Interactive textbook: Gas laws

The simplest state

In some ways the gaseous state is the most complex. All the particles are moving about in random motion unlike the particles of a solid that have fixed position. However, in terms of energy, gases are the simplest since they only have kinetic energy.

This may seem strange since, when a liquid changes to a gas at constant temperature, the heat added goes to increase potential energy (and not kinetic energy)! Remember that potential energy is the work done moving a body to a certain place from a position of zero potential energy. If we take two atoms an infinite distance apart and bring them close enough together to mimic a solid, we will not have to push them together since they will be attracted by the interatomic force. We would actually have to hold the atoms back to prevent them from gaining kinetic energy. The force we apply would be in the opposite direction to motion so work done is negative. The potential energy is also negative.

The situation is rather like a ball in a hole in the ground. The ball would have zero potential energy on the ground so has negative potential energy in the hole. To get the ball out of the hole, work would have to be done. The work done will increase the potential energy of the ball from some negative value to zero.  The "hole in the ground" is called a potential well. It is very useful for visualising the way a body will react as it is moved about.

This diagram represents the potential well close to an atom. It looks similar to the force vs distance graph but it is actually the area under the force vs distance graph. The equilibrium position of the atom is at the bottom of the well, the position where the force is zero on the force vs distance graph.

With gases, we don't have to worry about potential wells and the work done to transport the atoms from infinity.

 

The zero in potential is at:

The potential tends to zero at infinity

 

Kinetic model

We have already stated that a gas is made of a large number of small perfectly elastic spheres moving in random motion but let's be more specific.

Large number

What is a large number? \(24 \text{ litres}\) of air at normal room temperature and pressure contains about \(6 \times 10^{23}\) molecules. That's a very big number. If we make a simulation using Algodoo we can't use so many balls, but how many would be enough so that the balls behave like a gas? Enough so that the particles exert a constant force on the walls of the container. This is difficult to achieve because they also must have enough space to move around. When modelling a gas we combine mechanics with statistics (statistical mechanics). Statistics only work when you have a large amount of data, so we need enough particles to make the use of statistics reasonable.

This example does not have enough particles.

Small

An atom has a diameter of about \(10^{-10}\text{ m}\) so has a volume of the order of \(10^{-30}\text{ m}^3\). There are approximately \(6 \times 10^{23}\) molecules in \(24 \times 10^{-3}\text{ m}^3 \), which means a volume of \(4 \times 10^{-26}\text{ m}^3\) per molecule. So the volume of the molecules is about \(1\over10, 000\)th of the volume of the gas.

Perfectly elastic

In perfectly elastic collisions, no kinetic energy is lost when molecules collide. If it was, the temperature of the gas would continually decrease - and we know this doesn't happen. Using the simulation above we can deduce that the average distance travelled between collisions is about 10 times the size of the molecule. This is less than for a real gas where it is more like 100 times but remember this is a 2-dimensional model.

From this we can also deduce that the time colliding is negligible compared to the time between collisions.

Spheres

There is no point in making life difficult for ourselves. Collisions of spheres are easier to deal with than cubes.

There is this dairy with cows and everything. The dairy farmer wants to increase her production of milk. To do this, she hires three consultants - an engineer, a psychologist, and a physicist.

After a week, the engineer comes back with a report. He said: "If you want to increase milk production, you need to get bigger milk pumps and bigger tubes to suck the milk through."

Next came the psychologist. He said: "You need to make the cows produce more milk. One way to do this is to make them calm and happy. Happy cows produce happy milk. Paint the milking stalls green. This will make the cows think of grass and happy fields. They will be happy."

Finally, the physicist came to present her ideas. She said: "Assume the cow is a sphere...."

Real molecules are not spheres so our model will have its limitations. But this assumption works fine for monatomic gases.

Random motion

The fact that the motion is random means that the sum of the components of momentum in any given direction will be the same so the force exerted per unit area on each wall is the same. Due to all the collisions the particles will have different speeds, and we can represent the spread of speeds on a bar chart.

Each bar represents a range of velocity and the height of the bar is how many atoms have this range of velocity. Air is made of many different gases but is mainly nitrogen. All of the gasses have the same temperature so their average kinetic energy (\(E_\text{k}={1\over2}mv^2\)) is the same. The heavier molecules will therefore have lower velocity than the lighter ones. The peak of the velocity distribution chart for nitrogen at room temperature is around \(600\text{ m s}^{-1}\) (much faster than our animation where the particles have speed in the region of \(1\text{ m s}^{-1}\)).

