Question

A platform resistance thermometer reads \[0^\circ {\text{C}}\] when its resistance is \[80\,\Omega \] and \[100^\circ {\text{C}}\] when its resistance is \[90\,\Omega \]. Find the temperature at the platinum scale at which the resistance is \[86\,\Omega \].
A. \[30^\circ {\text{C}}\]
B. \[60^\circ {\text{C}}\]
C. \[20^\circ {\text{C}}\]
D. \[10^\circ {\text{C}}\]

Answer 1

Hint: Use the equation relating the resistance with the temperature. This equation gives the relation between the resistance at zero degree Celsius, the resistance at a known temperature and the resistance at an unknown temperature.

Formula used:
The formula for temperature when the resistances for the two temperatures are given is
\[T = \dfrac{{{R_T} - {R_0}}}{{{R_{{T_1}}} - {R_0}}}{T_1}\]

Here, \[T\] is the temperature for resistance \[{R_T}\], \[{R_{{T_1}}}\] is the resistance for temperature \[{T_1}\] and \[0^\circ {\text{C}}\] is the resistance for temperature \[{R_0}\].

Complete step by step answer:
The platform resistance thermometer reads \[0^\circ {\text{C}}\] when its resistance is \[80\,\Omega \].

The platform resistance thermometer reads \[100^\circ {\text{C}}\] when its resistance is \[90\,\Omega \].

Determine the temperature \[T\] for the resistance \[86\,\Omega \].

Substitute \[80\,\Omega \] for \[{R_0}\], \[86\,\Omega \] for \[{R_T}\], \[90\,\Omega \] for \[{R_{{T_1}}}\] and \[100^\circ {\text{C}}\] for \[{T_1}\] in the above equation.
\[T = \dfrac{{\left( {86\,\Omega } \right) - \left( {80\,\Omega } \right)}}{{\left( {90\,\Omega } \right) - \left( {80\,\Omega } \right)}}\left( {100^\circ {\text{C}}} \right)\]
\[\Rightarrow T = 60^\circ {\text{C}}\]

Therefore, the temperature shown by the platform resistance thermometer is \[60^\circ {\text{C}}\].

So, the correct answer is “Option B”.

Additional Information:
An instrument which measures the temperature comprising the resistance measuring device is known as the resistance thermometer.
The resistors used in the resistance thermometer are very high sensing resistors usually made of platinum.
The resistance increases with the increase in temperature.
The platinum resistance thermometers are very high sensing, stable, unreactive metal and can be drawn into very fine wires.
The platinum wire resistors measure the temperature by measuring the electrical resistance.
To measure the resistance of a wire, an electric current is passed through it and the potential difference across its ends is measured by a voltmeter and resistance is measured. The reading obtained is converted into temperature using the equation relating the voltmeter and temperature.

Note:
There is no need to convert the temperature of the thermometer in Kelvin. One may also use the equation relating the resistance at a temperature if the coefficient of resistance is given.