Physics Extension Edexcel

The following are examples of using radiation for diagnosis:

X-rays (ionising radiation) : A form of electromagnetic radiation which is used to take images of broken bones and some soft tissues (eg chest X-rays of lung cancer).

CAT scans : CAT stands for Computer Axial Tomography. It generates a 3-D image from a large series of two-dimensional X-ray images.

PET scanners (ionising radiation) : Radiopharmaceuticals (radioactive chemicals) are either injected or swallowed. External detectors called gamma cameras take images of the radiation emitted.

Endoscopes (non-ionising radiation) : An endoscope is a medical device which is used to look inside the body - commonly the reproductive and digestive systems, and the urinary and respiratory tracts. One bundle of fibre optics carries light to the object and another returns an image for the doctor to view.

Ultrasound :  Ultrasound has a frequency higher than human hearing. Some tissues and organs reflect these waves back and the time taken for them to return is used to measure their depth. It is often used to create images of unborn babies, but it is also used to create images of tendons, muscles joints and some internal organs.

The following are examples of using radiation for treatment:

Radiation therapy (ionising radiation) : When ionising radiation is used to treat diseases this is called radiation therapy. This is used to control or kill malignant cancer cells. It damages the DNA of the cells it is exposed to and therefore kills them.

Ultrasound : Ultrasound pulses can be used to break up kidney stones into small pieces that can be passed from the body without needing an operation. It can also be used to treat internal injuries to help them heal faster.

Lasers :  Lasers can be used in eye surgery to correct myopia and hyperopia. The laser is used to remodel the cornea after a flap of corneal tissue is folded back. The flap of tissue is then repositioned and seals naturally. Lasers are also used to remove tattoos, hair and birth marks.

Radiation is any form of energy originating from a source and includes both waves (eg light, sound or gamma rays) and particles (eg alpha and beta radiation).

As radiation spreads out from its source, it does so in all directions in a sphere of increasing size. The intensity of the radiation decreases the further it travels from its source.

Radiation intensity

The intensity of emitted radiation can be calculated by using the following equation:

intensity (W/m2) = power of incident radiation (W) ÷ area (m2)

I = P ÷ A

Calculate the intensity of a 200 W beam of radiation over an area 10 m2

I = 200 ÷ 10

Therefore the intensity = 20 W/m²

Reflection

Light rays reflect from surfaces. They behave like water in a ripple tank. The angle at which a ray hits a surface (the angle of incidence) is always the same as the angle at which it is reflected (the angle of reflection). This is called the law of reflection:

• smooth surfaces, such as mirrors, reflect all rays in parallel lines (specular reflection)

• rough surfaces scatter light rays in different directions (diffuse reflection) - but each ray still reflects at its own angle of incidence

Refraction

When light passes at an angle from one transparent medium to another of a different density it changes direction. This is called refraction and it happens because light travels at different speeds in materials of different densities.

Refraction is often seen when looking at a straw in a glass of water. Note that refraction doesn’t occur when light rays cross into a different medium at 90^o. Here the rays carry on in a straight line.

Waves going from a dense medium to a less dense medium speed up at the boundary. This causes light rays to change direction when they pass from glass to air at an angle other than 90º. This is refraction.

Beyond a certain angle, called the critical angle, all the waves reflect back into the glass. We say that they are totally internally reflected.

All light waves, which hit the surface beyond this critical angle, are effectively trapped. The critical angle for most glass is about 42°.