No forces

And another assumption! When deriving relationships between the different properties of the gas molecules it is simplest if we assume that the velocity of the particles is constant between collisions. This means that there is no force acting on them. Given the separation of the molecules, this seems reasonable. But if we were to compress the gas, the molecules would be closer together and exert more intermolecular forces so it would no longer be true, so our model only works for low pressure gases.

Ideal gas

The factors discussed here are the assumptions of the kinetic model of an ideal gas. Using this model we can explain the behaviour of a gas and make predictions as to the result of changing any of its properties. A gas that behaves in the way predicted by the model is called an ideal gas.

We have been looking at the microscopic properties of the particles that make up an ideal gas: velocity, kinetic energy and momentum of the particles. These give rise to the macroscopic properties of the gas: pressure, volume and temperature. Let's see how these are connected.

Volume

The volume of a gas is the same as the volume of its container. If the volume of the container increases, the volume of the gas increases. Here we will imagine that the container is a cylinder with the gas molecules held by a moveable piston (so that volume is easily changed).

Here you can see how the gas fills the container as its volume is increased.

Temperature

The temperature of a gas in Kelvin (absolute temperature) is proportional to the average kinetic energy of its particles. Temperature can be increased either by transferring heat or doing work. Work can be done on the gas by moving the piston.

Here you can see the piston hitting the particles like a tennis racket hitting a ball:

\(\bar{E_\text{k}}\propto T\)

\(\bar{E_\text{k}}= {3\over2}kT\)

  • \(\bar{E_\text{k}}\) is mean kinetic energy in \(\text{J}\)
  • \(k\) is the Boltzmann constant, \(1.38 \times 10^{−23}\text{ J K}^{−1}\)
  • \(T\) is absolute temperature in \(\text{K}\)

If there are \(N\) molecules then the total kinetic energy of the gas is \(​​​​N\times\bar{E_\text{k}}={3\over2}NkT\).

An alternative expression is \(E_\text{k} = {3\over2}nRT\) where \(R\) is the molar gas constant, and \(n\) is the number of moles.

If we compare two gases at the same temperature:

We can see that the particles with larger mass have lower speed. Since temperature is the same we can equate the two equations for kinetic energy:

\({1\over2}m_1{v_1}^2={1\over2}m_2{v_2}^2\)

\({v_1\over v_2}^2={m_2\over m_1}\)

\(v\propto {1\over \sqrt{m}}\)

If the mass is four times bigger the speed will be halved.

 

Gas A has atoms of mass m travelling with average velocity v. Gas B has atoms of mass 2m travelling at average velocity v/2.

The ratio av. KEA/av. KEB =

1/2mv2/1/2 x 2m/(v/2)2 = 2


 

Pressure

The pressure of a gas is the force per unit area exerted by the gas particles on the walls of the container. This is due to the particles hitting the walls. When a particle hits a wall it experiences a change in momentum so has experienced a force. This force was exerted by the wall so the wall must experience an equal and opposite force:

\(P={F\over A}\)

  • \(P\) is pressure in \(\text{N m}^{-2}\) or \(\text{Pa}\)
  • \(F\) is force in \(\text{N}\)
  • \(A\) is area in \(\text{m}^2\)

The size of the force is equal to the rate of change in momentum of the colliding particles. This depends on the change in velocity and the number of collisions per second:

\(F=\Delta v{\Delta m\over\Delta t}\)

By applying Newton's laws to one particle then averaging the result to take into account the large number it can be shown that:

\(P={1\over3}\rho\bar{c}^2\)

  • \(\rho\) is density in \(\text{kg m}^{-3}\)
  • \(\bar{c}\) is the average speed of the particles

From this we can deduce that both pressure and temperature of a fixed mass of gas with a constant density are proportional to the square of the speed. In other words, they are proportional to each other. Here we can see that the force required to keep the piston in place increases when the temperature increases.

The density of a sample of gas can be increased by reducing the volume. From this we can deduce that if the temperature is constant the pressure is inversely proportional to the volume. Here the volume increases and the pressure decreases:

 

 

Cylinder A contains atoms with av. velocity v and has a piston of area A. Cylinder B contains atoms with av. vel 2v and has a piston of area A/2.

What is the ratio Force on piston A/Force on piston B?

