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The exact relationship derived mathematically shows that the temperature T is actually proportional to the average kinetic energy of the gas particles, since the particles don’t all have the same speed. The complete result gives this relationship as

KE = 3 2 RT

where KE is the average kinetic energy of the gas particles and R is the same constant, which appears in the Ideal Gas Law.

This is a wonderful result for many reasons. The relationship of temperature to molecular kinetic energy is used constantly by Chemists in interpreting experimental observations. Even though we have worked it out for a gas, it turns out that this proportionality is also valid in liquids and solids.

Analysis of the ideal gas law

We can use the results of the previous section to understand many of the observations we have made about gases. For example, Boyle’s Law tells us that the pressure of a gas is inversely proportional to the volume of the gas, if we have a fixed number of molecules and a fixed temperature. Our results above show us that decreasing the volume for a fixed number of molecules increases the frequency with which the particles hit the walls of the container. This produces a greater force and thus a higher pressure.

We also know that the pressure of a gas increases with the number of particles, if the volume and temperature are fixed. Our results show us again that, for a fixed volume, more particles will create more collisions with the walls, producing a greater force and a higher pressure.

Finally, we know that the pressure of a gas increases with the temperature. Our results above show that increasing the temperature increases the speed of the particles. This increases the frequency of collisions and increases the force of each collision. Therefore, the increase in pressure is proportional to v 2 .

We can also interpret the deviations from the Ideal Gas Law observed in Figures 1, 2, and 3. Remember that a gas at high density may have a greater pressure or a lower pressure than predicted by the Ideal Gas Law. But the postulates of the Kinetic Molecular Theory lead us to predictions that match the Ideal Gas Law. This means that, if the pressure of a gas under some special conditions does not match the prediction of the Ideal Gas Law, then one or more of the postulates of the Kinetic Molecular Theory must not be correct for those conditions.

We only see deviations from the Ideal Gas Law at high particle density, and in this case, the particles are much closer together on average than at lower density. Looking back at the postulates above, this means that we can no longer assume that the gas particles do not interact with each other. If the particles do interact, they exert forces on each other, which will consequently change their speeds.

If the speeds are reduced by these forces, there will be fewer impacts with the wall, and each impact will exert a weaker force on the wall. Therefore the pressure will be lower than if the particles don’t exert these forces on each other. What force between the particles would cause the speeds to be lower? If the particles attract each other, then this attraction will cause each particle to slow down as it moves towards the wall, since it will be attracted to particles behind it. We can conclude that attractions between molecules will lower the pressure, which we called a negative deviation from the Ideal Gas Law. Therefore, when we see a large negative deviation from the Ideal Gas Law, we can conclude the molecules have strong intermolecular attractions.

If the speeds are increased by the intermolecular forces, there will be more impacts with the wall and each impact will exert a great force on the wall, causing a higher pressure. Following our reasoning above, a positive deviation from the Ideal Gas Law must be due to molecules with strong intermolecular repulsions.

Our experiments tell us that, as we increase the density of the gas particles to a high value, we first see negative deviations from the Ideal Gas Law. Therefore, as we increase the density, the molecules are on average closer together and the first intermolecular force they experience is attraction. This means that the attraction of particles is important when the particles are still rather far apart from each other. Only when the particle density gets very high and the particles are on average much closer together will repulsions become important resulting in positive deviations from the Ideal Gas Law.

This analysis of experimental observations from the Kinetic Molecular Theory can be applied to understanding properties of liquids and solids. In particular, it is very helpful in understanding why substances have high or low melting or boiling points. We begin our study of these different phases of matter in the next Concept Development Study.

Review and discussion questions

  1. Explain the significance to the development of the kinetic molecular model of the observation that the ideal gas law works well only at low pressure.
  2. Explain the significance to the development of the kinetic molecular model of the observation that the pressure predicted by the ideal gas law is independent of the type of gas.
  3. Sketch the value of PV/nRT as a function of density for two gases, one with strong intermolecular attractions and one with weak intermolecular attractions but strong repulsions.
  4. Give a brief molecular explanation for the observation that the pressure of a gas at fixed temperature increases proportionally with the density of the gas.
  5. Give a brief molecular explanation for the observation that the pressure of a gas confined to a fixed volume increases proportionally with the temperature of the gas.
  6. Give a brief molecular explanation for the observation that the volume of a balloon increases roughly proportionally with the temperature of the gas inside the balloon.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Concept development studies in chemistry 2013. OpenStax CNX. Oct 07, 2013 Download for free at http://legacy.cnx.org/content/col11579/1.1
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