Question

A non-conducting body floats in a liquid at ${20}^{\circ }C$ with $\frac{2}{3}$ of its volume immersed in the liquid. When liquid temperature is increased to ${100}^{\circ }C$, $\frac{3}{4}$ of body's volume is immersed in the liquid. Then the coefficient of real expansion of the liquid is (neglecting the expansion of the container):

• A. $15.6\times {10}^{-4}{}^{\circ }{C}^{-1}$
• B. $156\times {10}^{-4}{}^{\circ }{C}^{-1}$
• C. $1.56\times {10}^{-4}{}^{\circ }{C}^{-1}$
• D. $0.156\times {10}^{-4}{}^{\circ }{C}^{-1}$

Answer 1

Answer: (A)

$15.6\times {10}^{-4}{}^{\circ }{C}^{-1}$

From Archimedes' principle, weight of the body is balanced by the weight of the liquid displaced.

At ${20}^{o}C,{\rho }_b{V}_{b}g={\rho }_{l1}\times \frac{2}{3}{V}_{b}g.........\left(i\right)$

At ${100}^{o}C,{\rho }_b{V}_{b}g={\rho }_{l2}\times \frac{3}{4}{V}_{b}g........\left(ii\right)$

From (i) and (ii),

$\frac{{\rho }_{l1}}{{\rho }_{l2}}=\frac{9}{8}.........\left(iii\right)$

From Law of conservation of mass, total mass of liquid is constant.

$M_{l1}=M_{l2}$

${\rho }_{l1}{V}_{l1}={\rho }_{l2}{V}_{l2}..........\left(iv\right)$

Volume expansion of liquid is governed by,

$V_{l2}=V_{l1}(1+\gamma \Delta T)...........\left(v\right)$

From (iv) and (v),

$\frac{{\rho }_{l1}}{{\rho }_{l2}}=1+\gamma \Delta T...........\left(vi\right)$

Solving, (iii) and (vi),

$\gamma =15.6\times {10}^{-4}{}^o{C}^{-1}$