F = PA

ratio = 1/3ρv2A / 1/3ρ(2v2)A/2


 

Gas laws

The gas laws are a set of three relationships that model the relationship between the pressure, volume and temperature of a fixed mass of gas. To relate one of these with another, the third quantity must be kept constant (and hence why we have three!):

  • To find how pressure is affected by volume (or vice versa) we keep temperature constant
  • To find how pressure is affected by temperature (or vice versa) we keep volume constant
  • To find how temperature is affected by volume (or vice versa) we keep pressure constant

Boyle's law

Boyle's law states that the pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume:

\(P={1\over V}\)

\({P_1\over P_2}={V_2\over V_1}\)

One way to remember which one is Boyle's law is that 'Boyle' sounds like 'boil' and, during boiling, temperature doesn't change.

To verify that a gas obeys Boyle's Law you can measure the gas pressure in a syringe as the volume is reduced. This must be done slowly to keep the temperature constant.

If pressure is inversely proportional to volume then a graph of pressure vs volume will be a curve. This curve is called an isotherm as it represents a constant temperature. The graph below shows an isotherm for a sample of gas. By adjusting the slider you can see the curve for different temperatures:

 

When the volume of a gas is changed the temperature remains constant.

When a gas is compressed work is done so energy is transferred which inceases the temperature of the gas. The temperature only remains constant if heat can be lost to the surroundings.

 

Pressure law

The pressure law states that the pressure of a fixed mass of gas with constant volume is directly proportional to its absolute temperature:

\(P\propto T\)

\({P_1\over P_2}={T_1\over T_2}\)

The PASCO absolute zero sphere enables you to measure the pressure and temperature of a fixed volume of gas.

Since pressure is directly proportional to temperature, a graph of pressure vs temperature will be linear. The graph below shows an example of the sort of line you might achieve. Try changing the volume to see the effect on the line (given that you can't in the experiment!):

 

In the experiment above the volume is slightly bigger than the volume of the sphere.

This will cause:

P = nRT/V

Gradient = nR/V V is larger but still constant.

 

Charles' law

Charles' law states that the volume of a fixed mass of gas at a constant pressure is directly proportional to its absolute temperature:

\(V\propto T\)

\({V_1\over V_2}={T_1\over T_2}\)

To verify Charles' law you would have to measure the volume of a fixed mass of gas at different temperatures. This would require a piston that moved very easily. An ordinary syringe is no use as there is far to much friction but PASCO make a very good one.

Note that we are not sponsored by PASCO!

The temperature of the air in the metal can is changed by placing it in a water bath. As the water is heated, the gas expands pushing the piston up. The pressure of the gas equals atmospheric pressure plus the pressure due to the weight of the piston. To simplify matters the piston can be put on its side so the weight of the piston does not add to the pressure due to the atmosphere.

Since volume is proportional to temperature a graph of volume vs temperature will be linear. Use the slider to vary the pressure and note what effect this has on the gradient of the graph below:

 

The experiment above is carried out with the piston vertical and horizontal. The vertical experiment will have:

V = nRT/P

Vertical exp. will have greater P so smaller gradient


 

Ideal gas equation

The three laws can be represented by one equation

\(PV=nRT\)

  • \(n\) is the number of moles
  • \(R\) is the molar gas constant, \(8.31\text{ J mol}^{-1}\text{ K}^{-1} \)

Sometimes it can be more convenient in changing situations to use:

\({P_1 V_1\over n_1T_1}={P_2 V_2\over n_2T_2}\)

To represent this relationship on a graph we would need three axis for pressure, volume and temperature.

This surface represents all the possible states of a gas:

This is rather difficult to draw so we use the pressure vs volume graph. Different temperatures are represented by different isotherms. The isotherms here are plotted every \(500\text{ K}\).

All transformations of the gas can be represented on this graph. It is used extensively in the study of thermodynamics (Engineering) where the way that energy is distributed in thermal systems is considered in more detail.

Deviations from the ideal gas model

The ideal gas model is based on certain assumptions:

  • The volume of the particles are small compared to the volume of the gas.
  • No forces between particles.
  • Particles are perfectly elastic spheres.

These assumptions are true for low density monatomic gasses at high temperature but break down at high density and low temperature because:

  • the gas could change to a liquid or solid
  • the particles are so close that we can no longer ignore the interatomic force
  • the particles are no longer free to move about

The model would also break down for very small amounts of gas at low temperatures when the collisions between particles and the walls of the container and are too infrequent and we can no longer use statistics to average out the force.

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