## Sunday, March 27, 2011

### Converting Units of Temperature

In the course of writing articles for this blog, on occasion I write a post on the conversion of units.  Units can be confusing for some, and if I can add some clarity, I think it is worth doing.  This post addresses conversion between units of temperature.

The most common temperature conversion that one wants to do is to convert degrees Celsius to degrees Fahrenheit or vice versa.  I hope to explain how to do this conversion in such a way that one need not memorize a formula, even to get the exact answer.

First, it is worthwhile to mention a quick and dirty approximation that almost anyone can do in his or her head.  It is not exact, but it is useful, if you are traveling to a country that uses a scale that is different than the one you use to think about ambient temperatures.

Approximate Conversion Between Celsius and Fahrenheit

The approximate method of conversion is very simple.  To convert from degrees Celsius to degrees Fahrenheit, take the temperature in Celsius, double it, and add 30.

For example, suppose you live in the US but are visiting France.  Someone tells you that the temperature outside is 20 degrees, and you want to know what the temperature is in Fahrenheit.  Doubling twenty yields 40 and adding 30 yields 70.

The exact answer is 68; so you are off by 2 degrees Fahrenheit or about 1 degree Celsius.

The reverse calculation can be carried out as well.  Suppose someone gives you a temperature of 50 degrees Fahrenheit and you want to know the temperature in Celsius.  Subtract 30 and divide by 2.  20 divided by 2 is 10.

The exact answer in this case is also 10.  This approximation is only exactly correct at 10 degrees Celsius or 50 degrees Fahrenheit.  As the temperature gets colder or hotter, the approximation gets worse.

Between 5 and 95 degrees Fahrenheit, the error does not exceed 5 degrees Fahrenheit. Between 32 degrees Fahrenheit and 68 degrees Fahrenheit, the error does not exceed 2 degrees Fahrenheit.

The approximation works pretty well for most ambient temperatures.  One may desire to be more exact, however, or one may need to calculate a temperature that is outside the range where the approximation works well.  In either case, it is not difficult to calculate the answer as precisely as one desires.

Exact Conversion Between Celsius and Fahrenheit

First a caveat: the exact conversion is only exact if one does not round the answer one gets from the mathematical operations.  In some cases that math works out to provide an exact conversion.  For example 32 degrees Fahrenheit is exactly 0 degrees Celsius.

In other cases, one will round the result to the appropriate number of decimal places.  If one wishes to keep uncertainty to a minimum, one should perform the calculation with enough accuracy that the conversion itself is not the source of uncertainty in the temperature measurement. Rather, the uncertainty in the original measurement should be the source of uncertainty.

Some people like to memorize formulas and that is fine, but others find it easier to reason their way to an answer.  To convert between the Fahrenheit and Celsius scales, there is no need to memorize the formula.

It is sufficient to remember simple facts that you probably already know.  Water freezes at 0 degrees Celsius, which is 32 degrees Fahrenheit.  Water boils at 100 degrees Celsius, which is 212 degrees Fahrenheit.

These numbers are the only number you need to remember.

0 degrees Celsius is 32 degrees Fahrenheit.  This fact gives us an offset between the two scales of 32.  consider that fact and then set it aside.

Between 0 and 100 degrees Celsius, how many Fahrenheit degrees are there?  Well, we know that the Fahrenheit scale goes from 32 to 212 in that same span.  212 - 32 = 180.

So for every difference of 100 degrees Celsius, there is a difference of 180 degrees Fahrenheit.

Suppose we want to convert 20 degrees Celsius to Fahrenheit.  How far does the answer differ from 0 degrees Celsius or 32 degrees Fahrenheit?

In Celsius, the answer differs by 20 degrees.  Every difference of 100 degrees Celsius is a difference of 180 degrees Fahrenheit. 20 degrees is one-fifth of 100 degrees.  Our answer must differ from the freezing point by one fifth of 180.  One fifth of 180 is 36.  The freezing point in Fahrenheit is 32.  So the answer must be 68 Fahrenheit.

