Question

The measurement of the average coefficient of volume expansion $\beta$ for a liquid is complicated because the container also changes size with temperature. Figure shows a simple means for measuring $\beta$ despite the expansion of the container. With this apparatus, one arm of a U-tube is maintained at $0^{0}C$ in a water–ice bath, and the other arm is maintained at a different temperature $T_C$ in a constant-temperature bath. The connecting tube is horizontal. A difference in the length or diameter of the tube between the two arms of the U-tube has no effect on the pressure balance at the bottom of the tube because the pressure depends only on the depth of the liquid.Derive an expression for $\beta$ for the liquid in terms of $h_0,h_v$, and $T_C$.

Answer 1

Let $p_o$ represent the density of the liquid at $0^{0}C$. At temperature $T_c$, the volume of a sample has changed according to $\Delta V=\beta V\Delta T=\beta V{T}_c$ ,

so the density has become
$p=\frac{m}{V+\beta V{T}_c}=p_0\frac{1}{1+\beta T_c}$

So,$p(1+\beta T_c)=p_0$

Now the pressure at the bottom of the U tube is the same, whichever column it supports:

$P_o+p_{0}g{h}_0=P_0+pg{h}_t$
Simplifying,

$p_o{h}_o=p{h}_t$
and substituting,
$p(1+\beta T_c)h_0=p{h}_t$
$(1+\beta T_c)h_0=h_t\to \beta =\frac{1}{T_c}(\frac{h_t}{h_0}-1)$