Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
Which of the following statements is true about the effect of an increase in temperature on the dynamic viscosity of water and air, at room temperature?
Options:
(A) It increases for both water and air.
(B) It increases for water and decreases for air.
(C) It decreases for water and increases for air.
(D) It decreases for both water and air.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2020 Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to compare the effect of increasing temperature on the dynamic viscosity ($\mu$) of a liquid (water) versus a gas (air) at room temperature.
**2. Step-by-Step Mathematical Derivation**:
1. **Microscopic Mechanism of Viscosity in Liquids (Water)**:
In liquids, dynamic viscosity is dominated by **intermolecular cohesive forces** between closely packed molecules.
As temperature increases, thermal kinetic energy increases, weakening cohesive bonds and allowing molecules to slide past each other more easily:
\mu_{\text{liquid}}(T) \propto e^{B/T} \implies \frac{d\mu_{\text{water}}}{dT} < 0 \quad (\text{viscosity decreases})
2. **Microscopic Mechanism of Viscosity in Gases (Air)**:
In dilute gases, intermolecular distances are large, so cohesive forces are negligible. Dynamic viscosity is dominated by the **inter-layer transport of molecular momentum** via random thermal collisions.
From kinetic theory of gases, mean molecular speed is $\bar{c} = \sqrt{\frac{8RT}{\pi}}$, and viscosity follows Sutherland's law:
\mu_{\text{gas}}(T) \propto \sqrt{T} \implies \frac{d\mu_{\text{air}}}{dT} > 0 \quad (\text{viscosity increases})
3. **Conclusion**:
With rising temperature, dynamic viscosity decreases for water and increases for air.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) It increases for both water and air**: ❌ **INCORRECT**. Liquids exhibit opposite temperature dependence to gases.
- **(B) It increases for water and decreases for air**: ❌ **INCORRECT**. Inverts the physical mechanisms.
- **(C) It decreases for water and increases for air**: ✅ **CORRECT**. Thermal agitation disrupts liquid cohesive bonds (reducing liquid $\mu$) while enhancing gas molecular momentum exchange (increasing gas $\mu$).
- **(D) It decreases for both water and air**: ❌ **INCORRECT**. Gases do not decrease in viscosity with temperature.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Crucial GATE fundamental: Liquid viscosity decreases with $T$ (cohesion breakdown). Gas viscosity increases with $T$ (momentum exchange via Sutherland's law $\mu \propto T^{3/2} / (T + S)$).