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Theory [Custom Themometer Based on Customized Freezing Points and Boiling Points]

Consider a different temperature scale that a thermometer follows where the ice point [the temperature at which ice starts to melt at normal atmospheric pressure] is \(\text{X}_{\text{ice}}\) and that of vapor point [at which pure water starts to evaporate upon application of heat at normal atmospheric pressure] is \(\text{X}_{\text{vap}}\). We indeed want to find a relationship between this scale [that we have considered to be faulty or customized] with the Standard temperature scales that have been approved, here the Celsius. For that we consider a \(\text{C-X}\) graph, where the Celsius scale has been plotted along the Y-axis and the X- scale along the X axis. A straight line represents the equation, that joines two points \((0,\text{X}_{\text{ice}})\) and \((100,\text{X}_{\text{vap}})\). The equation of the straight line can be written as: $$\frac{\theta-\text{X}_{\text{ice}}}{\text{X}_{\text{ice}}-\text{X}_{\text{vap}}}=\frac{\text{C}-0}{0-100}$$ $$\Rightarrow\frac{\theta-\text{X}_{\text{ice}}}{\text{X}_{\text{vap}}-\text{X}_{\text{ice}}}=\frac{\text{C}}{100}$$ where plugging any values of \(C\), we can find the equivalent temperature in our new scale \(X\).