At what absolute temperature do the Celsius and Fahrenheit temperature scales give the same numerical value? What is the value?
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One way to solve this problem is to write both the Celsius and Fahrenheit temperatures in terms of the absolute temperature in Kelvin, then set these equal to each other and solve for the temperature in Kelvin. The Celsius temperature is T (°C) = T (K) – 273.15. The Fahrenheit temperature is T (°F) = 1.8*T (K) – 459.67. If we plot these two lines, they look like:
What we are asked to find is the point where these two lines cross. That is, where
T (K) – 273.15 = 1.8*T (K) – 459.67
Solving this for T (K) gives T (K) = (459.67 – 273.15) / 0.8 = 233.15 K. At this temperature (in K) the Fahrenheit temperature is 1.8 * 233.15 – 459.67 = -40 °F. Likewise, the Celsius temperature is 233.15 – 273.15 = -40 °C.
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