Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
The interaction energy E between two spherical colloidal particles as plotted against separation distance r according to DLVO theory displays:
Options:
(A) A primary minimum at close contact, an electrostatic repulsive barrier at intermediate distance, and a secondary minimum at larger distance
(B) Purely attractive van der Waals energy at all separation distances
(C) Purely repulsive electrostatic energy at all separation distances
(D) Constant potential independent of separation distance
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.50)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Biophysics / Colloid Science (Interparticle Interaction Potential Energy Curves)
**2. Detailed Step-by-Step Derivation**:
- The net interaction force $F(r)$ between two spherical particles is related to the interaction potential energy $E(r)$ by:
$F(r) = -\frac{dE}{dr}$
- Given the physical boundary conditions:
1. **For $r < a$ (Repulsion)**:
- The net force is repulsive ($F > 0 \implies -\frac{dE}{dr} > 0 \implies \frac{dE}{dr} < 0$).
- As the particles approach closer ($r \to 0$), interpenetration of electron clouds causes strong steric/Born repulsion, driving the potential energy $E(r)$ steeply positive ($E \to +\infty$).
2. **For $r > a$ (van der Waals attraction)**:
- The net force is attractive ($F < 0 \implies \frac{dE}{dr} > 0$).
- The interaction energy $E(r)$ is negative, asymptotically approaching zero from below as separation distance $r \to \infty$ ($E \propto -1/r^6$).
3. **At $r = a$ (Energy minimum)**:
- The transition between repulsion and attraction occurs at $r = a$, where the net force is zero ($F = 0 \implies \frac{dE}{dr} = 0$).
- This corresponds to a stable potential energy well (minimum $E$).
- The graph exhibiting steep positive energy for $r < a$, a minimum at $r = a$, and an asymptotic negative approach toward zero for $r > a$ is represented by **Plot D** (analogous to a Lennard-Jones 6-12 or DLVO potential profile).
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Shows purely positive potential energy without an attractive well.
- **Option (B) [INCORRECT]**: Fails to capture the steep short-range repulsive barrier.
- **Option (C) [INCORRECT]**: Inverts the sign of the interaction energy.
- **Option (D) [CORRECT]**: Accurately depicts the classical interparticle energy curve with short-range repulsion ($r < a$) and long-range van der Waals attraction ($r > a$) with an energy minimum at $r = a$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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