensions, and the flux in each direction is proportional to the rate of concentration change in that direction.
Fick's first law is very precise in normal applications because molecules are small, and even in tiny volumes, there are still a large number of molecules. This law is excellent for everyday applications in the lab. Examples of Fickian diffusion that follow Fick's first law include the spread of a drop of ink in water, the respiration of plants, and the way baking bread smells travels through the air.
Fick's Second Law
Fick's second law predicts the change in concentration gradient with time due to diffusion. It is a special case of the convection-diffusion equation, with no advective flux and no net volumetric source. It is derived from the continuity equation, where the total flux is the only source of diffusive flux.
Fick's second law is used to model transport processes in various fields, including food, neurons, pharmaceuticals, and nuclear materials. It is also used to predict the initial and steady-state adsorption rates of a system.
Applicability
Fick's first law is applicable to static systems, where the conditions are constant, and there is no change in concentration over time. It describes the movement of particles from high to low concentration, resulting in a uniform concentration throughout the substance.
Fick's second law, on the other hand, is applicable to dynamic systems, where there is a change in concentration over time. It predicts how the concentration gradient evolves with time, allowing for the calculation of the proper time to stop the diffusion process.
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Solving Fick's second law: erf tables and numerical analysis
Fick's second law of diffusion involves an equation that can only be solved numerically with a computer. This is because the integral in the equation is complex and requires numerical methods for its solution. To solve this equation, erf tables are used to address the diffusion equation where necessary. The erf{x} in the equation is known as the error function, which is the summation of thin sources at the end of the bar.
The diffusion equation is solved using Fick's second law, which describes the diffusion of particles in a system over time. The law states that the concentration of particles at a given point in space and time is equal to the initial concentration minus the difference between the initial and final concentrations multiplied by the error function. This error function takes into account the distance travelled and the square root of the diffusion coefficient multiplied by time.
The error function, erf{x}, is a critical component of Fick's second law. It represents the cumulative distribution function of the normal distribution, which is a bell-shaped curve that describes the spread of data in a set. By using erf tables, scientists and engineers can look up values of the error function for specific inputs and apply them to the diffusion equation. These tables provide a convenient way to approximate the error function without needing to calculate it directly.
For more complex scenarios, numerical analysis techniques are employed. This involves using computational methods to approximate the solution to the diffusion equation. While there may not be an exact analytical solution, numerical analysis offers a way to estimate the behaviour of the system. Various algorithms and methods are utilised to perform these calculations, enabling a deeper understanding of diffusion processes beyond what closed-form solutions can provide.
Fick's second law, with its reliance on erf tables and numerical analysis, provides a powerful framework for studying diffusion. It allows researchers to model and predict how particles disperse over time, contributing to advancements in fields such as materials science, chemistry, and physics.
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Fick's laws: describe diffusion and can be used to solve for the diffusion coefficient (D)
Fick's laws of diffusion describe how gases and fluids spread and mix. They were first posited by Adolf Fick in 1855 based on experimental results. Fick's laws can be used to solve for the diffusion coefficient, D.
Fick's first law states that the movement of particles from high to low concentration (diffusive flux) is directly proportional to the concentration gradient. In other words, solutes in a solvent rush from a region of higher concentration to a region of lower concentration, and their rushing speed is proportional to the negative concentration gradient. This law assumes that temperature, pressure, and other external forces are either not present or negligible. It can only be accurately applied when the conditions within the system are constant, with the flux going in equal to the flux going out.
Fick's second law predicts the change in concentration gradient over time due to diffusion. It describes the time-dependent behaviour of the concentration profile during diffusion. Fick's first law can be used to derive the second law, which is identical to the diffusion equation. The second law is a special case of the convection-diffusion equation, where there is no advective flux and no net volumetric source.
Fick's laws have wide-ranging applications in various scientific and engineering fields. For example, they are used in pharmaceutical sciences to model drug release and diffusion processes in drug delivery systems. They also play a role in chemical engineering, facilitating the design of various reaction systems.
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Frequently asked questions
Fick's laws describe diffusion and were first proposed by Adolf Fick in 1855.
Fick's first law states that the movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient.
Fick's second law predicts the change in concentration gradient over time due to diffusion. It is a special case of the convection-diffusion equation where there is no advective flux and no net volumetric source.
Fick's first law can be applied to systems where the conditions remain the same, i.e., the flux coming in equals the flux going out. Fick's second law is more applicable to physical science and other changing systems.
Fick's laws can be used to solve for the diffusion coefficient, D. The first law can be used to derive the second law, which is identical to the diffusion equation.
