The average time it takes for a molecule to diffuse a...

The average time it takes for a molecule to diffuse a distance of $x \mathrm{cm}$ is given by $$ t=\frac{x^{2}}{2 D} $$ where $t$ is the time in seconds and $D$ is the diffusion coefficient. Given that the diffusion coefficient of glucose is $5.7 \times 10^{-7} \mathrm{cm}^{2} / \mathrm{s},$ calculate the time it would take for a glucose molecule to diffuse $10 \mu \mathrm{m},$ which is roughly the size of a cell.

The average time it takes for a molecule to diffuse a distance of
$x \mathrm{cm}$ is given by
$$
t=\frac{x^{2}}{2 D}
$$
where $t$ is the time in seconds and $D$ is the diffusion coefficient. Given that the diffusion coefficient of glucose is $5.7 \times 10^{-7} \mathrm{cm}^{2} / \mathrm{s},$ calculate the time it would take for a glucose molecule to diffuse $10 \mu \mathrm{m},$ which is roughly the size of a cell.

Key Concepts

Diffusion

Diffusion is the process by which molecules move from areas of high concentration to areas of low concentration, driven by their random thermal motion. This fundamental transport mechanism plays a critical role in various fields such as physics, chemistry, and biology, and is essential for processes ranging from mixing of fluids to intracellular transport.

Diffusion Coefficient

The diffusion coefficient is a parameter that quantifies the rate at which particles spread out in a medium due to diffusion. It depends on factors such as temperature, the properties of the medium, and the size of the molecules involved. This coefficient is central to the mathematical description of diffusion, linking the mean squared displacement of molecules to time.

Mean Squared Displacement and Diffusive Timescale

The concept of mean squared displacement is used to describe how far particles, on average, move during diffusion. In many systems, this displacement is proportional to time, with the proportionality constant involving the diffusion coefficient. This relationship implies that the time required for a particle to diffuse a certain distance scales with the square of that distance, a key insight in understanding how quickly diffusion occurs over different length scales.

Unit Conversion and Scaling

Correctly converting units and applying scaling relations is essential when dealing with diffusion calculations. By ensuring that distance, time, and the diffusion coefficient are expressed in consistent units, one can accurately estimate timescales for processes such as mass transport in cells or other systems, illustrating the practical application of scaling laws in physical problems.

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