Conversion of compound temperature units

Question

In general one can convert from one unit to another (of the same dimension) by using (($U$,$A$) and ($V$,$B$) being compatible unit pairs in the following)

$xU=(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A=yA$

Oftentimes the offset $o_{U\rightarrow A}$ will have a value of zero, which reduces the conversion to a simple multiplication with a factor $f_{U\rightarrow A}$. For some units like temperatures, however, we need that offset as well.

There are a few compound units in use, which are combinations of a temperature and some other unit like joule per kelvin or degree Celsius day.

My question is how would I convert those compound units unambiguously?

To be more specific: Assume we want to convert from a (compound) unit $UV$ to a unit $AB$ and are given the following conversion formulas:

• $xU=(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A$
• $xV=(x\ast f_{V\rightarrow B})B$

The conversion formula from $x UV = y AB$ seems to depend on the order of conversions:

If converting first from $U$ to $A$ and then from $V$ to $B$ we end up with the following calculation

$xUV\\ =(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})AV \\ =((x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A\ast f_{V\rightarrow B})B \\ =(x\ast f_{U\rightarrow A}\ast f_{V\rightarrow B}+o_{U\rightarrow A}\ast f_{V\rightarrow B})AB$

Doing it the other way around, we end up with

$xUV\\ =(x\ast f_{V\rightarrow B})UB\\ =(x\ast f_{V\rightarrow B}\ast A\ast f_{U\rightarrow A}+o_{U\rightarrow A})B\\ =(x\ast f_{V\rightarrow B}\ast f_{U\rightarrow A}+o_{U\rightarrow A})AB$

So depending on the order of conversions we end up in results, which differ by

$\begin{array}{l}(x\ast f_{U\rightarrow A}\ast f_{V\rightarrow B}+o_{U\rightarrow A}\ast f_{V\rightarrow B})-(x\ast f_{V\rightarrow B}\ast f_{U\rightarrow A}+o_{U\rightarrow A})\\=o_{U\rightarrow A}\ast f_{V\rightarrow B}-o_{U\rightarrow A}\\=o_{U\rightarrow A}\ast(f_{V\rightarrow B}-1)\end{array}$

Can somebody explain to me, where I'm going wrong here or how this ambiguity is resolved in practice?

Answer 1

Note that those compound units are used to express rates of change: change in energy per unit of temperature, or change in temperature per unit of time (day). That is, $\Delta E/\Delta T$ and $\Delta T / \Delta t$. The offset term cancels out when taking the differences.

Answer 2

Most of the time, the temperature quantity is a difference between two temperatures, not an absolute value. So, the offset is zero: \begin{align} \Delta T(F) &= T_2(F) - T_1(F) \\&= \left(\frac{9}{5}T_2(C) + 32\right) - \left(\frac{9}{5}T_1(C) + 32\right) \\&= \frac{9}{5}\left(T_2(C) - T_1(C)\right) \\&=\frac{9}{5}\Delta T(C) \end{align} This is true for quantities like entropy changes, heat capacities, and coefficients of thermal expansion.

An exception to this would be the universal gas law $PV = nk_BT.$ In this case, you need to use an absolute temperature scale where $T=0$ actually means absolute zero. So, your choices are Kelvin or Rankine, and the offset is zero, since zero Kelvin and zero Rankine mean the same thing.