Question

The lengths of the mercury column of a mercury thermometer are 1.26cm and 20.86cm respectively at the standard fixed points. What is the temperature of body, which produces 9.0cm of this mercury column?

Answer 1

We can use that the linear size of body (liquid or solid) are increasing with temperature and this dependence is approximately linear. It means that

$$\Delta l = \alpha \Delta T$$

Where $\alpha$ - some coefficient. We can find ${l_0}$ and $\alpha$ by using the given data (fixed points are freezing and boiling points of water). The temperature difference between these two points are $100[^\circ C]$, thus

$$20.86[{\text{cm}}] - 1.26[{\text{cm}}] = \alpha \cdot 100[^\circ C]$$

And

$$\alpha = \frac{{19.6}}{{100}}[\frac{{{\text{cm}}}}{{^\circ C}}] = 0.196[\frac{{{\text{cm}}}}{{^\circ C}}]$$

Then

$$9.0[{\text{cm}}] - 1.26[{\text{cm}}] = 0.196[\frac{{{\text{cm}}}}{{^\circ C}}](T - 0)[^\circ C]$$

And

$$T = \frac{{7.74[{\text{cm}}]}}{{0.196[\frac{{{\text{cm}}}}{{^\circ C}}]}} \approx 39.49[^\circ C]$$

This is the answer.