- INTRODUCTION
- IDEAL GAS TEMPERATURE SCALE
- THE IDEAL GAS EQUATION
- THE HISTORICAL GAS LAWS
- KINETIC THEORY OF MATTER
- KINETIC THEORY OF AN IDEAL GAS
- GRAHAM'S LAW AND AVOGADRO’S HYPOTHESIS

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

This law is obeyed by real gases at low pressures.

*An ideal gas is an imaginary gas that obeys the law always*

For a real gas a low pressure, we get a straight line. For an
ideal gas this straight line can be extended till it meets the axis.

The temperature at which the line cuts the axis is called
*absolute zero* = - 273.15^{0}C.

**IDEAL GAS
TEMPERATURE SCALE **(return
to start of page)

Kelvin suggested that an ideal gas temperature scale could be based on using the product PV for an ideal gas as the thermometric property. He suggested that a temperature T on this scale be given by:

**Fixed points on the ideal gas temperature scale**

- The
*lower fixed point is absolute zero*, defined as zero kelvin (0K) - The
*upper fixed point is the triple point of water*, which is the only temperature at which ice, water and water vapour co-exist in equilibrium

K (not ^{0}K) is the symbol for the unit of temperature
on the ideal gas scale, the kelvin.

It would be possible to define the triple point in kelvins to
be any desired value, but defining it as above means that a temperature
*difference* of 1K equals a temperature *difference* of
1^{0}C, making it is easy to change between K and ^{0}C:

Modifying the above graph, we get:

**Measuring a temperature on the ideal gas scale**

We have,

Thus, we just
need to determine the pressure of an ideal gas at the triple point and at the
unknown temperature T to then determine the unknown temperature using the above
equation.

The following set-up can be used to establish the triple point:

Clearly, the bulb must contain a real gas and not an ideal gas. However, we can make use of the fact that a real gas at lower and lower pressure acts more and more like an ideal gas. Thus, we:- measure P and P
_{tr} - determine 273.16 P/P
_{tr} - extract some gas from the bulb, and redetermine 273.16
P/P
_{tr} - extract some more gas from the bulb, and redetermine 273.16
P/P
_{tr}and so on - the limiting value of (273.16 P/P
_{tr}) is the value of T on the ideal gas scale

Note: The kelvin (K) is the SI unit of temperature, and the
Kelvin temperature scale is also called the *absolute thermodynamic
temperature scale*

**THE IDEAL GAS
EQUATION **(return
to start of page)

A couple of preliminary definitions:

Now, we have seen that: PV = a constant * T

By experiment it is found that for one mole of any gas at low
pressure, the constant has the same value, denoted by R, called the *universal
molar gas constant*.

A real gas obeys this law well at low pressures, when the
molecules are relatively far apart, and do not interfere strongly with each
other. *An ideal or perfect gas is an imaginary gas that obeys it under all
conditions*.

**Example**

What volume is occupied by one mole of an ideal gas at
STP?

Note: ‘STP’ º ‘Standard
Temperature and Pressure’ º 0^{0}C and 76cm Hg
(=1.013 * 10^{5} Pa)

R = 8.31 J mol^{-1} K^{-1}

**Equation for a fixed mass of gas**

The ideal gas equation can be expressed as:

Note: When applying the above equation:

- Temperatures must be in kelvins (kelvins =
^{ 0}C +273) - The pressures and volumes can be in any units, as long as they are the same on both sides of the equation

A cycle pump has its exit sealed. It contains 50cm^{3}
of a gas at 20^{0}C and a pressure of 1.0*10^{5 }Pa . Find the
new pressure when the gas is compressed to 10cm^{3} and the temperature
rises to 25^{0}C.

**THE HISTORICAL GAS
LAWS **(return
to start of page)

Boyle’s law (1660), Charles’ law (1787) and the pressure law (Amontons, 1702) can all be inferred from the ideal gas equation: PV = nRT

Each law applies to a fixed mass of trapped gas, so the number of molecules, and therefore the number of moles, n, is constant.

**Boyle’s Law**

If T is constant, nRT is constant, so PV = a constant, or P = a
constant/V, so *P is proportional to 1/V*.

P
against 1/V is a straight line for a fixed mass of gas at low pressure. If we
plot P against V we get a graph like:

If P is constant, V = (nR/P)*T = a constant * T, so *V is
proportional to T*.

**Pressure Law**

If V is constant, P = (nR/V)*T = a constant * T, so *P is
proportional to T*.