Optical fibres

An optical fibre is a thin rod of high-quality glass. Very little light is absorbed by the glass. Light getting in at one end undergoes repeated total internal reflection, even when the fibre is bent, and emerges at the other end.

optical fibres are used in endoscopes that allow surgeons to see inside their patients

optical fibres can also carry enormous amounts of information as pulses of light

Snell’s Law is a formula used to describe in detail the relationship between the angles of incidence and refraction when light passes through a boundary of two materials with different densities.In this example the light is travelling from water into air. The equation is:

sin (angle of incidence) ÷ sin (angle of refraction) = refractive index of air ÷ refractive index of water

The refractive index is the extent to which light is refracted when it enters a medium. The greater the index the more refraction occurs. Air has an index of 1.0, water is 1.3 and many type of glass are around 1.5. Snell’s Law can be used to calculate the critical angle (c). At the critical angle (c) the angle of refraction (r) must be 90°. Here the angle of incidence is equal to the critical angle, so:

sin critical angle (c) ÷ sin 90°_(r) = refractive index of air ÷ refractive index of water

sin critical angle (c) ÷ sin 90°_(r) = 1.0 ÷ 1.3

sin 90°_(r) is equal to 1 so:

sin critical angle (c) = 1.0 ÷ 1.3

sin critical angle (c) = 0.77

Therefore the critical angle = 50.4°

Light can be refracted by lenses. A lens is a piece of glass or plastic that has been shaped to refract light in a particular direction. There are two types that you need to know about:

• converging lenses

• diverging lenses                                                                         

Converging lenses

These are thicker in the middle and focus light on a specific point called the focus. When the object being viewed is close to a converging lens, it acts like a magnifying glass. When the object is further away it appears smaller and upside down. The distance from the focus to the middle of the lens is called the focal length. The power of any lens is calculated by this equation:

power of lens (dioptre, D) = 1 ÷ focal length (metre, m)

Diverging lenses

These are thinner in the middle and are used to spread out light as it passes through the lens. To determine the focus for a diverging lens, you trace the path of the light rays that have left the lens backwards to the point at which they converge. The distance from this point to the middle of the lens is the focal length. When an object is placed within the focal length of a diverging lens it is magnified and remains the correct way up.

The lens equation allows us to calculate where an image will occur after passing through a lens. The equation is:

1 ÷ focal length (m) = 1 ÷ object distance (m) + 1 ÷ image distance (m)

1 ÷ f = 1 ÷ u + 1 ÷ v

This can be rearranged to:   1 ÷ image distance = 1 ÷ focal length – 1 ÷ object distance

Worked example - A converging lens has a focal length of 40 cm. An object is placed 50 cm from it. How far away will the image be formed? (Remember to change cm into m.)

• 1 ÷ image distance = 1 ÷ 0.40 m – 1 ÷ 0.50 m

• 1 ÷ image distance = 2.5 – 2

• 1 ÷ image distance = 0.5

This can be rearranged to:            1 = 0.5 x image distance         image distance = 1 ÷ 0.5 = 2 m

This means the image will be formed on the other side of the lens at a distance of two metres. This is called a real image. By convention, if v is positive then the image is a real image. If the focal length is greater than the object distance, the image distance will be a negative number. This is called a virtual image and will be formed on the same side as the object.

Cornea : Front part of the tough outer coat, the sclera. It is convex and transparent : Refracts light - bends it as it enters the eye.

Iris : Pigmented - decides the colour of your eyes - so light cannot pass through. Its muscles contract and relax to alter the size of its central hole or pupil : Controls how much light enters the pupil.

Lens : Transparent, bi-convex, flexible disc behind the iris attached by the suspensory ligaments to the ciliary muscles : Focuses light onto the retina.

Retina : The lining of the back of eye containing two types of photoreceptor cells - rods - sensitive to dim light and black and white - and cones - sensitive to colour. A small area called the fovea in the middle of the retina has many more cones than rods : Contains the light receptors

Optic nerve : Bundle of sensory neurones at back of eye : Carries impulses from the eye to the brain.

Pupil : A small opening at the front of the eye : Allows light to enter the eye and be focussed onto the retina.

Ciliary muscles : A ring of muscle that surrounds the lens : Changes the shape of the lens to focus light on the retina (a process which is called ‘accommodation’)

Component parts of the eye

Short sightedness - Someone with short-sightedness can see near objects clearly, but cannot focus properly on distant objects. This is caused by the eyeball being elongated, so that the distance between the lens and the retina is too great. It can be corrected by placing a concave lens in front of the eye.

