# Interactive textbook: Particle nature of matter

### Atoms and molecules

In mechanics we accounted for the loss of energy when a block is pulled along a rough surface by assuming that the block is made of particles called atoms. Once this assumption was made it was easy to explain where the energy was going; it was increasing the kinetic and potential energies of the atoms by causing them to move faster and change their position.

We also explained how a fuel such as petrol can contain energy because work was done when the atoms were arranged as molecules. This sounds straight forward but some details have been left out. If the atoms have to be pushed together to make a petrol molecule why don't they just fly apart? A collection of particles will always tend to move to a position of lowest potential energy - that's why balls roll down hills. There must be something stopping them from moving apart. This is too difficult to explain with the physics covered so far but we can use an analogy.

Imagine there were two balls on the hill shown below:

Work is required to push the balls up the hill but, due to the crater, the balls will be trapped on the top. They have potential energy but it can't be used to do work unless the balls get out of the crater. This could be achieved by giving them kinetic energy.

When the atoms that make up petrol (carbon, oxygen and hydrogen) get together in the right position their potential energy is lower than when they are slightly apart. This means they will stay together. If the molecule is given kinetic energy and more oxygen is added, the atoms move apart far enough so that they are no longer attracted to each other. They then fly apart, hitting other molecules and causing them to also fall apart.

This is a rather rough-and-ready explanation of what happens when petrol is burnt. If you study chemistry, you will know more about the different amounts of energy stored in different molecules and why changing one molecule into another either requires energy or releases it.

Different materials are made of different molecules that have different shapes. For the purpose of this course we will consider them all to be very small, perfectly elastic spheres. This isn't true but it leads to a model that works well enough. If you get to something that you can't explain remember this simplification.

### States of matter

Matter can exist in three states or phases: solid, liquid and gas. The differences between these states can be explained in terms of particles; solids have particles that are fixed in position, liquids have particles that are able to move around, gases have particles that are completely free to move around.

The way the same particles can behave in different ways can be explained by looking more closely at the force between them.

### Interatomic force

The interatomic force is due to the property of charge. We will deal with that properly in the section on electric fields but, for now, it is enough too know that the force can be attractive and repulsive. If you try to compress a solid the particles repel each other but if you try to stretch it they attract. It is rather like the particles have springs between them.

In the gaseous state, when the particles are far apart, the force is very small. It is as if the springs have broken. If we were able to measure the force between two particles we would get a force vs distance graph like the one below.

When the particles are close together they repel, so the force on the blue ball is to the right (positive). When the particles are pulled apart they attract, so the force is to the left (negative). If left alone the particles will have a separation equal to that when the force is zero - this is called the equilibrium position. If the particles are a long distance apart then the force is very small.

#### Solid

A solid is made of particles that are held together by the interatomic force. We can think of this as being like eggs in an egg box.

• A solid has a fixed shape. This is because the particles have a fixed position due to the interatomic force.
• A solid is difficult to stretch but even more difficult to compress. This is due to the increasing gradient of the force vs distance curve.
• A solid has a fixed volume. This is because the particles will tend to rest at the point where the force is zero.
• Solids obey Hooke's law for small extensions and compressions. For small distances either side of the equilibrium position, the force vs displacement graph is linear.

#### Liquid

A liquid is made of particles that are held together by the interatomic force but are able to move around from one atom to another.

• A liquid has a fixed volume since the equilibrium separation of the particles is constant.
• The shape is not fixed since the particles can move relative to each other.
• Liquids are difficult to compress because the interatomic forces keep the particles apart.

#### Gas

A gas is made of particles that are free to move about, there is no force between the particles except when they collide. This is because the particles are either far apart or moving so fast that the effect of the force is negligible.

• A gas has no fixed volume or shape since the particles are free to move around.
• A gas exerts a force on its container due to the collisions between the particles and the walls.

When heated a solid can change into a liquid.

This is because:

When water changes to a liquid the molecules get closer together.

The ineratomic force does not changes so the potential well doesn't change either.

The atoms of a gas are further apart than the atoms of a liquid.

Not necessarily. If a gas is compressed at high temperature the atoms can be closer together than in a liquid.

#### Alternative explanations

Even if you have never studied physics before this year, it is unlikely that you did not know that matter was made up of atoms and the way they are used to model the states of matter. After all, what is the alternative? We are so used to thinking that everything is made up of atoms that we can't think any other way.

The alternative to discrete particulates is continuous. No gaps; just matter. But how could you explain the three states of matter?

Simple - as soft, medium and hard. What about gas pressure? This is just another property of matter like mass. What we need is a piece of evidence that would be impossible to explain if matter wasn't made of particles...

