Ideal Gases and the Equation of State

Review the motion of gas particles

Our model of gas molecules is fast moving particles (~ 400m/s in air at STP), bouncing off the walls of the container and each other in perfectly elastic collisions.

Evidence for this is Brownian motion.

Recap the ideal gas laws: Boyle's, Charles' and Pressure Laws.

• Boyle's Law: pV = constant.

• Charles' Law: V/T = constant

• Pressure Law: p/T = constant

Exercise:

What properties of a gas depend on temperature? Use the heat/ molecular motion applet under this link to find out and explain:

Pressure- What causes it? How is it affected by temperature? Why?

Number of molecules of gas- How does this affect pressure? Why?

Important note: This is where our concept of absolute temperature comes from.

In order to exert pressure, gas particle must be moving, and thus have a kinetic energy. The average kinetic energy of a particle of a substance defines that substance's temperature. Thus when the particle's kinetic energy drops to zero (absolute zero), it can no longer exert a pressure on its container walls.

The dependance of the behaviour of a gas is obviously linked to how much gas there is. Measuring the mass of a gas is not easy (or practical given the significant buoyancy force on a light container of a gas). Volume is not a reliable measure of how much gas there is either, since it will change with termperature and pressure.

So we define the mole:

A mole of gas is 6.02 x 10²³molecules - this is Avogadro's number.

Avogadro defined the mole as the number of molecules in at standard temperature (273K/ 0^oC) and pressure (100,000Pa/ 1atm) one mole of any gas occupies 0.0224m³.

Question:

What is molar mass? Why do you think this is important for use with gases, but not for solids and less so for liquids?

The Equation of State:

To combine these 3 equations that describe gases gives us the following equation...

p₁ x V₁ /T₁ = p₂ x V₂ /T₂

p x V / T = constant

pV/T = nR

pV = nRT

You may already know this as the ideal gas equation, because...

For these laws to apply the gas must be ideal, which means making a few assumptions:

1. Gases consist of large numbers of molecules (or atoms, in the case of the noble gases) that are in continuous, random motion

2. The volume of all the molecules of the gas is negligible compared to the total volume in which the gas is contained

3. Attractive and repulsive forces between gas molecules is negligible

4. The average kinetic energy of the molecules does not change with time (as long as the temperature of the gas remains constant). Energy can be transferred between molecules during collisions (but the collisions are perfectly elastic)

5. The average kinetic energy of the molecules is proportional to absolute temperature. At any given temperature, the molecules of all gases have the same average kinetic energy. In other words, if I have two gas samples, both at the same temperature, then the average kinetic energy for the collection of gas molecules in one sample is equal to the average kinetic energy for the collection of gas molecules in the other sample.

Task:

Questions on the equation of state of an ideal gas (teaching questions).

Kirk and Hodgeson pp116-117

Back to topic 10

The First Law of Thermodynamics

Recall our definitions of heat, temperature and internal energy.We need a good handle on this to understand what comes next.

Questions:

Why is an ideal gas ideal?

What implications does this have for internal energy when we consider a gas?

Although the ideal gas model is not "ideal", it does begin to break down at very low temperatures, it will do us for a year or two more!

Since one of our assumptions of an ideal gas is that intermolecular forces are negligible, we can neglect potential energy in our calculations of internal energy.

Thus...

U_internal=S (¹/₂mv²)

A molecule in an ideal gas can gain kinetic energy by compression or heating.

Therefore the change in internal energy of a gas = heat supplied to the gas + work done on the gas.

This is the 1st Law of Thermodynamics...

The internal energy of a system depends only on it's state; any increase in the internal energy of a system is the sum of the heat supplied to the system and the work done on the system.

Important note: the + sign hold because work is done on the system, thus energy enters the system. In many cases the 1st law is quoted with a - sign and refers to work done by the system, in which case energy is leaving the system.

You can see that to fully appreciate this statement we must define the difference between a system and its surroundings, and distinguish between work done and heat supplied.

Applying this idea to volume changes in an ideal gas at constant pressure...

If the volume of a gas increases, then work is done by the gas in expanding.

W = F x

So for a gas in a cylinder expanding by a small distance, dx...

dW = F dx

and F = p A

dW = p A dx

and A dx = dV

dW = p dV

integrating this experssion for a change in volume from V₁ to V₂ ...