Long sightedness - Someone with long-sightedness can see distant objects clearly, but cannot focus properly on near objects. This is because the lens focuses the sharpest image behind the retina, instead of on it. This defect is often age-related, and due to a loss of elasticity in the lens. It is corrected by putting a convex lens in front of the eye.

TreatmentAdvantagesDisadvantages Glasses

• No operation required

• Can be easily changed if eyesight gets worse

• Not a permanent solution

• Initial expense to buy glasses

• Some people don’t like wearing glasses

Contact lenses

• No operation required

• Lightweight

• Can be easily changed if eyesight gets worse

• Almost invisible

• Not a permanent solution

• On-going cost

• Can irritate eyes

Laser eye surgery (need to know this for Higher tier)

• Permanent solution

• No need for glasses or contact lenses

• Expensive procedure

• Slight medical risk

X-rays are one type of electromagnetic radiation. They are commonly used in hospitals to produce photographs of bones which can be checked for breaks or fractures.

X-rays have the following properties:

• they can penetrate less dense matter such as skin and body tissue (but not bone)

• they have a range of frequencies (lower frequency X-rays have less energy and are therefore less penetrating and ionising)

• they leave an image on photographic paper

More dense areas in our bodies, such as bone, allow fewer X-rays through and so appear paler on X-ray images.

Less dense areas, such as skin and many tissues, allow more X-rays though. These appear black on X-ray images.

X-rays give more reliable results than ultrasound when checking for broken bones and so X-rays are used more often to check for breaks. Although X-rays used to take photographs are relatively safe, they can cause cells to become cancerous. Therefore precautions are taken in hospitals to limit the doses received by patients and staff.

The low risk posed by X-rays to a mother and her unborn baby means that ultrasound is used instead of X-rays for pre-natal checks.

Other uses of X-rays

Before the use of gamma rays, a high-energy, concentrated dose of X-rays was used to treat some cancers. X-rays are still used to identify tumours.

X-rays are also used in CAT scans. ‘CAT’ stands for Computer Axial Tomography and generates a three-dimensional image from a large series of two-dimensional X-ray images.

X-rays are also used in fluoroscopy. This allows doctors to see moving images of the internal structures of patients. The patient stands between an X-ray source and a fluorescent screen and a large number of images are taken to form a short film. These are often used to investigate problems with the digestive system.

The inverse square law

The intensity of all electromagnetic radiation, including X-rays, obeys the inverse square law.

This scientific law states that the further the point source of radiation is from the point at which you are measuring it, the lower the dose.

If the distance is doubled, the intensity is reduced by four times (two squared). If the intensity is trebled, the intensity is reduced by nine times (three squared).

An X-ray machine consists of an evacuated tube in which two electrodes are found. The negative electrode (also called the cathode) is a wire filament that emits electrons when heated. This process is called thermionic emission and the wire filament is called an ‘electron gun’.

When there is a large potential difference between the two electrodes, electrons are accelerated from the cathode to the anode (the positive electrode). The evacuated tube means that the electrons do not collide with any other particles. This movement of electrons from the cathode to the anode is equivalent to an electric current.

When the electrons collide with the metal anode, kinetic energy is transferred into thermal energy. At this point, some kinetic energy is also transferred into X-rays.

A higher potential difference between the two electrodes produces X-rays with more energy – these X-rays are more penetrating and more ionising.

An x-ray machine

Kinetic energy

The kinetic energy of an electron can be calculated using the following equation:                                                                 kinetic energy (joule, J) = 0.5 x mass (kilogram, kg) x velocity (metres / second, m/s)²

KE = ½mv²

This equation can also be written as:

kinetic energy (joule, J) = charge on an electron (1.6 x10^-19 coulomb) x potential difference (volts, V)

KE = e x V

Current

The size of the current in an X-ray machine can be calculated using the following equation:                                            current (ampere, A) = number of particles per second (1 / second, 1/s ) x charge on each particle (coulomb, C)

I = N x q

The heart is a major organ in the circulatory system. It is called a ‘double pump’ because blood travels through it twice in one circuit around the body - once to the lungs to receive oxygen and remove carbon dioxide and water, and once again to be pumped around the body.