### Brownian motion

If you view smoke under a microscope, you will see that it is made up of small particles of carbon. These particles aren't atoms - they are about 10, 000 times bigger than an atom - but they are still quite small and even under a microscope look like tiny dots. The interesting thing about smoke particles is their jiggling motion.

Notice how the smoke particles move in random directions. This is Brownian motion. They don't hit each other every time they change direction but instead seem to be pushed about by something else - molecules in the air! These have been hidden in this simulation but they can be revealed...

Brownian motion is not possible to explain in any other way - except perhaps invisible fairies playing volleyball with the smoke.

The fairies can't be seen or detected by anyone so we can never prove that they exist or do any experiment to see if fairies are actually the right size to play volleyball with the smoke. It is important that a scientific theory can be falsified

Particle theory could be falsified if we calculated the momentum of an air molecule and found it did not have enough momentum to move a smoke particle. Pleasingly, it does.

Brownian motion is the random motion of gas atoms

It's the random motion of smoke particles.

A balloon with the same mass as a smoke particle would not exhibit Brownian motion because:

If the air had very low density then the balloon would move about in a random way.

If iron is made of atoms then $$2\text{ kg}$$ of iron must contain twice as many atoms as $$1\text{ kg}$$. Iron is one of 118 elements that exist on Earth.

Elements are substances that are made of the same type of atom. When certain elements are mixed, the atoms join to form molecules and the new substance is called a compound. For example, iron (Fe) and sulphur (S) are elements that can be combined to give iron sulphide (FeS).  To convert all of a mixture of iron and sulphur into iron sulphide, all of the iron atoms must join with a sulphur atom. Mixing 56g of iron with 56 kg of sulphur will result in some iron sulphide with a lot of sulphur left over. This is because 56 g of sulphur contains more atoms, as each atom of sulphur has a smaller mass.

To get all the sulphur atoms to combine with the iron you would need the same number of atoms of each. This happens if you have $$56\text{ g}$$ of iron and $$32\text{ g}$$ of sulphur. We can deduce from this that the ratio of mass of an iron atom to the mass of sulphur atom is $$56\over32$$.

Each element has atoms of different mass - so the same number of atoms of each element will have a different mass.

Avogadro's constant is defined as the number of atoms in $$12\text{ g}$$ of carbon-12. This number is approximately $$6.02\times10^{23}$$. The mass of an Avogadro's number of atoms is proportional to the mass of each atom. For example:

• Hydrogen: $$1\text{ g}$$
• Helium: $$4\text{ g}$$
• Oxygen: $$16\text{ g}$$
• Sulphur: $$32\text{ g}$$
• Iron: $$56\text{ g}$$
• Uranium: $$236\text{ g}$$

#### Mole

A mole contains the same number of atoms as $$12\text{ g}$$ of carbon-12. The masses of elements above are therefore the mass of one mole of these elements.

12 g of carbon 12 contains exactly Avogadro's number of atoms?

Theoretically this is true  since avagadro's constant is defined in terms of 12 g of carbon 12. In practice it would not be possible to obtain exactly 12 g.

16 g of sulphur react with 32 g of oxygen to produce 32 g of a new compound.

The chemical symbol for that compound is

1 mole of O and 1/2 mole S

The relative molecular mass of the compound in the previous question is:

32 g contains 1/2 a mole of molecules.

How many particles does 28 kg of Iron contain?.

1000 x 1/2 mole

### Temperature

Whether we are observing the universe or a container of gas, physicists define quantities that describe the state of a system of bodies and the relationship between them. If we know the state now then we can calculate how it will be in the future. So for a system of balls, if we know their position, mass and velocity we can use the conservation of energy and momentum to calculate their position, mass and velocity at any time in the future (or past).

If we were to touch the balls we would notice another difference between them - some are hot and some are cold. We need to add a new quantity to our model, temperature, but first we need to define a scale.

#### Defining a scale

To define the scale for length we take the distance between the ends of a one metre ruler. It is important that everyone uses the same units so all rulers are the same length (or at least calibrated with the same intervals). This length must be defined by something that never changes.

Originally, the metre was defined as one ten-millionth of the distance from the equator to the North Pole along a great circle which made the Earth's circumference approximately $$40,000\text{ km}$$. But this distance was found to change so a better definition was made based on a quantity we believe to be constant, the speed of light in a vacuum:

The metre is defined as the length of the path travelled by light in a vacuum in 1 299 792 458th of a second.

Measuring length is fairly straightforward. We simply compare an unknown length to a known one. Measuring temperature is not so simple. We could make a simple scale based on touch: very cold - cold - just right - hot - very hot - ouch (!). But this wouldn't be much use when we tried to add it to our model, as we need to give temperature a number.

