Bette:  Oh Prof, can we talk with you a minute? 
Prof:  Hi Bette; sure. What seems to be the problem? 
Alf:  We're getting confused about converting temperatures. 
Bette:  Yeah. Say, we need to convert the gas constant R from one set of units to another. 
Prof:  Ok, you've proposed a situation; can you state the problem? 
Alf:  Well, let's say we have R = 8.314 J/(mol K), but we need it in Btu/(lbmol °R). The problem is, how do we convert the temperature in R? 
Prof:  What temperature? There is no temperature in the gas constant: the Kelvin or Rankine in the gas constant is a temperature unit, not a temperature. 
Alf:  I don't get it. 
Prof:  Ok, let's leave temperature aside for a few minutes and begin by reviewing what you know about conversion factors. How long have you been doing unit conversions? 
Bette:  Gee, Prof, it seems like forever; I mean, we're always doing it. 
Prof:  So, you must have organized your thinking about them, right? 
Alf:  Organized? 
Prof:  A purpose of education, Alf, is to organize your thinking into economical and useful forms. 
Bette:  What's to organize about conversion factors? You just do them. 
Prof:  But you do seem to have a difficulty distinguishing temperatures from temperature units, so something in your thinking seems lacking. 
Alf:  Well … 
Prof:  Indeed. Ok, consider this: one way to organize is by classification. Can we organize conversions into classes? If we look at the conversions we routinely do, they divide into three classes. 
First are those in which we merely rename the unit. For example,
 
These are called isomorphic transformations; in such conversions, nothing changes numerically, we only change the name. The conversion factor is unity.  
Bette:  Why do we bother to change a name? 
Prof:  Usually, it's a matter of economy; I'd rather write one symbol (1 N) instead of half a dozen (1 kg m/s^{2}). Sometimes there are political motivations, such as the use of Celsius instead of the older term centigrade. 
Alf:  Ok, Prof, what other classes are there? 
Prof:  You tell me, Alf. 
Alf:  Well, there are those in which the name changes and in which the conversion factor is not unity. 
Prof:  Good. These are called similarity transformations. For us, similarity simply means a scaling; we scale one unit by a factor to get another unit. Most conversions are like this. Can you give some examples? 
Bette:  1 foot = 12 inches; 1 minute = 60 seconds; 1 meter = 100 cm. 
Prof:  Good. Now, Alf, can you identify the third class? 
Alf:  Well, we haven't gotten to temperature conversions yet. 
Prof:  Right. The third class is composed of linear transformations; these involve a shift in the reference point (origin), with or without a scaling. For example, conversions from Celsius to Fahrenheit involve both a shift in origin and a scaling: 
T[°F] = (9/5) T[°C] + 32
 
But conversions from Celsius to Kelvin only involve a shift in the reference point:  
T[K] = T[°C] + 273.15
 
Alf:  Are there other linear transformations, besides those in temperature? 
Prof:  Bette? 
Prof:  Wait a minute … oh, there's the conversion from gage pressure to absolute, 
P[absolute] = P[gage] + P[atmosphere]
 
Prof:  Good. 
Alf:  Ok, Prof, we've got three kinds of conversions: isomorphic transformations, similarity transformations, and linear transformations. So what? 
Prof:  So, let's consider what's really happening in each of these. To begin, consider the similarity transform; we've said this is always a scaling, but a scaling of what? 
Alf:  I'm not sure … 
Prof:  Look at one of Bette's examples: 1 foot = 12 inches. What is being scaled when we change a measured length from feet to inches? Does the length change? 
Bette:  Of course not, Prof. 
Prof:  Then what? 
Bette:  The units. 
Prof:  And, what about the unit changes from 1 to 12? 
Alf:  Ah! The size of the unit. 
Prof:  Exactly. Similarity transforms scale the size of the unit; and when we change the size of the unit we also give the new size a different name. In contrast, in an isomorphic transformation, the size of the unit remains the same. 
Alf:  But in a linear transform we change the zero for the scale of unit and we may or may not also change the size of the unit? 
Prof:  You've got it. 
Bette:  Ok, Prof. But how does this help us with the conversion of R problem? 
Prof:  Remember my remark that R contains temperature units, not temperatures? 
Alf:  So? 
Prof:  So, converting a temperature unit is never a linear transformation; it may be isomorphic or similarity. For example,

And since converting among values of the gas constant R involves converting among temperature units, only scalings of those units are required. So your conversion of R requires only similarity transforms:  
Bette:  Ok, but what about converting temperatures?  
Prof:  I'm afraid we have so many possible temperature scales and units, that any kind of conversion can arise. A temperature conversion
 
Alf:  But can we make a general rule?  
Prof:  Sure Alf, here it is: Whenever you convert any derived quantity that contains a temperature unit, the conversion requires only a scaling of the temperature unit—never a linear conversion. You merely scale by a factor; it might be that the factor is unity (isomorphic) or not unity (similarity).  
Bette:  So our organization of conversion factors is this:  
 
Prof:  Exactly! 
Alf:  Ok, Prof, what about converting temperature changes? 
Prof:  Good question, Alf. A change is a difference, for example (recall that Δ is the "change" operator), 
ΔT = T_{2}  T_{1}
 
And the subtraction removes the reference point from the change. So converting a temperature change is never a linear transform; it may be similarity or isomorphic. It is similarity if the sizes of the new and old temperature units differ, but it is isomorphic if the sizes are the same.  
Bette:  What about some examples, Prof? 
Prof:  Ok, Bette. Since Celsius and Kelvin have the same size unit, conversions of changes between these two are isomorphic: 
ΔT[Kelvin] = 1 ΔT[Celsius]
 
But the sizes of Celsius and Fahrenheit units differ, so conversions of changes between these two are similarity transforms:  
ΔT[°F] = 1.8 ΔT[°C]
 
Alf:  Then conversions of changes between Fahrenheit and Rankine must also be isomorphic? 
Prof:  That's right, Alf. 
Bette:  Great Prof. Thanks a lot. 
Prof:  No problem. 