Our heart beats - or contracts - approximately 70 times a minute. Each contraction is started by a small group of cells that creates a wave of electricity. This spreads across our heart and causes the muscle fibres to contract. An electrical discharge that spreads across any membrane is called an action potential.

Doctors can use an electrocardiogram (ECG) machine to measure the action potentials that spread across the heart muscle. A healthy heart gives the characteristic shape shown in the diagram below for each heartbeat.

The three main electrical potentials are:

• P wave – de-polarisation of the atria (upper heart chambers)

• QRS wave – de-polarisation of the ventricles (lower, larger heart chambers)

• T wave – re-polarisation of the ventricles

Heart problems such as damaged muscles or blockages are easily diagnosed using an ECG.

The frequency of a beating heart can be calculated using this equation:

Frequency (hertz, Hz) = 1 / time period (second, s)

f = 1/T

After exercising, the time period between peaks of an ECG of a healthy person is 500ms.

So f = 1/T

= 1 / 500ms

= 1 / 0.5s = 2Hz (or 120 beats per minute)

Pacemakers :  The small bundle of cells that begins the action potentials that spread across the heart muscle is our ‘natural pacemaker’.

These action potentials are changes in electrical potential across cell membranes and cause the muscle cells in our hearts to contract forcing blood around our bodies.

Some heart problems require people to have artificial pacemakers inserted above their hearts to create these action potentials for them. This is usually because a patient’s heart beats too slowly or there is a block in the heart’s conductive tissue.

Pulse oximetry is a medical way of determining how much oxygen there is in a patient’s blood. There are many occasions when patients wear pulse oximeters, including during intensive care, operations and in recovery.

The device passes two beams of light (a red and an infrared beam) through the patient to a sensor.

The difference in the absorbance of these two beams of light determines the amount of oxygen carried in the blood. This is called percentage oxygen saturation. If this drops below a certain level an alarm sounds, alerting medical staff.

Every atom has the same number of positive protons and negative electrons - meaning it has no overall charge. When the nucleus of an atom possesses either too many or too few neutrons compared to the number of protons, it becomes unstable. These are called radioactive isotopes. Unstable nuclei split up in a process called radioactive decay and emit radioactive radiation.

There are five types of radioactive radiation:

Alpha(α) - Particles made from two protons and two neutrons (description) - Positive (charge) - Low (penetration) - High (ionising abilty)

Beta(β-) -  Electrons - Negative - Medium - Medium

Gamma(γ) - Not particles but electromagnetic waves - None - High - Low

Positron(β+) - Particles with the same mass as electrons - Positive -  Medium (excluding annihilations) - Mediuk

Neutron(n) -  Neutrons - None - High - Not directly

Radiation and living cells

When radiation collides with molecules in living cells it can damage them. If the DNA in the nucleus of a cell is damaged, the cell may become cancerous. The cell then goes out of control, divides rapidly and causes serious health problems. The greater the dose of radiation a cell gets, the greater the chance that the cell will become cancerous. However, very high doses of radiation can kill the cell completely. We use this property of radiation to kill cancer cells, and also harmful bacteria and other microorganisms.                                                                                                                                Alpha, beta and gamma radiation - The degree to which each different type of radiation is most dangerous to the body depends on whether the source is outside or inside the body. If the radioactive source is inside the body (perhaps after being swallowed or breathed in), then:

• alpha radiation is the most dangerous because it is easily absorbed by cells    

• beta and gamma radiation are not as dangerous because they are less likely to be absorbed by a cell and will usually just pass right through it

If the radioactive source is outside the body:

• alpha radiation is not as dangerous because it is unlikely to reach living cells inside the body

• beta and gamma radiation are the most dangerous sources because they can penetrate the skin and damage the cells inside

High-powered gamma rays are used to kill cancer cells inside the body. As the gamma rays strong enough to kill cancer cells would also kill healthy cells around the tumour, several weaker sources are used and arranged so the gamma rays are focused on the tumour. This concentrates the gamma rays on the cells that need to be killed.

