Gay-Lussac’s Law and Temperature
Boyle’s Law is the relationship between Pressure and Volume but does not address temperature. How does temperature effect a sample of gas? First, let’s remember what temperature
actually is. If you remember from our Thermodynamics unit, temperature is a measure of the
average kinetic energy of the particles in a sample. Or put more simply, temperature approximates how fast the particles are moving. The faster the particles move, the higher the temperature.
How would this effect a system where the volume is closed and constant? Have you ever
thrown an aerosol can into a campfire? You shouldn’t because it is extremely dangerous as the
aerosol can explodes, throwing shards of metal and combustible material in all directions! Why
does it explode, though?
http://www.unizar.es/lfnae/luzon/CDR3/images/gay-lusac.jpg
The answer is explained by Gay-Lussac’s Law. Gay-Lussac’s Law says that for a closed system where the volume is constant, the temperature and pressure are directly proportional. In
other words, if you heat up an aerosol can, the gas inside can’t escape but it moves faster and
builds up pressure.
In the first example, the particles are in an ice bath and so the temperature is low. The particles
are therefore not moving very fast and subsequently have a low pressure. In the second picture,
the flask is now in boiling water so the particles are moving much faster. They have much
more kinetic energy in the same volume so the pressure is increased.
Experiment
In this experiment we will be graphing Gay-Lussac’s Law. An empty beaker of air is sealed
with a stopper and then attached to a pressure sensor. The beaker will then be put in boiling
water and then in an ice bath. Pressure readings will be taken at each of the three temperatures.
http://web.njit.edu/~grow/sensors/Pprobe_files/image005.gif
Pressure (mm Hg)
Temperature (oC)
703
0
760
22
961
100
Pressure vs Tem perature
1200
Pressure (mm Hg)
1000
800
600
400
200
0
0
20
40
60
Tem perature (Celcius)
80
100
120
Results
We can see from the graph that there is a direct relationship between the temperature and the
pressure. As the temperature went up, so did the pressure. As the temperature went down, so
did the pressure. Using the same idea as before in Boyle’s Law:
y = mx + b
P = mT + b
In the Boyle’s Law graph, it was obvious that the y-intercept, b, was zero. But look again at
this graph, when the temperature is zero, there is still pressure.
Note that the line doesn’t
cross at 0,0
Even if we ignore that, we have another problem. The ratio of P/T should be constant
amount
P=m
T
But if we do the math, it doesn’t work out:
Temperature
(Celcius)
Pressure
(mm Hg)
P/T
100
961
9.61
22
760
34.5
0
703
???
We need to fix these problems. But to do so is going to require rectifying the math with the
theory.
Rectification
The first thing we have to address is the theory behind this relationship. We are saying that the
hotter the particles get, the faster they move in a closed container, and thus the more pressure
they exert. What if we took that in the other direction? What happens to the particles as they
get colder? They should move slower. Ultimately, the coldest they can get is if there is no molecular motion at all. What should the pressure be at that point? If there is no molecular motion, there should be no pressure. The coldest a particle can get is if there is no motion at all.
What is this temperature? We can estimate it by using our graph from before:
We can see that by this graph, there is still pressure at 0 oC. Particles must be able to get colder
than this. How cold? As said above, if the pressure is 0, there should be no molecular motion
and therefore no temperature. So if we extend out the graph until the line touches 0 mm Hg,
that should approximately be the coldest possible temperature:
Pressure vs. Tem perature
1200
1000
y = 2.5792x + 703.11
Pressure (mm Hg)
800
600
Pressure is zero here so this is the
coldest possible temperature
400
200
0
-300
-250
-200
-150
-100
-50
Tem perature (Celcius)
0
50
100
Absolute (Kelvin) Temperature
The graph on the previous page leads us to an important distinction about temperature. If we
remember, the two temperature scales we are used to, Fahrenheit and Celcius, were arbitrarily
picked by their creators. Daniel Fahrenheit picked 32 oF as the point that water freezes and 212
o
F as the point that water boiled. Celcius picked 0 oC as the point that water freezes and 100 oC
as the point that water boils. They are both temperature scales that allow negative numbers. To
fix this, William Thomson, aka Lord Kelvin, devised a temperature scale that makes sense. In
his scale, the lowest possible temperature should be zero. This is the absolute lowest temperature that could be reached. He called this temperature absolute zero and it is the temperature
where there is no molecular motion at all. This temperature, if you look at the graph on the previous page is –273 oC. So to make sure there are no negative numbers, the Kelvin scale simply
adds 273 to any Celcius temperature.
o
C + 273 = K
You’ll notice that there is no o symbol by the Kelvin in the above equation. Because Kelvin is
the absolute scale and not a relative scale like Celcius and Fahrenheit, it doesn’t need the o symbol!
http://www.lyc-schweitzermulhouse.ac-strasbourg.fr/spip/IMG/
jpg/image004-4.jpg
http://
www.biografiasyvidas.com/
biografia/c/fotos/celsius.jpg
http://sol.sci.uop.edu/
~jfalward/
temperatureandexpansion/
LordKelvin.jpg
Daniel Fahrenheit lived from 1686 to
1736. He was German by birth but
lived most of his life in the Dutch Republic. In 1724 he proposed his scale
that water boils at 32 oF and freezes at
212 oF. It was later reversed for obvious reasons. Why the temperatures of
32 and 212 were picked is not known
and is the subject of much debate.