In this case the gas is expanding, so work is being done by the system.

So the term W = p (V₂ -V₁) must be negative thus the internal energy of the gas decreases.

Questions:

Topic 10 teaching questions 13-15

Hutchings pp446-468, Kirk and Hodgeson pp122

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Thermodynamic Processes

We derived an experssion for the expansion of an ideal gas at constant pressure. This is an isobaric process.

W = p (V_f -V_i)

It can be seen the work done is equal to the area under the p-V graph.

This principle holds for non-isobaric situations...

Isometric Processes

This is a change that happens at constant volume...

The work done in an isometric process is zero. Therefore any change in internal energy of a gas in an isometric change is wholy due to heat exchange between a system and it's surroundings.

DU = Q

Isothermal Processes

This is a change that happens at constant temperature

In this case neither pressure or volume are constant.

We can use the ideal gas equation to find a value of p with respect to V at any time...

Now we need to integrate with respect to V to determine the area under the graph...

NB: We refer to the line on a pV graph that describes a change at constant temperature as an 'isotherm'.

In order for the temperature to remain constant so must the internal energy of the system. Therefore...

Questions:

Hutchings 26.1 and 26.2 pp450

Adiabatic Processes

These are defined as processes where no heat enters or leaves the system (ie: DQ = 0).

Therefore the 1st law statement for an adiabatic process is...

This is difficult to achieve practically. it requires that the system is well insulated, and the process is carried out quickly.

The combustion stroke of an internal combustion engine is approximately an adiabatic process.

Questions:

Topic 10 teaching questions 15-21

Kirk and Hodgson pp125

A note on calculations.

You will be asked to determine heats supplied and lost be gases.

You will recall that this can be done by , where m is mass, and c is specific heat capacity.

We can also do this in different conditions by using molar heat capacity, C_m.

In this equation n, refers to the number of moles of gas, and v, suggests molar heat capacity at constant volume (isometric).

C_p,m, refers to molar heat capacity at constant pressure.

Hutchings pp446-468, Kirk and Hodgeson pp122-123

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2nd Law of Thermodynamics

Heat Engines and Heat Pumps

You are already familiar with heat engines and pumps, such as the internal combustion engine or the refrigerator. A heat engine converts thermal heat energy into mechanical work, and a heat pump visa versa.

To make the best possible heat engine one must maximize efficiency.

We represent such changes in a Sankey diagram .

You can see that the energy supplied to the heat engine is in the form of heat from the hot source, Q_H, and the heat lost to the cold surroundings is Q_C.

Thus the work done by the heat engine is the difference between the two, W = Q_H -Q_C.

The efficiency of a heat engine is thus,

We can reduce this equation to a form that is easier to deal with from the absolute temperature scale...

Q = mcT

Therefore efficiency becomes,

You can see that by the zeroth law of thermodynamics, an efficiency of 1 can never be achieved.

A heat pump works basically in the opposite direction.

This time you can see that work is done in order to make heat flow in the opposite direction.

The second law of thermodynamics gives us another way of seeing this, but is more fundamental, and wide reaching. It deals with an idea called entropy. The easiest way to think of entropy is as a measure of disorder.

The Second Law

In any process, the entropy of the universe must either stay the same or increase

This is equivilent to saying that a system always tends to move towards a state of greater disorder. Analogising this with the break in a game of pool...

This statement has massive implications. It means that we could never construct a 100% efficient heat engine.

If this were possible then we could take a gas at some temerature T_H, and convert all it's internal energy to work.

Question:

If this were possible, what would be the final temperature of the gas?

0K. But since the ambient temperature of deep space is 2.7K at the moment, this is impossible because to do. Anywhere in the universe we would have to do work to remove energy from particles.

If we have to do work to a machine, then it can't be 100% efficient.

We can quantify a change in entopy, DS, in terms of the heat energy supplied to a system, and the absolute temperature of the system, using the relationship,

Questions:

Complete the questions from the topic 10 question sheet.

Complete questions from Kirk and Hodgson pp124-126

Hutchings pp455-457, Kirk and Hodgson pp125.

Back to topic 10

anrophysics 2007