Estimate the time needed for a glycine molecule (see Table 18–3) to diffuse a distance of 25μm in water at 20°C if its concentration varies over that distance from 1.00 mol/m³ to 0.50 mol/m³. Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u.

Verified step by step guidance

1. Step 1: Identify the key parameters for diffusion. The diffusion process can be analyzed using Fick's second law of diffusion. The diffusion coefficient (D) for glycine in water at 20°C can be estimated using empirical data or provided values. If not given, approximate values for small molecules in water are typically in the range of 10⁻⁹ m²/s.
2. Step 2: Use the diffusion equation to estimate the time required for the molecule to diffuse. The relationship between diffusion distance (x), diffusion coefficient (D), and time (t) is given by the formula: $\sqrt{x²}=\sqrt{Dt}$. Rearrange this equation to solve for time: $t=\frac{x²}{D}$, where x = 25 μm = 25 × 10⁻⁶ m.
3. Step 3: Calculate the root mean square (rms) thermal speed of the glycine molecule. The rms speed is given by the formula: $\sqrt{\frac{3kT}{m}}$, where k is the Boltzmann constant (1.38 × 10⁻²³ J/K), T is the temperature in Kelvin (20°C = 293 K), and m is the mass of the glycine molecule in kilograms. Convert the molecular mass of glycine (75 u) to kilograms using the conversion factor 1 u = 1.66 × 10⁻²⁷ kg.
4. Step 4: Compare the diffusion speed to the rms thermal speed. The diffusion speed can be approximated as the distance traveled divided by the time calculated in Step 2. The rms thermal speed represents the average speed of the molecule due to thermal motion. Analyze the ratio of these speeds to understand the relative significance of diffusion versus thermal motion.
5. Step 5: Summarize the findings. Diffusion is typically much slower than the rms thermal speed because diffusion involves random motion and collisions, which significantly slow down the net movement of molecules over macroscopic distances. This comparison highlights the difference between molecular-scale dynamics and macroscopic transport processes.

Key Concepts

Diffusion

Diffusion is the process by which molecules spread from areas of high concentration to areas of low concentration. This movement occurs due to the random thermal motion of particles, leading to a net movement that continues until equilibrium is reached. The rate of diffusion can be influenced by factors such as temperature, concentration gradient, and the size of the molecules involved.

RMS Speed

The root mean square (RMS) speed is a measure of the average speed of particles in a gas, calculated from the kinetic theory of gases. It is defined as the square root of the average of the squares of the speeds of the particles. For a molecule, the RMS speed can be determined using the formula v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the molecule.

Concentration Gradient

A concentration gradient refers to the gradual change in the concentration of solutes in a solution as a function of distance. In the context of diffusion, a steeper concentration gradient (larger difference in concentration over a distance) typically results in a faster rate of diffusion. Understanding the concentration gradient is crucial for estimating how long it will take for a molecule to diffuse across a given distance.