The Celsius and Fahrenheit Scales — Question 1

Question 1

Two scales can show the same number at one temperature

Use numerical readings $F=\frac95C+32$ and $T/\mathrm K=C+273.15$; these conversions are exact. Two correctly calibrated displays view the same bath: one reports Celsius and the other Fahrenheit.

1. Find the temperature at which their numerical readings coincide, and express it in kelvins.

2. Find the temperature at which the Fahrenheit number is twice the Celsius number. Does that relation mean one physical temperature is twice another?

Question 1 - Solution

Two scales can show the same number at one temperature

(a) Method and calculation

At coincidence $F=C$, so$$C=\frac95C+32\ \Longrightarrow\ C=-40.$$$$F=-40,\qquad T=233.15\,\mathrm K.$$

(b) Interpretation and further calculation

$$2C=\frac95C+32\ \Longrightarrow\ C=160,\quad F=320.$$$$T=160+273.15=433.15\,\mathrm K.$$Both numbers label the same state. Their factor of two is a relation between coordinates, not two absolute temperatures.

Physical check

The offset is crucial for the coincidence calculation; equal-sized numerical readings are generally not equal physical temperatures across scales.