Radiotherapy can also be given internally. This means that the source of the radiation is inside the patient. A beta emitter, often iodine-131, is used.

The effects of both external and internal tumour treatment are usually restricted to the area in which the radiation source is applied. Treatments often require repeated doses. A potential side effect is damage to surrounding healthy tissue.

Tracers

A radioactive tracer is used to investigate a patient’s body without the need for invasive surgery. A small amount of radioactive material is put into the patient’s body and a radiographer puts a detector around the body to detect any gamma rays that pass out of the patient’s body.

The source used is usually a gamma emitter and, depending on which part of the body is being investigated, is put into a drink, ingested or injected. The radioactive material is given enough time to move around the body before a radiographer positions a detector - which can produce a picture of the patient’s internal organs - outside the body.

Gamma emitters are used as this type of radiation passes easily through the body and causes little ionisation. The half-life of the tracer should be short enough to minimise the exposure to the patient, but long enough for doctors to see the results.

PET Scans

PET (Positron Emission Tomography) scanning uses radioisotope tracer drugs such as fluorine-18. The tracer is usually injected into the patient’s blood. Gamma rays emitted by the tracer are detected by the PET scanner and multiple images are taken and then analysed by computers.  The half-lives of radioisotopes used in PET scans are very short. This means that they often have to be produced in hospitals or at a nearby location.

Alpha (α) decay

When an alpha particle is emitted from a nucleus the nucleus loses two protons and two neutrons. This means the atomic mass number decreases by 4 and the atomic number decreases by 2. A new element is formed that is two places lower in the Periodic Table than the original element.

Beta (β-) decay

In Beta (β-) decay, a neutron changes into a proton plus an electron. The proton stays in the nucleus and the electron leaves the atom with high energy, and we call it a beta particle.

When a beta particle is emitted from the nucleus the nucleus has one more proton and one less neutron. This means the atomic mass number remains unchanged and the atomic number increases by 1.

Beta (β+) decay

Unlike β- decay, which occurs when radioisotopes have too many neutrons, β+ decay occurs when they have too many protons. In this case, a proton is converted into a neutron and a positive beta particle of β+. This is called a positron and has the same mass as an electron but the opposite charge (positive). The mass number (A) of a radioisotope undergoing β+ decay remains the same but the atomic number (Z) reduces by one. (Z increases by one in β- decay.)

We can determine whether a radioisotope is likely to decay - and what type of decay is likely to occur - by looking at its position on an N-Z curve graph.

This type of graph shows the number of neutrons (N) against the number of protons (Z). The number of neutrons can be calculated by taking the atomic number (Z) from the mass number (A).

The blue dotted line is called the stability line. If a radioisotope does not lie on this line, it is unstable and likely to decay to become stable.

• if a radioisotope lies to the left of the stability line, it has too many neutrons and is likely to undergo β- decay (electrons)

• if a radioisotope lies to the right of the stability line, it has too many protons and is likely to undergo β+ decay (positrons)

If particles have high numbers of protons (usually more than 82), they are likely to undergo alpha decay.

Quarks are tiny particles that make up protons and neutrons. There six types of quarks (known as flavours): up, down, strange, charm, bottom and top. They each have different properties including charges, masses and spins.

Up quarks and down quarks are generally stable and the most common. They are therefore the two most important ones. It is not possible to separate quarks from each other.

Protons are made from two up quarks and one down quark.

Neutrons are made from one up quark and two down quarks.

Up quarks have an electrical charge of +⅔. Down quarks have an electrical charge of -⅓. This explains why protons have a positive charge and neutrons have no charge.

