Question

The volume coefficient of expansion of mercury is 20 times the linear coefficient of expansion of glass. What volume of mercury should be poured into a glass vessel of volume V, so that the volume above mercury in the vessel may remain constant at all temperatures?

• A. $\frac{V}{3}$
• B. $\frac{3V}{20}$
• C. $\frac{V}{20}$
• D. $\frac{7V}{20}$

Answer 1

Answer: (B)

$\frac{3V}{20}$

Let x be the volume of mercury poured into vessel. The volume unoccupied by $Hg=(V-x).$
Let the temperature be raised through ${\theta }^{0}C.$ Then V becomes $V(1+3\alpha \theta )=V^{\prime }$
x becomes x$(1+\gamma \theta )=x^{\prime }$
$V^{\prime }-x^{\prime }=V-x$

$V(1+3\alpha \theta )-x(1+\gamma \theta )=V-x$

$x=\frac{3\alpha }{\gamma }V,but\frac{\alpha }{\gamma }=\frac{1}{20}$

$\therefore x=\frac{3V}{20}$