The Fahrenheit scale is used only in
the US and a few other countries.
Anders Celcius lived from
1701 to 1744 in Sweden. In
1742 he proposed a scale
where water boils at 0 oC and
freezes at 100 oC. One year
later it, like the Fahrenheit
scale, was reversed to the scale
we now know. The Celcius
scale is used throughout most
of the world.
William Thomson, aka
Lord Kelvin, lived from
1824 to 1907 in Belfast,
Ireland. His Kelvin temperature scale is one of
many different contributions to the field of Thermodynamics and other
aspects of Science and
Engineering.
Putting it All Together
We now need to combine all of the information from the previous sections. Let’s quick review:
Gay-Lussac’s Law: Pressure is directly related to Temperature for a system of constant volume
Temperature is a measure of average kinetic energy
Temperature can be measured in an absolute scale by adding 273 to any Celcius temperature
Now let’s examine the relationship from before but this time use Kelvin Temperature
P = mT + b
But since b is now zero:
P=m
T
Pressure (mm Hg)
Temperature (Kelvin)
P/T
961
373
2.58
760
295
2.58
703
273
2.58
This one works perfectly! As long as there are no negative numbers allowed, the ratio in this
relationship is right on target. You will notice that the slope of the line is the 2.58 that we predicted!
Conclusion & Gay-Lussac’s Law
The relationship between pressure and temperature is a direct one that works perfectly when
Kelvin temperatures are used. We come to the conclusion:
P=m
T
P = 2.58
T
The fact that it is 2.58, though, shouldn’t even come up. Look at experiments #1 & 3:
961 = 2.58
373
703 = 2.58
273
Let’s cut out the middle-man:
961 = 703
373
273
Or to put another way:
P1 = P2
T1 T 2
Temperature here MUST
be converted to Kelvin for
this to work!
This is a statement of Gay-Lussac’s Law. The pressure of a gas at constant volume is directly
related to its Kelvin temperature.
Standard Temperature and Pressure (STP)
With all these pressures and temperatures flying about, it was decided to have a standard temperature and pressure. To keep things easy, the standard was set at 0 oC (273 K) and a pressure of 1 atm (14.7 psi or 760 mm Hg or 760 torr or 101 kPa).
Joseph Louis Gay-Lussac was a French chemist who lived from 1778 to
1850. Aside from the law that bears his name he was the co-discoverer of
the element Boron and coined the terms ―pipette‖ and ―burette‖.
http://creationwiki.org/pool/images/
thumb/2/2f/Gaylussac.jpg/150pxGaylussac.jpg
Gay-Lussac’s Law Questions:
A) A sample of gas in a closed, rigid container at 750 mm Hg is at a temperature of 22
o
C. What pressure would the gas exert at a temperature of 100 oC?
P1 = 750 mm Hg
T1 = 22 oC = 295 K
P2 = X
T2 = 100 oC = 373 K
750 mm Hg
295 K
=
X
373 K
Cross multiply and divide:
(750 mm Hg)(373 K) =
(750 mm Hg)(373 K) =
(295 K)
X(295 K)
X
X = 948 mm Hg
B) At what temperature will a sample of gas in a closed container reach a pressure of 5
atm if at standard temperature (0 oC) the gas is at 1 atm?
P1 = 5 atm
T1 = X
P2 = 1 atm
T2 = 0 oC = 273 K
Be careful, the X is in the denominator here!
5 atm =
X
1 atm
273 K
Cross multiply and divide:
(5 atm)(273 K)
=
(5 atm)(273 K)
(1 atm)
=
X(1 atm)
X
X = 1365 K
If you want it in oC then you must subtract 273
X = 1365—273 = 1092 oC
Questions
1.
2.
3.
4.
5.
Gay-Lussac’s Law is the relationship between what two variables?
Which variable must be constant in Gay-Lussac’s Law?
Why should you never throw a closed container like an aerosol can in a campfire?
What is different about the Kelvin temperature scale compared to Celcius and Fahrenheit?
For each temperature below convert either from Celcius to Kelvin or the reverse:
a) 50 oC to K
b) 110 oC to K
c) -200 oC to K
d) 0 C to K
e) 400 K to oC
f) 100 K to oC
g) 1250 K to to oC
6. Sketch the curve of the graph of Pressure vs. Temperature:
P
T
7. What are standard temperature and pressure?
8. A sample of gas at 1.5 atm and 300 K is heated to 500 K. What is the new pressure?
9. A sample of gas at 750 mm Hg and 20 oC is cooled to –40 oC. What is the new pressure?
10. A sample of gas at 120 kPa and 25 oC is heated until the pressure is 200 kPa. What is the
temperature (in both K and oC) that this occurs?
11. A sample of gas will explode out of its container if the pressure reaches 3 atm. What temperature (in both K and oC) will this occur if the can is currently at standard temperature and
pressure?