Question

How is the concentration of a commodity defined? How is the concentration gradient defined? How is the diffusion rate of a commodity related to the concentration gradient?

Short answer

Question: Define concentration, concentration gradient, diffusion rate, and explain the relationship between the diffusion rate of a commodity and the concentration gradient. Answer: Concentration is the measure of the amount of a substance (commodity) in a given volume, usually expressed in terms of mass or moles per unit volume. The concentration gradient refers to the rate at which concentration changes over a distance, pointing in the direction of the greatest increase in concentration. Diffusion rate refers to how quickly particles spread out through a medium, mainly driven by a concentration gradient. The diffusion rate of a commodity is directly proportional to the concentration gradient, as stated in Fick's first law of diffusion (J = -D * ∇C). This means that when there is a larger difference in concentration over a given distance, the particles spread out faster to equalize the concentration throughout the medium. The diffusivity 'D' depends on factors such as temperature and the properties of the medium through which the particles are diffusing.

Step-by-step solution

Definition of Concentration

Concentration is a measure of the amount of a substance (commodity) in a given volume. It is usually expressed in terms of mass per unit volume (e.g., grams per liter) or moles per unit volume (e.g., moles per liter). In general, the concentration of a commodity (C) can be represented as the ratio of the quantity of the commodity (Q) to the total volume (V) in which it is present: C = Q/V

Definition of Concentration Gradient

The concentration gradient refers to the rate at which the concentration of a substance changes over a distance. It is a vector quantity that points in the direction of the greatest increase in concentration. Mathematically, the concentration gradient is represented as the spatial derivative of the concentration function: ∇C = (∂C/∂x, ∂C/∂y, ∂C/∂z) Here, ∇C is the concentration gradient, and (∂C/∂x, ∂C/∂y, ∂C/∂z) are the partial derivatives of the concentration with respect to the spatial coordinates x, y, and z.

Definition of Diffusion Rate

Diffusion is the process by which molecules or particles move from an area of high concentration to an area of low concentration. The diffusion rate refers to how quickly the particles spread out through the medium, mainly driven by a concentration gradient. It is typically expressed in terms of mass or moles per unit time per unit area. The diffusion rate (J) depends on the diffusivity (D) of the substance and the concentration gradient: J = -D * ∇C Here, the negative sign indicates that the diffusion occurs in the direction opposite to the concentration gradient (i.e., from high to low concentration).

Relationship between Diffusion Rate and Concentration Gradient

The diffusion rate of a commodity is directly proportional to the concentration gradient. This relationship is given by Fick's first law of diffusion: J = -D * ∇C As the concentration gradient (∇C) increases, the diffusion rate (J) also increases. This means that when there is a larger difference in concentration over a given distance, the particles spread out faster to equalize the concentration throughout the medium. The proportionality constant 'D' is the diffusivity of the substance, which depends on factors such as temperature and the properties of the medium through which the particles are diffusing.