Question

At high temperatures, which one of the following empirical expressions correctly describes the variation of dynamic viscosity \(\mu\) of a Newtonian liquid with absolute temperature \(T\)? (Given: \(A\) and \(B\) are positive constants.)

• A. \(\mu = A + BT\)
• B. \(\mu = A \exp\!\left(-\dfrac{B}{T}\right)\)
• C. \(\mu = A \exp(BT)\)
• D. \(\mu = A \exp\!\left(\dfrac{B}{T}\right)\)

Hint:

Remember: For \emph{liquids} \(\mu \downarrow\) with \(T\) (Arrhenius/Andrade: \(\ln\mu = \ln A + B/T\)); for \emph{gases} \(\mu \uparrow\) with \(T\) (power law like Sutherland's formula).

Answer 1

The Correct Option is D

Step 1: Physical trend for liquids.
For liquids, viscosity decreases with temperature: \( \dfrac{d\mu}{dT} < 0 \). Empirically (Andrade/Arrhenius form), \[ \mu = A\,\exp\!\left(\frac{E}{RT}\right) \equiv A\,\exp\!\left(\frac{B}{T}\right), \quad A,B > 0, \] so as \(T \uparrow\), the exponent \(\tfrac{B}{T}\downarrow\) and \(\mu\) decreases, approaching \(A\) at very high \(T\). Step 2: Test each option against the trend and limiting behavior.
(A) \(A+BT\): linear increase with \(T\) \(\Rightarrow\) contradicts \(d\mu/dT < 0\).
(B) \(A\exp(-B/T)\): as \(T\uparrow\), \(-B/T\) becomes less negative, so the exponential increases; hence \(\mu\) increases with \(T\), opposite to liquids.
(C) \(A\exp(BT)\): grows explosively with \(T\).
(D) \(A\exp(B/T)\): as \(T\uparrow\), \(B/T\downarrow\Rightarrow \mu\downarrow\), and \(\lim_{T\to\infty}\mu=A\). Matches liquid behavior. Step 3: Conclude.
The only expression consistent with the observed decrease of \(\mu\) with \(T\) for liquids and with the high-\(T\) limit is option (D). \[ \boxed{\mu = A \exp\!\left(\dfrac{B}{T}\right)} \]