Suppose that a new temperature scale has been devised on...

Suppose that a new temperature scale has been devised on which the melting point of ethanol $\left(-117.3^{\circ} \mathrm{C}\right)$ and the boiling point of ethanol $\left(78.3^{\circ} \mathrm{C}\right)$ are taken as $0^{\circ} \mathrm{S}$ and $100^{\circ} \mathrm{S},$ respectively where $S$ is the symbol for the new temperature scale. Derive an equation relating a reading on this scale to a reading on the Celsius scale. What would this thermometer read at $25^{\circ} \mathrm{C}$ ?

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Key Concepts

Linear Relationship

Temperature scales are typically related by a linear equation of the form y = ax + b, where a represents the scaling factor between the two scales and b represents the offset due to the different zero points. This concept is fundamental to converting measurements from one scale to another through a direct, proportional relationship.

Fixed Reference Points

Fixed or reference points are standard values (such as melting and boiling points for a substance) that are used to define a temperature scale. By assigning specific values on each scale to these reference points, a conversion formula can be derived, ensuring that the scales are aligned accurately with each other.

Scaling Factor Calculation

Determining the scaling factor involves calculating the ratio of the interval spans on the two scales. This factor is obtained by dividing the difference between the two fixed reference points on one scale by the corresponding difference on the other scale. It serves as the multiplier that converts temperature differences from one scale to the other.

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