Ideal Gases

Heat

IDEAL GASES (return to CONTENTS)

INTRODUCTION (return to start of page)

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

We can determine the quantity PV for a fixed mass of gas at various temperatures and plot a graph of PV against temperature:

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 = - 273.15⁰C.

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

The triple point is defined as 273.16K (= 0.01⁰C).

K (not ⁰K) 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 1⁰C, making it is easy to change between K and ⁰C:

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

This is indicated below, where the temperature determined is the temperature of the steam-point using a variety of gases:

The solid lines above would be experimental, produced as described. When continued to where P_tr = 0, the dotted lines, they all cut the vertical axis at the same point. This indicates the ideal gas temperature. Thus, by this means, any real gas can be used to determine a temperature 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.

For n moles of an ideal gas:

This is called the ideal gas equation (or the universal gas law equation).

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 0⁰C and 76cm Hg (=1.013 * 10⁵ 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 =⁰C +273)
The pressures and volumes can be in any units, as long as they are the same on both sides of the equation

Example

A cycle pump has its exit sealed. It contains 50cm³ of a gas at 20⁰C and a pressure of 1.0*10⁵Pa . Find the new pressure when the gas is compressed to 10cm³ and the temperature rises to 25⁰C.

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:

Charles’ Law

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 –273⁰C 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 (-273⁰C, 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

The above is a version of an experiment performed by the Scottish Botanist Robert Brown in 1827. He observed that fine pollen grains suspended in water were in a constant state of irregular motion - believed to be due to pollen grains being bombarded by invisible water molecules.

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

For example, the spreading out of perfume in still air in a room is believed to be due to molecules of the perfume gradually 'working' their way between the molecules of the air.

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

Derivation of equation for pressure

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 u₁, u₂, u₃, ……, 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 c₁, c₂, c₃, ……, 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_rmsfor molecules with speeds: 1, 2, 3, 4, 5 ms^-1

Average KE per molecule

Temperature and kinetic theory

For one mole (i.e. N_Amolecules) 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

Consider a container holding a gas, with a single small hole in it through which gas molecules can pass:

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

Now, for two ideal gases, 1 and 2, we can write the previous equation (2) as:

Applying the conditions stated in the hypothesis:

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.

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