Question

The molecular diffusivity of a liquid:

• A. decreases with temperature
• B. increases with temperature
• C. may increase or decrease with temperature
• D. is independent of temperature

Hint:

Temperature dependencies of Diffusivity: - In Gases: \( D_{AB} \propto T^{1.5} \) to \( T^{1.75} \) (increases with temperature because molecular velocities increase). - In Liquids: \( D_{AB} \propto \frac{T}{\mu} \) (increases with temperature because liquid viscosity drops sharply as temperature rises).

Answer 1

The Correct Option is B

Concept: Molecular diffusivity ($D_{AB}$) characterizes the rate at which a solute molecule diffuses through a solvent medium under the driving force of a concentration gradient. The behavior of molecular diffusivity in liquids as a function of temperature can be accurately analyzed using the hydrodynamic theory of diffusion, represented by the classic Stokes-Einstein Equation, or empirical correlations such as the Wilke-Chang Correlation.

Step 1: Analyzing the Stokes-Einstein Relationship.
For a spherical molecule diffusing through a liquid solvent, the Stokes-Einstein equation relates the liquid molecular diffusivity ($D_{AB}$) to system properties as follows: \[ D_{AB} = \frac{k_B \cdot T}{6 \cdot \pi \cdot r \cdot \mu} \] Where:
• \( k_B \) represents the Boltzmann constant.
• \( T \) represents the absolute thermodynamic temperature measured in Kelvin.
• \( r \) represents the hydrodynamic radius of the diffusing solute molecule.
• \( \mu \) represents the dynamic viscosity of the liquid solvent.

Step 2: Evaluating the combined effect of temperature changes.
Let us analyze how an increase in absolute temperature ($T$) affects the parameters in the Stokes-Einstein equation:
• Direct Temperature Term: The numerator contains the absolute temperature $T$ explicitly. As $T$ increases, the thermal kinetic energy of the molecules increases, which directly increases the value of the numerator.
• Liquid Viscosity Term (\(\mu\)): In liquids, cohesive intermolecular forces hold the molecules together. When the temperature of a liquid increases, thermal vibrations weaken these cohesive bonds, causing the dynamic liquid viscosity ($\mu$) to drop sharply with temperature. This relationship can be modeled by an exponential Arrhenius-type equation: \[ \mu \propto \exp\left(\frac{E_{\text{visc}}}{R \cdot T}\right) \quad \Rightarrow \quad T \uparrow \implies \mu \downarrow\downarrow \]

Step 3: Determining the final trend for liquid diffusivity.
Let us substitute both temperature dependencies back into the diffusivity expression: \[ D_{AB} \propto \frac{T}{\mu} \] When temperature ($T$) increases, the numerator ($T$) increases, and the denominator ($\mu$) decreases significantly. Both effects work together to increase the molecular diffusivity value ($D_{AB}$). Therefore, the molecular diffusivity of a liquid always increases with temperature.