• β- decay involves a down quark changing into an up quark (a neutron becomes a proton and an electron).

• β+ decay involves an up quark changing into an down quark (a proton becomes a neutron and a positron)

Particle accelerators are machines that accelerate charged particles to extremely high energies before smashing them into each other. They are a new, modern technology that is being used to help physicists explain the properties of fundamental particles such as quarks. In previous years and centuries, scientists have used equipment or machines like microscopes and seismometers to explain the physical world - just as particle accelerators are being used now.

 Medical uses

There are over 25,000 particle accelerators worldwide. More than half of them are used for medical purposes. Often this is to produce the radioactive isotopes that are needed for PET scanning and radioactive tracers. Particle accelerators can convert some stable elements into radioactive isotopes by bombarding them with protons. This makes their nuclei unstable and radioactive decay follows.

Centripetal force : is the force that makes an object follow a curved path. This can be bucket of water on a length of string, a charged particle in a cyclotron, or Earth orbiting the Sun.

The equation which explains this is:   force = mass x acceleration   This is Newton’s second law of motion.

Even if the object is moving at a constant speed, because it is moving in a circle it is changing direction and so changing velocity. An example of this would be Earth moving around the Sun. Any change in velocity is acceleration. For any object to move in a circle, there must be a resultant force acting towards the centre of the circle. This is called the centripetal force. For Earth this force is supplied by gravity. For a bucket of water on the end of length of string this force is supplied by the tension of the rope.

A collision is an event in which two moving objects exert forces on each other for a short period of time. We tend to think of this as car crashes or nasty tackles in football games, but even things like a cat’s paws walking though grass cause collisions (between the blades of grass and paws).   In any collision, the total momentum is always the same before and afterwards. Momentum is calculated using the following equation:   momentum = mass x velocity     The fact that momentum is not lost in collisions is called conservation of momentum.

If two cars with momentums of 10,000 kg m/s and -5,000 kg m/s collide, their total momentum before the collision is: 10,000 + -5,000 = 5,000 kg m/s. This must also be their total momentum after the collision.

Note - one momentum needs to be positive and the other negative to show that the cars are travelling in opposite directions.

There are two types of collisions:

• in elastic collisions, momentum is conserved and there is no loss of kinetic energy

• in inelastic collisions, momentum is conserved but some of the kinetic energy is transferred to other forms (commonly thermal or sound energy)

This means that there is a loss of kinetic energy in inelastic collisions. The collisions we see on a daily basis are almost always inelastic. However, collisions of particles are elastic. If a ball is dropped to the ground from a height, an inelastic collision will occur. That is, some of the kinetic energy will be transferred. You may be asked to investigate some of the factors that affect the height of the bounce of a ball.

During β+ decay, a proton is converted into a neutron and a positive beta particle or β+. This is called a positron and has the same mass as an but the opposite charge (positive).

When an electron and a positron collide an annihilation event occurs and gamma rays are produced. This is a frequent occurrence in PET scanning in hospitals.

The positron and electrons have opposite charges and so the overall charge before annihilation is zero. The resulting gamma rays have no charge. So charge is conserved in this collision.

In annihilation, the positron and electron collide head on moving at the same speed. The overall momentum is therefore zero. The resulting gamma rays move in opposite directions with equal and opposite momentum. So momentum is also conserved.

Einstein’s famous equation E = mc² means that the mass of an object is a measure of its energy content, and that mass and energy can be converted into each other (mass energy).

In annihilation, the masses of both the positron and electron are converted into energy (gamma rays). The energy of the gamma rays is the same as the mass energy of the original positron and electron and so mass energy is also conserved. E = mc² can be used to calculate the energy of gamma rays following annihilation:

The most commonly used measurement of temperature is the Celsius scale. The units of this scale are degrees Celsius (^oC).

There is no upper limit, but the lowest temperature possible is -273^oC. Here, almost all matter is solid and the vibrations of particles are very small. The temperature cannot go any lower. The theoretical temperature at which particles would stop moving is called absolute zero.

