Question

At what temperature do the Celsius and Fahrenheit scales coincide?

Answer 1

Recall that the Celsius and the Fahrenheit scales differ in terms of the numerical values of the freezing ($0^{\circ}C, 32^{\circ}F$) and boiling points ($100^{\circ}C, 212^{\circ}F$) of water respectively. In such a case, determine the relation between the two scales and assume the same value for the temperature in both scales, determine the appropriate result.

Formula used: Temperature conversion: $T^{\circ}F = \left(\dfrac{5}{9} \times T^{\circ}C\right)+32$

Complete step by step answer:
Let us begin by understanding the distinction between the Celsius and Fahrenheit scales.
In the Celsius scale, the lower fixed point is considered as $0^{\circ}C$ and the upper fixed point is considered $100^{\circ}C$. The region between these two temperatures is divided into 100 divisions such that each division signifies $1^{\circ}C$. The upper and lower fixed points indicate the boiling point and freezing point of water. The absolute zero value on the Celsius scale is $-273.15^{\circ}C$.
In the Fahrenheit scale, the lower fixed point is considered as $32^{\circ}F$ and the upper fixed point is considered $212^{\circ}F$. The region between these two temperatures is divided into 180 divisions such that each division equals $1^{\circ}F$. The upper and lower fixed points indicate the boiling point and freezing point of water. The absolute zero value on the Fahrenheit scale is $-459.67^{\circ}F$
The formula for the conversion of temperature from the Celsius scale to Fahrenheit scale is:
$T^{\circ}F = \left(\dfrac{5}{9} \times T^{\circ}C\right)+32$
Let us assume the temperature at which the two scales coincide be T, such that $T^{\circ}C = T^{\circ}F$
Applying this to the conversion formula above, we get:
$T^{\circ}C = \left(\dfrac{9}{5} \times T^{\circ}C\right)+32$
$\Rightarrow 5T = 9T +160 \Rightarrow 4T = -160 \Rightarrow T = \dfrac{-160}{4}$
$\Rightarrow T = -40^{\circ}C$

Therefore, the temperature at which both the Celsius and Fahrenheit scales are the same in $-40^{\circ}$.

Note: The modern Celsius scale has been adopted based on the triple point of Vienna Standards Mean ocean water which has optimized the value of absolute zero. According to this, the Celsius scale is not based on the boiling point or freezing point of water but on the triple point of water. The triple point of water is defined as the temperature at which the three phases of water (gas, liquid and solid) coexist in thermal equilibrium. Following this, the absolute zero is denoted by $-273.15^{\circ}C$ which is used to define Kelvin as $T\;K = T^{\circ}C+273.15$.