Question

A platinum resistance thermometer reads ${{0}^{0}},{{100}^{0}}C$ at ice point and boiling point of water. Resistance varies with wire as ${{R}_{i}}={{R}_{0}}(1+\alpha \theta +\beta {{\theta }^{2}})$, what is the reading of thermometer if it is placed in a liquid at ${{50}^{0}}C$?$(\alpha =3.8\times {{10}^{-3}}{{C}^{-1}},\beta =5.6\times {{10}^{-7}}{{C}^{-2}})$

Answer 1

Hint: Find the resistance of the platinum wire at the given temperatures zero, fifty and hundred degrees. Now the temperature reading of the thermometer when it is placed in a liquid of temperature fifty degrees is found by substituting the derived values in the formula used for relative temperature.
Formula used:
$T=\dfrac{{{R}_{t}}-{{R}_{0}}}{{{R}_{100}}-{{R}_{0}}}\times 100..(1)$

Complete answer:
Let us first find the temperature of the platinum wire at given temperatures,
\[\begin{align}
  & {{R}_{50}}={{R}_{0}}(1+\alpha 50+\beta {{(50)}^{2}}) \\
 & {{R}_{100}}={{R}_{0}}(1+\alpha 100+\beta {{(100)}^{2}}) \\
\end{align}\]
Substituting these values in the equation 1,
$\begin{align}
  & T=\dfrac{{{R}_{50}}-{{R}_{0}}}{{{R}_{100}}-{{R}_{0}}}\times 100 \\
 & \Rightarrow T=\dfrac{{{R}_{0}}(1+\alpha 50+\beta {{(50)}^{2}})-{{R}_{0}}}{{{R}_{0}}(1+\alpha 100+\beta {{(100)}^{2}})-{{R}_{0}}}\times 100 \\
 & \Rightarrow T=\dfrac{50\times 3.8\times {{10}^{-3}}+2500\times (-5.6\times {{10}^{-7}})}{100\times 3.8\times {{10}^{-3}}+10000\times (-5.6\times {{10}^{-7}})}\times 100 \\
 & \Rightarrow T={{50.4}^{0}}C \\
\end{align}$
Therefore, we have found the temperature reading of the thermometer when it is dipped in a liquid of temperature fifty degrees.

Additional Information:
Electrical resistance thermometers characteristically use platinum metal for their construction and this thermometer operates on the principle that electrical resistance varies every minute change in temperature. These types of resistors are usually called resistance thermometers. These are used in industries. Thermocouples are among the most widely used industrial thermometers. These electrical resistance thermometers are used basically for metals that change their temperatures rapidly in seconds. For accurate results, these are used in industrial purposes. Temperature compensation is not at all necessary for a resistance thermometer. These are available in a wide range. The stability can be maintained for long periods. These are also suitable for remote indication and these can be easily installed and repaired.

Note:
The major disadvantages of resistance thermometers are they are very costly. These have large bulb size. These easily get affected by shocks and vibrations, and are less sensitive. If temperature is more than the melting point of the material used, the problem of self-heating raises. In order to connect to power supply, a bridge circuit is always important.