Question

In a mercury thermometer, the lower fixed point is marked as $10{}^\circ C$ and the upper fixed point is marked as $130{}^\circ C$. The interval between the fixed points being divided into $120$ equal divisions. At $40{}^\circ C$ temperature, what would be the reading indicated by the thermometer.

Answer 1

Hint: The relation between the temperatures is measured by taking the ratio of the difference in the temperature to be calculated and the lower fixed point to the number of divisions between the upper fixed points and the lower fixed points. This will be equal in every thermometer. These all may help you to solve this question.

Complete step-by-step answer:
let the temperature in the mentioned scale be $x$. That is the relation between the temperature values in two thermometers are given as,
$\dfrac{x-LFP}{UFP-LFP}=\dfrac{40-LFP}{UFP-LFP}$
Where $LFP$ be the lower fixed point, $UFP$ be the upper fixed point
It is mentioned in the question that the lower fixed point is
$LFP=10{}^\circ C$
And the upper fixed point is given as,
$UFP=130{}^\circ C$
The number of divisions between this upper fixed point and lower fixed point has been given in the question. Therefore this has to be used in the equation instead of the difference of the fixed points.
The value of the number of divisions is given as,
$n=UFP-LFP=120$
The temperature shown in the second thermometer is given as $40{}^\circ C$
As we all know, the lower point and upper fixed point of a degree celcius thermometer are given as,
$\begin{align}
  & UFP=100{}^\circ C \\
 & LFP=0{}^\circ C \\
\end{align}$
Substituting this in the equation will give,
\[\begin{align}
  & \dfrac{x-10}{130-10}=\dfrac{40-0}{100-0} \\
 & \Rightarrow \dfrac{x-10}{120}=\dfrac{40}{100} \\
\end{align}\]
Simplifying the equation will give,
\[\begin{align}
  & 100x-1000=4800 \\
 & \Rightarrow 100x=5800 \\
\end{align}\]
From this, we will get the value of the temperature in the faulty thermometer equivalent to the \[40{}^\circ C\]thermometer.
That is,
\[x=58{}^\circ C\]
Therefore the correct answer has been obtained.

Note: The lower fixed point is also known as the ice point. It is the temperature of the melting ice at normal atmospheric pressure. The upper fixed point is otherwise called a steam point. It is defined as the temperature of the boiling water at normal atmospheric pressure.