**KINETIC THEORY OF
MATTER **(return
to start of page)

**Solids, liquids and gases**

The kinetic theory of matter asserts that matter is made up of particles (atoms, molecules) that are in constant motion. We can interpret the solid, liquid and gas state in terms of the kinetic theory:

*A solid has a fixed volume and shape*. Its atoms are close together, and cannot move around - their motion consists of continuous vibrations*A liquid has a fixed volume, but no fixed shape*- it takes on the shape of any container it is placed in. Its atoms are also close together, but they can move around inside the liquid*A gas has no fixed volume and no fixed shape - it fills all the space available to it*. Its atoms are far apart and move around freely - they collide with each other and with the walls of the container, if enclosed

Since the atoms/molecules in a substance are constantly moving,
they possess kinetic energy. *It is believed that temperature is a measure of
molecular kinetic energy.* If a substance is cooled, then the molecules move
slower, and so lose kinetic energy

Looking at the previous volume-temperature graph, we can see
that, going down in temperature, the volume of the gas gets smaller and smaller.
At –273^{0}C the volume is zero, which implies that the gas has
disappeared! In reality this does not happen, because at some point the gas will
change to a liquid, and is then almost incompressible. However, the liquid can
still be cooled, and eventually will become solid. In the solid, the molecules
do not move around, but they do constantly vibrate, so they have vibrational KE.
Absolute zero (-273^{0}C, OK) is the
temperature at which, it is believed, that all molecular motion stops.
Researchers have got close to absolute zero, but have not quite reached
it.

**Brownian motion and diffusion**

Though the kinetic theory can help explain some observations, we do not have
*direct* proof that atoms and molecules actually exist, since they are too
small to see directly. However,

*Brownian motion gives indirect evidence for the existence of atoms and molecules*

- Some smoke is introduced into a transparent container, containing air
- The smoke particles appear as spots of light when illuminated and viewed through a microscope
- The smoke particles are seen to move in a very erratic, irregular, ‘jiggly’ manner
- Their motion is believed to be due to being repeatedly hit by (invisible) molecules of the air

*Diffusion is also taken as indirect evidence for the existence of atoms and molecules*

Bromine is a dark red volatile liquid (volatile means that it vaporises easily at room temperature). A capsule of bromine is broken by squeezing the rubber tubing in the following set-up:

After about 10 to 15 minutes, the bromine vapour has filled the container, showing that the bromine molecules have worked their way between the air molecules, i.e. diffused, from the bottom to the top.**KINETIC
THEORY OF AN IDEAL GAS **(return
to start of page)

It is believed that the pressure of a gas is due to molecules
in the gas repeatedly striking surfaces.

Here we want
to derive an expression for the pressure of an ideal gas in terms of the motion
of its molecules.

**Assumptions of the theory**

- The forces between molecules are negligible except during collisions
- The duration of collisions is negligible compared to the time between collisions
- Between collisions molecules move with uniform velocity
- The volume of the molecules themselves is negligible compared to the volume of the container

Suppose that a cubical box of sides L contains N molecules of a gas, each of mass m. Consider one molecule with velocity c, which can be resolved along Ox, Oy and Oz, the components being denoted by u, v and w.

First consider motion in the Ox direction. The molecule strikes side X and rebounds:

If the components of velocity in the same direction of *all*
molecules are u_{1}, u_{2}, u_{3}, ……,
u_{N}, then:

Consider now the single molecule and the components u, v, w of its
velocity c:

All the triangles are right-angled triangles. Using Pythagoras’ Theorem:

(this is Pythagoras’ Theorem in three dimensions)

This applies to all the molecules, so if we denote the speeds
by c_{1}, c_{2}, c_{3}, ……, c_{N}, then:

Now, since the molecules do not move in any preferred direction, we can
infer that:

Thus, we have three equivalent equations for the pressure of an ideal gas.

**Root mean square speed**

**Example**

Calculate (a) <c> and (b) c_{rms }for molecules
with speeds: 1, 2, 3, 4, 5 ms^{-1}

**Average KE per molecule**

**Temperature and kinetic theory**

For one mole (i.e. N_{A
}molecules) of an ideal gas:

Now, both R and N_{A} are constant, and \ so is R/N_{A}, and we define this as Boltzmann’s
constant, k_{B}, so:

A monatomic gas is one whose molecules consist of single atoms, for example, an inert gas such as helium. For an ideal monatomic gas:

Note:

The potential energy ('PE') component is due to forces between molecules
(*none* for an ideal gas, since the forces are zero)

**Speed distribution of molecules in a gas**

As, the graphs
show, the mean speed etc. increase with temperature.

**GRAHAM'S LAW AND AVOGADRO'S HYPOTHESIS **(return
to start of page)

Clearly, the derivation leading to the above equation(s) for the pressure of an ideal gas contains approximations (after all it deals with an imaginary ‘ideal’ gas, and not a real gas). Here we consider how well the derived equation(s) relate to the well-established Graham's law and Avogadro's hypothesis.

**1. Graham’s law of diffusion**

*The rate of diffusion of a gas is inversely proportional to the square root of its density*

We may define:

Now, the greater the average speed of the molecules, the more often they hit the side with the hole and the more likely they are to escape (i.e. diffuse).

So we assume:

So, from the derived expression for the pressure of an ideal gas we infer that the rate of diffusion of a gas is inversely proportional to the square root of its density, just as Graham’s law says.

**2. Avogadro’s hypothesis**

*If any two gases have the same pressure, volume and temperature then they contain the same number of molecules*

If the gases have the same pressure and volume:

Thus, the derived expression for the pressure of an ideal gas is
consistent with Avogadro’s hypothesis.

A Level Physics - Copyright © A
C Haynes 1999 & 2004