Could you have a quartic or square degree Celsius or other degree on a temperature scale raised to any power?

Question

The units of the Stefan-Boltzmann constant are watts per square meter per quartic kelvin, so it is possible with kelvins. But a kelvin is not a type of degree, although it is a unit of temperature, and is in fact the SI unit of temperature.

Please note that I'm not sure about any of this, your feedback in the comments would be welcome. I'm not sure "per quartic kelvin" is optimal, though it is sometimes used; perhaps "per kelvin to the power four" is better. I know that nomenclature can be a controversial and emotive subject, even in science.

Searching the Internet turned up no trace of a square degree of temperature, and Wikipedia has no disambiguation page for "square degree" and says, "A square degree (deg2) is a non-SI unit measure of solid angle. Other denotations include sq. deg. and (°)2. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere."

So it's possible to write it, but it only means square degree of angle, according to Wikipedia. https://en.wikipedia.org/wiki/Square_degree

So my question is: could you have a quartic or square degree Celsius or other degree on a temperature scale raised to any power?

This is a valid and reasonable question and should not be closed. — John Alexiou, Dec 02 '21 at 13:11
Please note that you can only apply non-linear mathematical functions such as $\exp(x)$, $\ln(x)$, $\sin(x)$ etc to dimensionless quantities, such as a ratio of temperature to a reference temperature. — John Alexiou, Dec 02 '21 at 13:19
@JohnAlexiou What do you mean by non-linear? Clearly raising to a power is legal. — J.G., Dec 02 '21 at 17:45
@J.G. - raising to a rational power is ok, but not for an irrational power. By non-linear, I really meant transcendental function. — John Alexiou, Dec 02 '21 at 21:11
@J.G. - hmm how can that be? What kind of physical meaning would $\text{(length)}^{0.69314718055..}$ have? At least with rational powers, you can construct relations with integer powers in both sides of the equation. Like $$y=x^{3/7} ;\Rightarrow , y ,x^7 = x^3$$ — John Alexiou, Dec 03 '21 at 05:11
@JohnAlexiou Why should I have to do that? Your original criterion was only dimensional analysis, so $y=kx^p$ will always be fine for $[k]=[y][x]^{-p}$. — J.G., Dec 03 '21 at 07:47

score 4 · Answer 1 · answered Dec 02 '21 at 13:02

Let's focus on two example equations where temperature enters. The first example is Newton's law of cooling: $$ J = c (T_h-T_c) $$ where $J$ is the heat flux from a hot reservoir at temperature $T_h$ and a cold reservoir at temperature $T_c$ and $c$ is some constant that depends upon the interface details. It should be clear that in this equation we could (if we wished) write instead: $$ J = c\left((T_h-T_r)-(T_c-T_r)\right)$$ and insert some reference temperature $T_r$. Thus this equation makes sense whether or not we use absolute temperature (measured in Kelvin) or relative temperature (measured in degree Celsius or Fahrenheit, and which has precisely such a reference temperature).

Instead consider the Stefan-Boltzmann equation: $$ P = A\sigma(T_h^4-T_c^4) $$ because $x^4$ is a non-linear function, we can no longer do the same shifting trick. Thus, the Stefan-Boltzmann law can only hold in absolute units. This should make it clear that any quantity where the units include Kelvin raised to any non-linear power can't make much sense when written in Celsius.

Dimensionally speaking the coefficient $\sigma$ also contains a Kelvin to the negative four $K^{-4}$ term inside here, so you can argue that only dimensionless quantities can and should be raised to any power other that 1 or -1. — John Alexiou, Dec 02 '21 at 13:14

score 1 · Accepted Answer · answered Dec 03 '21 at 15:48

If you were measuring coefficients of thermal expansion, where the length at some temperature is

$$ L(T) = L(T_\text{ref}) \times \big( 1 + (T-T_\text{ref}) \cdot\alpha\big) $$

then the unit for $\alpha$ is $\rm K^{-1}$ or $\rm(ºC)^{-1}$. This online reference uses units of $\frac{\rm \mu m}{\rm m\cdot K}$, because the typical scale for $\alpha$ is a part-per-million expansion per degree.

If you were interested in the rate of change of $\alpha$ with temperature, you'd measure $\frac{\mathrm d\alpha}{\mathrm d T}$, which would have units of $\rm K^{-2}$.

score 1 · Answer 3 · answered Dec 03 '21 at 16:03

Degrees Celsius are problematic in general, unless a temperature difference is used so that Kelvin offset is cancelled. If for example a temperature is in denominator, you would have big problem if temperature turns out to be $0 \, ^\circ\text{C}$.

You could probably find some examples where temperature in degrees Celsius is used, but that is probably limited to some numerically-tuned equation which works only in some specific temperature interval.

As for the power of the temperature in Kelvins, you already pointed out one example where power 4 is used.

Another example is Steinhart-Hart equation which describes resistance of a semiconductor at different temperatures:

$$\frac{1}{T} = A + B \ln R + C (\ln R)^3$$

where $A$, $B$, and $C$ are equation parameters. The inverse Steinhart-Hart equation is:

$$R = R_{25} \exp\left(1 + \alpha \frac{1}{T} + \beta\frac{1}{T^2} + \gamma\frac{1}{T^3}\right)$$

This is just one of many examples that shows that temperature variable can undergo any nonlinear function which depends on the underlying process. Sometimes it has physical meaning, but more often it is numerically-tuned equation.

Could you have a quartic or square degree Celsius or other degree on a temperature scale raised to any power?

3 Answers3