Some scientists use the Kelvin scale instead. The units are Kelvin (K). This begins at absolute zero and so there are no minus numbers in the Kelvin scale. An increase in one degree Celsius is the same as one Kelvin.

Conversion between the scales is as follows:

• temperature in degrees Celsius = temperature in Kelvin - 273

• temperature in Kelvin = temperature in degrees Celsius + 273

Solid

Liquid

Gas

Arrangement of particles

Close together

Regular pattern

Close together

Random arrangement

Far apart

Random arrangement

Movement of particles

Vibrate on the spot

Move around each other

Move quickly in all directions

The kinetic theory model can be used to explain the properties of solids, liquids and gases by considering the molecules they are made from and their motion.

For gases, the kinetic theory model explains that gas pressure is caused by the collisions between the particles and their container. This is called the outward pressure and is usually greater than normal atmospheric pressure outside the container.

Examples of this can be found in balloons and car tyres. Atmospheric pressure is measured as one bar (or one atmosphere). Maintaining the correct pressure in car tyres is important. Typically this is two to three bar.

• When the temperature is increased, the gas particles move faster and the collisions become harder and more frequent. This means that the pressure also increases.

• When the temperature is decreased, the gas particles move more slowly and the collisions are less hard and less frequent. This means that the pressure also decreases.

The average kinetic energy of the particles in a gas is directly proportional to the Kelvin temperature of the gas.

Charles’ Law

Charles’ Law states that:        volume₁ = volume₂ x temperature₁ / temperature₂           V₁ = V₂ x T₁ / T₂

Where volume₁ and temperature₁ are the initial temperature and volume, and volume₂ and temperature₂ are the final volume and temperature.  Here volume is measured in metres cubed (m³) and temperature in Kelvin (K).

This means that if a gas is heated up and the pressure does not change, the volume will. So, for a fixed mass of gas at a constant pressure, volume / temperature will remain the same.

Boyle’s Law

Boyle’s Law states that:           volume₁ x pressure₁ = volume₂ x pressure₂              V₁P₁ = V₂P₂

Where volume₁ and pressure₁ are the initial volume and pressure, and volume₂ and temperature₂ are the final volume and pressure.  Here volume is measured in metres cubed (m³) and temperature in Kelvin (K).  This means that for a gas at a constant temperature, pressure x volume is also constant. So, increasing pressure from pressure₁ to pressure₂ means that volume₁ will change to volume₂, providing that the temperature remains constant.

A third equation involving gases states that:                                                                                                                         pressure₁ x volume₁ / temperature₁ = pressure₂ x volume₂ / temperature₂               P₁ V₁ / T₁ = P₂ V₂ / T₂

Where pressure₁, volume₁ and temperature₁ are the initial pressure, temperature and volume, and pressure₂, volume₂ and temperature₂ are the final pressure, volume and temperature. Here pressure is measured in pascal (Pa), volume in metres cubed (m³) and temperature in Kelvin (K).  This means that at a constant volume, pressure / temperature remains constant.

Worked example : A 0.5m³ gas cylinder can stand a pressure of 15 x 10⁷ Pa before it explodes. If it is filled with gas to 5 x 10 Pa at 25^oC, what temperature can it stand before exploding?

1. Using the original equation P₁V₁/T₁ = P₂V₂/T₂, this can be rearranged to give us the value for T₂ – the second temperature (which is what we have been asked to calculate).

2. The rearranged equation is T₂ = P₂V₂T₁/P₁V₁

3. Substituting in the values gives us T2 = 15 x10⁷ Pa x 0.5m³ x 298K/ 5 x 10⁷ x 0.5m³

4. Therefore T2 = 894K

This equation is used to fill bottled gases for use in hospitals and scientific research. In these bottles, the pressure is above atmospheric. The gas equation can be used to calculate the volume of gas released from the bottle at atmospheric pressure.