Surfaces, Interfaces and Colloids Questions and Answers

Answer: C

The correct relation is θ = V_ads at P_A / V_max.

The fractional coverage is the volume of gas adsorbed at the given pressure divided by the volume needed for a complete monolayer. It lies between 0 and 1, so V_max must be in the denominator; the inverted ratio would exceed 1.

The rate statements are the wrong way round. In the Langmuir model, adsorption needs a vacant site and a colliding molecule, so its rate is proportional to (1 - θ)P_A; desorption depends only on the occupied sites, so its rate is proportional to θ, with no pressure term.

Answer: B

The weight of the drop equals 2πrγ.

The drop hangs from the tip of the capillary, held by surface tension acting along the circumference of the tip. Surface tension γ is a force per unit length, and the length of the contact line is the circumference 2πr. At the moment of detachment the upward force just balances the weight (Tate's law):

\[ mg = 2\pi r\gamma \]

This is the basis of the drop-weight (stalagmometer) method. πr²γ uses an area where a length is needed, and the other two forms are not dimensionally a force.

Answer: A

The keyed form is ΔP = r[1/R₁ + 1/R₂], where the factor printed as r stands for the surface tension γ.

The Young-Laplace equation gives the pressure difference across a curved interface with principal radii of curvature R₁ and R₂:

\[ \Delta P = \gamma (\frac{1}{R_{1}} + \frac{1}{R_{2}}) \]

For a sphere, R₁ = R₂ = R and this becomes 2γ/R. The curvatures add, which rules out the forms with a minus sign, and the surface tension multiplies the curvature rather than dividing it, which rules out the forms with 1/r. Read with the printed r as γ, the first option is the Young-Laplace equation.

Answer: B

According to Langmuir, the heat of adsorption is independent of coverage.

The Langmuir model assumes a uniform surface of equivalent sites, a monolayer, and no interactions between adsorbed molecules. Because every site is the same and neighbours do not affect each other, the heat of adsorption is the same whether the surface is nearly empty or nearly full.

A heat of adsorption that changes with coverage (A) and interacting adsorbates (C) are the deviations that the Temkin and Freundlich treatments allow for. Dissociative adsorption (D) can be handled within the Langmuir framework, so 'always non-dissociative' is not part of the theory.

Answer: A

The correct statements are (I) and (III).

At equilibrium the rates of adsorption and desorption are equal:

\[ \begin{aligned} k_{a}[\mathrm{A}][\mathrm{S}] &= k_{d}[\mathrm{A{-}S}] \\ K = \frac{k_{a}}{k_{d}} &= \frac{[\mathrm{A{-}S}]}{[\mathrm{A}][\mathrm{S}]} \end{aligned} \]

So (I) is right and (II), its reciprocal, is wrong. A large K means adsorption is fast compared with desorption and most of the adsorbate stays bound, which is strong adsorption (III), not strong desorption (IV).

Answer: C

The pan should be coated with a low energy substance.

From Young's equation, food or water sticks least when it wets the surface least, which means a large contact angle and a small cosθ. That requires a small γ_SV, a low-energy solid surface. This is why non-stick pans are coated with PTFE (Teflon), whose surface energy is very low.

A high-energy coating would raise γ_SV and increase wetting and sticking. Surfactants in the water lower γ_LV and γ_SL, which improves wetting, the opposite of what is wanted.

Answer: B

The correct statements are (I) and (III).

A surfactant forms micelles when the gain from hiding its hydrocarbon tails outweighs the cost of crowding its head groups together. Ionic head groups repel one another electrostatically, which makes micellization harder, so an anionic surfactant needs a higher concentration to form micelles than a non-ionic surfactant with the same tail. Statement (I) is true and (II) false; for C₁₂ chains the CMC of sodium dodecyl sulfate is about 8 mM, while non-ionic C₁₂ surfactants lie near 0.1 mM.

Added electrolyte screens the repulsion between ionic head groups (and its counter-ions bind to the micelle surface), so micelles form more easily and the CMC of an ionic surfactant decreases. Statement (III) is true and (IV) false. The options that include (II) or (IV) contain one of the false statements.

Answer: C

The phenomena directly related to surface tension are (I), (III) and (IV).

• (I) A free droplet takes a spherical shape because surface tension minimizes its surface area, and a sphere has the least area for a given volume.
• (III) A steel needle laid gently on water floats, supported by the surface film, although steel is far denser than water.
• (IV) The rise of water in the narrow xylem vessels of plants is treated as capillary action, which depends on surface tension together with the adhesion of water to the vessel walls.

(II) A sailing boat floats by buoyancy (Archimedes' principle) and moves by wind on its sails, neither of which involves surface tension. Every option containing (II) is therefore wrong.

Answer: B

For this adsorption ΔH < 0 and ΔS < 0.

Nitrogen adsorbs readily on activated carbon at 77 K, so the process is spontaneous and ΔG < 0. Gas molecules that settle on the surface lose most of their freedom to move, so the entropy falls: ΔS < 0. With ΔG = ΔH − TΔS, a negative ΔG and a negative ΔS are possible only if ΔH is negative:

\[ \begin{aligned} &\Delta G = \Delta H - T\Delta S \lt 0,\ \Delta S \lt 0 \\ &\Rightarrow \Delta H \lt 0 \end{aligned} \]

Adsorption is therefore exothermic; this physical adsorption releases heat comparable to the heat of condensation of nitrogen.

ΔH = 0 or ΔH > 0 combined with ΔS < 0 would make ΔG positive and the adsorption non-spontaneous, and ΔS = 0 ignores the loss of freedom.

Answer: B

The Langmuir assumptions here are (I) and (IV).

The Langmuir model treats the surface as a set of identical, equivalent sites (I), each holding at most one molecule, so adsorption stops at a monolayer (IV). It also assumes no interaction between adsorbed molecules, so the chance of adsorbing on a site does not depend on whether neighbouring sites are filled.

(II) is wrong: in the model adsorption and desorption are independent, non-cooperative events whose rates balance at equilibrium. (III) as worded is also wrong: the probability of adsorption on a site does depend on its own occupancy, since an occupied site cannot take another molecule; the rate of adsorption is proportional to the fraction of empty sites, (1 − θ). So every option containing (II) or (III) is wrong.