We did not make any approximations in our math; so this answer is exact, assuming our input was exactly 20 degrees Celsius.

This method effectively derives the formula for converting between the Celsius and Fahrenheit scales.  180 divided by 100 is equal to 9/5.

To convert from Celsius to Fahrenheit, one multiplies by 9/5 and adds 32.

9/5 multiplied by 20 is 36, and 36 + 32 is 68.

If one wishes to convert from Fahrenheit to Celsius, simply reverse the logic.

To convert from Fahrenheit to Celsius, subtract 32 and multiply by 5/9.

Note again that there is no need to memorize these numbers if you know the temperatures at which water freezes and boils.

The Kelvin Scale

The third law of thermodynamics provides an absolute zero for temperature, and tells us that it is impossible to get there in a finite number of steps.  The Kelvin scale is defined so that 0 kelvin (K) is absolute zero.

Note that in the Kelvin scale, one does not refer to "degrees Kelvin."  Because the scale is an absolute scale it is recognized by making the unit of temperature be the kelvin (K).  Unlike degrees Fahrenheit (ºF), there is no degree (º) sign.  The unit is the kelvin (K).

Zero degrees Celsius is defined to be exactly 273.15 K.  Conversion between the Kelvin and Celsius scales is easy.

To convert from kelvin to degrees Celsius, simply subtract 273.15.

To convert from degrees Celsius to kelvin, add 273.15.  So,  25.00  ºC = 298.15 K.

The conversion is exact; so one may maintain as many significant figures as are available in the original measurement.

300.19478 K = 27.04478  ºC.  One may keep all of the significant figures.

If one wishes to convert from the Kelvin to the Fahrenheit scale, the easiest method to remember is to convert from Kelvin to Celsius first and then to Fahrenheit.  The reverse is also true.  To convert from the Fahrenheit scale to the Kelvin scale, first convert to Celsius, and then convert from Celsius to Kelvin.

The Rankine Scale

Just as the Kelvin scale is an absolute temperature scale in which a difference of one kelvin is a difference of one degree Celsius, there is an absolute temperature scale that is offset from the Fahrenheit scale such that one degree Fahrenheit is one unit on that scale.

The Rankine scale is offset from the Fahrenheit scale by (9/5)*273.15 - 32 = 459.67.

Zero degrees Fahrenheit is defined to be exactly 459.67 degrees Rankine (ºR). Inconsistent with the designation of the kelvin (K) as a unit, the Rankine scale uses degrees Rankine (ºR).

To convert from degrees Fahrenheit to degrees Rankine, add 459.67.

To convert from degrees Rankine to degrees  Fahrenheit, subtract 459.67.

Again, the conversion is exact.

Sources

NIST: International System of Units

Anonymous said...

I am looking for guidance on significant figures in temperature readings. The statement in the conversion of celsius to kelvin seems incorrect, as the Kelvin temperature has one more significant figure than the celsius temperature. The decimal places are the same...

Sign me,

Confused...

Rich said...

Here is why it is confusing. 0 degrees Celsius is exactly 273.15 Kelvin by definition. Even though 273.15 might seem to have only five significant figures, it is an exact number. The significant figures are therefore limited by your measurements, not the use of an exact number. Read up on exact numbers in significant figures. Other examples of exact numbers are the speed of light and also exact integers. If I have exactly 2 masses that each have a mass of 1.23 g, then 2 x 1.23 = 2.46, not 2.

Rich said...

Also, bear in mind that addition and subtraction work differently than multiplication and addition. When I add 25.00 to an exact number, no matter how many digits it has, I get to keep two places after the decimal. 273.15 + 25.00 = 298.15. 273.15 + 25.00000 = 298.15000.
If both numbers are inexact, it is different: 200.01 + 25.00 = 225.01, but 200.0 + 25.00 = 225.0.