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Proportionality and Absolute Temperature

Date: 06/06/99 at 18:38:32
From: Sean
Subject: Variation and polynomial equation word problems

I don't understand how to do the word problem below:

Absolute temperature is measured in degrees Kelvin (K). A degree on 
the Kelvin scale is the same size as the Celsius degree, but 0 K is 
-273 C (Celsius). [This is a part that puzzles me; please explain]. 
The volume of a fixed amount of gas kept at constant pressure varies 
directly as its absolute temperature. If the gas occupies 100 liters 
at -13 C what is its volume at 26 C?

I have tried to do -13/100 is equal to 26C/x as a proportion but I 
have failed to reach the answer: 115 liters. I was given a clue to 
convert the Kelvin to Celsius, but have failed to see how. (That is 
why I wrote - I don't understand the puzzling part above.)

Thank you for taking the time to solve this. My thanks to you are 
square root pi x infinity.

Date: 06/07/99 at 09:22:59
From: Doctor Rick
Subject: Re: Variation and polynomial equations word problems

Hi, Sean.

I'm glad you are persistent - that's the way to learn.

Here is what is going on with Celsius and Kelvin. I'll draw a Celsius 
thermometer and a Kelvin thermometer:

                                A     B
C +-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
 -273  -223  -173  -123  -73   -23    27    77    127   177   227

                                A     B
K +-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
  0     50    100   150   200   250   300   350   400   450   500

You see that 0 K = -273 C (actually, I think it's -273.15 C or 
something like that, but we're rounding). The difference between A and 
B is 27 - (-23) = 50 C. On the Kelvin thermometer the difference 
between A and B is also 50 K (300-250). That's why the problem can 
state that a degree on the Kelvin scale is the same size as the 
Celsius degree. All you do to convert Celsius to Kelvin is to add 273 
degrees; for instance:

  A = -23 C
    = -23 + 273 K
    = 250 K

The problem stated that the volume of a fixed amount of gas kept at 
constant pressure varies directly as its absolute temperature. 
Absolute temperature is the temperature on the Kelvin scale, or 
another scale that has the same zero point. (The Rankine scale, for 
example, is to the Fahrenheit scale what Kelvin is to Celsius: its 
degrees are the same size as Fahrenheit degrees, but 0 R = 0 K.)

So, convert the two Celsius temperatures to Kelvin before you work out 
the proportionality.

  -13 C = -13 + 273 = 260 K
   26 C =  26 + 273 = 299 K

   x L    299 K
  ----- = ----- = 1.15
  100 L   260 K

  x = 1.15 * 100 L = 115 L

Think about WHY we must express the temperatures in Kelvin (or 
Rankine) for this proportionality to work. Let's invent a new scale 
for measuring people's height, the Sean scale. This scale uses feet 
and inches, but we call a person who is 5 feet tall "0 feet on the 
Sean scale." I am 6 feet tall on the "absolute" height scale, but I 
would be called 1 foot tall on the Sean scale (because I'm 1 foot more 
than 5 feet tall). A child 3 feet tall would be called -2 feet S. 

Sounds bizarre, doesn't it? But that's what Celsius temperature is 
like. Could you tell that I (1 foot S) am twice as tall as that -2 
foot S child? You would have to convert both of our heights to 
"absolute height" by adding 5 feet: I am 1 + 5 = 6 feet (absolute) and 
the child is -2 + 5 = 3 feet (absolute). Then you can divide and find 
that the ratio of our heights is 6/3 = 2.

When the common scales (Fahrenheit and Celsius) were developed, no one 
knew yet what the lowest possible temperature was, or even that there 
was a lowest temperature. The proportionality rule stated in the 
problem wasn't known yet. It seemed that one could choose a zero point 
arbitrarily, the way a zero point of longitude was chosen arbitrarily 
at the meridian of the Royal Observatory in Greenwich, England. 
Indeed, Fahrenheit and Celsius chose different zero points. But when 
experiments with gases produced the law that is used in this problem, 
the zero point was no longer arbitrary. That's why the Kelvin scale 
was developed.

I hope this helps you understand the why and the how of absolute 
temperatures and proportions. If not, keep at it!

- Doctor Rick, The Math Forum
Associated Topics:
High School Physics/Chemistry

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