To do this we use some measurable quantity that changes with temperature, like the length of a column of mercury. It's a bit like measuring distance using the time for a ball with velocity $$1\text{ m s}^{-1}$$ to travel between two points. It's time that is actually measured and, because it is proportional to distance, can be used instead.

#### Fixed points

To measure temperature we need some physical property that varies with temperature and two temperatures that are fixed. Temperature itself isn't visible, so we need to establish two observable events that always take place at the same temperature.

We use the freezing and boiling points of pure water at normal atmospheric pressure. How did we know these events always happened at the same temperature before we had a thermometer? Put a tube of mercury into freezing water many times and check it's always the same length

#### Celsius scale

Once we have marked our two fixed points we define the size of a unit by splitting the length into a convenient number of degrees. The Celsius scale defines the upper fixed point as $$100\text{ }℃$$ and the lower as $$0\text{ }℃$$ so the length is divided into 100 degrees:

By fixing the scale in this way we have made the length of the mercury column proportional to temperature.

What if it isn't? Well it is now!

We have defined temperature in such a way that, by definition, the length is proportional to temperature. Anyone doing an experiment where they measured the length of a column of mercury at different temperatures would find that it was proportional to temperature. However if they used a different way of measuring temperature, for example with an electrical resistance thermometer, they would probably find that it wasn't. It is important therefore to state what sort of thermometer is being used. All thermometers will agree at the fixed points but may not agree for any other temperature.

#### Absolute zero and the Kelvin scale

Zero degrees Celsius is not the lowest possible temperature; it's just some arbitrary point based on water.

If we extended the graph backwards we would get to a point where the length of mercury column was zero, this is at $$-273.15\text{ }℃$$. This doesn't make any practical sense as the mercury can't simply disappear but it does give an indication of the existence of a lowest possible temperature.

In the next section we will find that temperature is related to the kinetic energy of atoms and that a high temperature means high kinetic energy. The lowest possible temperature is therefore the point at which the atoms stop moving. The Kelvin scale takes this as the zero: absolute zero. The upper fixed point is the triple point of water, this is the temperature at which water can be solid, liquid and gas all at the same time (and only at one possible pressure). This happens at a more precisely defined temperature than freezing ($$0.01\text{ }℃$$). For convenience this is taken as $$273.16\text{ K}$$ which makes the intervals of temperature the same in both scales.

$$0\text{ }℃ = 273.15\text{ K}$$

To convert from $$℃$$ to $$\text{K}$$ just add $$273.15$$. Temperature changes do not need to be converted.

The length of a column of mercury is proportional to temperature.

Thermometers based on different physical properties only agree at the fixed points.

### Kinetic energy

When dealing with work done against friction, we said that the energy transferred goes to the internal energy of the body (increasing the potential and kinetic energies of the atoms). If we measured the temperature of the body we would notice an increase. There seems to be a connection between internal energy and temperature.

When we say something is hot we mean it is hotter than we are. If we touch it we will get hotter and it will get colder. If temperature is connected to energy, it means that energy has been transferred from the hot object to the cold.

The only way energy can be transferred from one particle to another is if they hit each other. Kinetic energy is transferred from the faster to the slower particle during the collision.

We can explain what happens when energy is transferred from a hot body to a cold one if we assume temperature is related to the kinetic energy of the atoms.

Here you can see how the faster moving red particles transfer energy to the lower energy blue ones until it is evenly spread.

The temperature in Kelvin is proportional to the average kinetic energy of the atoms.

Temperature is not related to change in the PE of the atoms because:

Temperature is all about direction of energy transfer, if no energy is transferred the temp. is equal. If the red balls were further apart than the blue but not moving, no energy would be transferred.

### Heat

We know that energy can be transferred by doing work but it is also transferred when a hot body is in contact with a cold one. We call this transfer of energy heat.

When a hot body is in thermal contact with a cold one, heat is transferred from the hot body to the cold one. This increases the internal energy of the cold one because bodies contain internal energy (the sum of potential and kinetic energy of the atoms) and because temperature is related to the average kinetic energy of the atoms.

Bodies do not contain heat; they contain internal energy. Heat is the transfer of energy.

### Thermal equilibrium

Bodies in contact are said to be in thermal equilibrium if no heat is transferred between them, this means they are at the same temperature.

Which statement is true about 1kg of water at 0℃ and 1 kg of ice at 0℃

Same temp means same average KE but to turn water into ice heat must be added.

Which statement is true about 1 litre of water at 20 ℃ and 2 litres of water at 20℃

Same temp but twice as much internal energy.

Which statement is true about 1 litres of water at 20 ℃ and 1 litre of water at 40℃

av. KE is proportional to temp. in Kelvin not Celsius.

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