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Denote by $a$ and $b$ temperatures measured in $^\circ \rm C$. My aim is to find their difference in Kelvin ($\rm K$). I thought of this question for fun after noticing that I can approach this problem in two different ways, resulting in two different answers.

Recall that $\mathrm K = \mathrm C + 273$. The difference of $a$ and $b$ in $^\circ \rm C$ is $|a - b|$.

  1. Convert the difference, namely $|a-b|$, to $\rm K$. $$|a-b|_\rm C = (|a-b|+273)_\rm K.$$ $\therefore$ The difference in $\rm K$ is $|a-b|+273$.

  1. By converting $a^\circ \,\rm C$ and $b^\circ \,\rm C$ to $\rm K$, we obtain $(a+273)\, \rm K$ and $(b+273)\,\rm K$ respectively. Therefore the difference in $\rm K$ is $$|(a\require{cancel}{\cancel{+\,273}})-(b\cancel{+\,273})|=|a-b|.$$ $\therefore$ The difference in $\rm K$ is $|a-b|$, equal to the difference in $^\circ \rm C$.

The approaches in both these calculations make total sense to me, but clearly each one yields a different answer.

What is the correct answer (if there is one) and what is the correct approach to solving it? Is there a general case as regards $a$ and $b$, or would such an approach only apply to specific examples of fixed quantities (e.g. $a=30$ and $b=60$).

I don't have to convert from $^\circ \rm C$ to $\rm K$, but similarly, I could work in Fahrenheit ($\rm F$); however I am not sure what would happen if I let $a$ or $b$ equal to $-40$, given that $-40^\circ \, \rm C = -40^\circ \, \rm F$.

Qmechanic
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Mr Pie
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    I've removed a comment that answered the question. Please use comments to improve and clarify posts. – rob Feb 19 '20 at 04:32

3 Answers3

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Temperature difference is not the same thing as temperature. If two objects are at the same temperature, then their temperatures differ by $0^\circ$C. Would you say, "aha! $0^\circ$C = $32^\circ F$, so the two objects differ in temperature by $32^\circ$F"?

The correct approach is, of course, the second one. If you want to find the difference between two temperatures in a particular temperature scale, then express the two temperatures in that scale and then subtract.

J. Murray
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  • But in the correct approach, wouldn't the difference always be the same in every temperature conversion, then? – Mr Pie Feb 19 '20 at 01:23
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    @MrPie I don't know why you think that. A temperature difference of $1^\circ$C is equal to a temperature difference of $1.8^\circ$F. – J. Murray Feb 19 '20 at 01:25
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Here is a visual for what’s happening:

enter image description here Temperature can be represented on a 1D number ray. The transformations that can be done on a ray is scaling and shifting. And this is exactly what the different temperature scales are related to each other by. Scaling and shifting.

Now the conversion between celsius and kelvin is just shifting thus the temperature differences are preserved. This is what your second approach tells you.

Superfast Jellyfish
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    I think your diagram has all of the "number rays" aligned by their values, with all the zeros vertically aligned, all the 100s vertically aligned, etc., and the physically identical values connected by crooked lines. From the perspective of an experimental physicist who recognizes that the different scales are arbitrary labels for the same physical reality, I'd expect to see the physically identical temperature labels vertically aligned. But maybe your illustration speaks better to the asker's confusion. It's an interesting presentation. – rob Feb 19 '20 at 04:45
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    See also https://i.imgur.com/94IdP0x.jpg – rob Feb 19 '20 at 04:52
  • Yes. I agree that the numerical value is arbitrary. What I am trying to show is given a scale (kelvin) how the other scales are related to it. And thanks for the image! Pure gold. – Superfast Jellyfish Feb 19 '20 at 04:57
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The problem with your method is that temperature difference and temperature cannot be used in similar fashion as you do with meter, etc.

Anyway $1° \text C=1 \text K$ when these represent temperature difference. Now this might be be a bit confusing if the context isn't given as $1°\text C =274.15\text K$.

So to resolve such confusion my book (Halliday Resnick Walker) has suggested that we represent temperature difference in non-absolute temperature scales such as Celsius and Fahrenheit by $\text C°$ and $\text F°$.

Then $1\text C° = 1\text K$ and $1°\text C =274.15\text K$ hence solving the confusion. It would be great if it is taken as convention by SI.