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The Ideal Gas Law

Learn more about the ideal gas law and its main components: pressure, volume and temperature.
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In this first revision lecture,  we’re going to recap the ideal   gas law and revise and expand on what  we know about pressure and energy  We should already know about  things like the basic states   of matter. There are solids, liquids and gases. We’ll mostly focus on gases for now, but going   through thermodynamics, it will be useful to  remember that changes between these states   are thermodynamic transitions, and have  energy changes associated with them.  You should also have some grasp of  what pressure is, and what it means.   We’re mostly aware of it being force spread over  an area. Force also appears in the second law of   motion, which links it movement of molecules.
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So  movement, pressure and speed, can all be related,   though not necessarily very simply. By the end of this video we should know   more about the components of the ideal gas  law. This will be revision for most of you,   but we’re going to unpack and  explore these terms again anyway.  So, what is pressure? Pressure, as we measure it,   comes from the collision that molecules have with  containers and, also, each other. If there are   more collisions, the pressure is higher.
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So if we pump in more molecules   into the same space, there are more collisions,  more force applied per area, so a higher pressure.   The rest of this first part of thermodynamics  will look at what this force can do.  Dimensionally, there’s also  a connection with energy.  Pressure, multiplied by volume, will get energy.  And pressure multiplied by area gets force.   There’s a difference of only  one length dimension there.   Force and energy are then connected by that length  because dividing energy by a length gets force.  This is actually reflected in the ideal  gas equation, as both sides of it,   as it’s usually written, are equal to energy. 
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Next, we’re going to look at  temperature. And what is temperature?  Well, it turns out that’s a more complicated  question than you might think. It has a number   of definitions, and some almost sound circular,  like how temperature is a measure of “hotness”   of something. It also pops up later  in statistical thermodynamics,   where it’s interpretation is about how particles  are distributed around different energy states.  For our purposes, right now, we’re  interested in temperature as a function of   speed or velocity of particles. Therefore,  it is a measure of average kinetic energy.  Remember that term average, there. Atoms and  molecules fly around at different speeds,   and that’s described by a longer equation  called the Maxwell-Boltzmann distribution. So,   molecules run at different speeds.
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Their  average speed gets us temperature, among a   few other useful properties in  kinetics and thermodynamics.  Now, finally volume. You might think that there isn’t anything   interesting to say about volume… and mostly you’d  be right. Volume is just a three-dimensional   space. X times Y times Z, or length times breadth  times depth. BUT, everything we will be talking   about with ideal gases can be made two-dimensional When about surface tension, or surface pressure,   we’re dealing with areas. And, rather  usefully, there is an ideal gas analogue   in two dimensions. This can describe things like  surfactants or other materials that interact   on the surface of a liquid. You’ll find out more  about that when you do materials chemistry later.
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If we combine all those things together,  we end up with the ideal gas law.  This tells us that pressure, times  volume, divided by temperature,   the things we’ve just revised,  should always be constant.  That constant is the number of moles of the gas  in the sample, and the universal gas constant.   When we’re talking about a single molecule,  that’s the Boltzmann constant. So you can   easily tell if an equation is talking about  moles or molecules by whether it uses K or R.  So, if we fix one of those three variables,  the temperature, pressure or volume,   the other two will always be perfectly related. This is what the technical detail of   thermodynamics is based around. What conditions  are we working in?
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Is temperature fixed?   Are we fixing a volume by trapping  everything in a container?   Are we fixing pressure by leaving everything  open to the atmosphere where it can expand? 
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Over the next few lectures, we’ll look at  these equations, and how to apply them.

This video and the notes below cover details about the ideal gas law, and the variables that make it up.

Gas Laws

The ideal gas law is:

[pV=nRT]

Where the variables are:

symbol description
p pressure
V volume
n moles
R the universal gas constant
T temperature

The units of these variables will be discussed over the next steps, and are the key to applying this equation correctly.

Empirical Gas Laws

The gas law leads to a number of relationships that can be empirically justified – the ideal gas law was historically assembled from these empirical laws. At constant temperature, pressure multiplied by volume is constant, so a change in pressure results in a predictable change in volume.

[p_1 V_1=p_2 V_2]

At constant pressure, a change in volume results in a predictable change in temperature and a change in temperature results in a predictable change in volume.

[frac{V_1}{T_1}=frac{V_2}{T_2}]

And at constant volume a change in pressure results in a predictable change in volume.

[frac{p_1}{T_1}=frac{p_2}{T_2}]

What is Pressure?

Pressure comes from the force felt due to microscopic particles colliding with a container or the surrounding gases. Each particle carries momentum, and the momentum is felt as pressure after the collision.

It is quantified in two ways. First, as a force spread out over an area, and also the amount of energy divided by its volume (energy density). These concepts are related by how energy and force are related by distance.

Pressure = Force over area

Equation: (P=frac{F}{A})

Justification of the units: (kg:m^{-1}s^{-2}=frac{kg:m:s^{-2}}{m^2})

Pressure = Energy over volume

Equation: (P=frac{E}{V})

Justification of the units: (kg:m^{-1}s^{-2}=frac{kg:m^2:s^{-2}}{m^3})

Force = energy over distance

Equation: (F=frac{E}{d})

Justification of the units: (kg:m:s^{-2}=frac{kg:m^2:s^{-2}}{m})

What is Temperature?

Temperature is an SI base unit. It is measured in Kelvin. It is the manifestation of sub-microscopic kinetic energy of particles.

These particles do not all have the same kinetic energy, and so they don’t have the same speed. Temperature is a measure of the average amount of energy of the particles. Hotter gases have more particles that travel faster, cooler gases have more particles that travel slower.

What is Volume?

Obviously, volume is a three-dimensional space. However, it’s worth bringing up because the ideal gas law has a two-dimensional analogue that relates to concepts in surface tension. So we can replace “V” with “A”, and pressure with “surface pressure”, which is related to surface tension measurements. This is useful in materials chemistry, should you study that later.

Ideal gas law Surface pressure
(pV=nRT) (pi A=nRT)
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Introduction to Thermodynamics

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