3.2.9 Define pressure.
3.2.10 State the assumptions of the kinetic model Of an ideal gas.
3.2.11 State that temperature is a measure of the average random kinetic energy of the molecules of an ideal gas.
3.2.12 Explain the macroscopic behaviour of an ideal gas in terms of a molecular model.
© IBO 2007
3.2.9 PRESSURE
Investigations into the behaviour of gases involve measurement of pressure, volume, temperature and mass. Experiments use these macroscopic properties of a gas to formulate a number of gas laws.
In 1643 Torricelli found that the atmosphere could support a vertical column of mercury about 76 cm high and the first mercury barometer became the standard instrument for measuring pressure. The pressure unit 760 mm Hg (760 millimetres of mercury) represented standard atmospheric pressure. In 1646, Pascal found that the atmosphere could support a vertical column of water about 10.4 m high.
For our purposes in this section, pressure can be defined as the force exerted over an area.
Pressure = Force / Area
P = F /A
The SI unit of pressure is the pascal Pa.
1 atm = 1.01 × 105 Nm-2 = 101.3 kPa = 760 mmHg
3.2.10 THEKINETIC
MODEL
OFAN
IDEALGAS
An ideal gas is a theoretical gas that obeys the ideal gas equation exactly. Real gases conform to the gas laws under
certain limited conditions but they can condense to liquids, then solidify if the temperature is lowered. Furthermore, there are relatively small forces of attraction between particles of a real gas, and even this is not allowable for an ideal gas.
Most gases, at temperatures well above their boiling points and pressures that are not too high, behave like an ideal gas. In other words, real gases vary from ideal gas behaviour at high pressures and low temperatures.
When the moving particle theory is applied to gases it is generally called the kinetic theory of gases. The kinetic theory relates the macroscopic behaviour of an ideal gas to the behaviour of its molecules.
The assumptions or postulates of the moving particle theory are extended for an ideal gas to include
• Gases consist of tiny particles called atoms (monatomic gases such as neon and argon) or molecules.
• The total number of molecules in any sample of a gas is extremely large.
• The molecules are in constant random motion. • The range of the intermolecular forces is small
compared to the average separation of the molecules.
• The size of the particles is relatively small compared with the distance between them.
• Collisions of short duration occur between molecules and the walls of the container and the collisions are perfectly elastic.
• No forces act between particles except when they collide, and hence particles move in straight lines. • Between collisions the molecules obey Newton’s
Laws of motion.
Based on these postulates the view of an ideal gas is one of molecules moving in random straight line paths at constant speeds until they collide with the sides of the container or with one another. Their paths over time are therefore zig-zags. Because the gas molecules can move freely and are relatively far apart, they occupy the total volume of a container.
The large number of particles ensures that the number of particles moving in all directions is constant at any time.
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3.2.11 TEMPERATUREANDAVERAGE
RANDOMKINETIC
ENERGY
Temperature is a measure of the average random kinetic energy of an ideal gas.
At the microscopic level, temperature is regarded as the measure of the average kinetic energy per
molecule associated with its movements. For gases,
it can be shown that the average kinetic energy,
Ek 1
2
---mv2 3
2 ---kT
= = where k = Boltzmann constant
v2∝T
∴
The term average kinetic energy is used because, at a particular temperature different particles have a wide range of velocities, especially when they are converted to a gas. This is to say that at any given temperature the average speed is definite but the velocities of particular molecules can change as a result of collision.
Figure 325 shows a series of graphs for the same gas at three different temperatures. In 1859 James Clerk Maxwell (1831-1879) and in 1861 Ludwig Boltzmann (1844-1906) developed the mathematics of the kinetic theory of gases. The curve is called a Maxwell-Boltzmann
speed distribution and it is calculated using statistical
mechanics. It shows the relationship between the relative number of particles N in a sample of gas and the speeds v that the particles have when the temperature is changed. (T3 > T2 > T1)
The graphs do not show a normal distribution as the graphs are not bell-shaped. They are slightly skewed to the left. The minimum speed is zero at the left end of the graphs. At the right end they do not touch the x-axis because a small number of particles have very high speeds.
The peak of each curve is at the most probable speed vp a large number of particles in a sample of gas have their speeds in this region. When the mathematics of statistical mechanics is applied it is found that mean squared speed
vav2 is higher than the most probable speed. Another
quantity more often used is called the root mean square speed Vrms and it is equal to the square root of the mean
squared speed.
vrms = v2
The root mean square is higher than the mean squared speed.
Other features of the graphs show that the higher the temperature, the more symmetric the curves becomes. The average speed of the particles increases and the peak is lowered and shifted to the right. The areas under the graphs only have significance when N is defined in a different way from above.
Figure 326 shows the distribution of the number of particles with a particular energy N against the kinetic energy of the particles Ek at a particular temperature. The shape of the kinetic energy distribution curve is similar to the speed distribution curve and the total energy of the gas is given by the area under the curve.
T1
T2
T2>T1
N
Ek
Figure 326 Distribution of kinetic energies for the same gas at different temperatures.
The average kinetic energy of the particles of all gases is the same. However, gases have different masses. Hydrogen molecules have about one-sixteenth the mass of oxygen molecules and therefore have higher speeds if the average kinetic energy of the hydrogen and the oxygen are the same.
Because the collisions are perfectly elastic there is no loss in kinetic energy as a result of the collisions.
Figure 325 Maxwell-Boltzmann speed distribution for the same gas at different temperatures.
vmpv vrms T1 T2 T3 T3>T2>T1 N v ms–1
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3.2.12 MACROSCOPICBEHAVIOUROF
AN
IDEALGAS
Robert Boyle (1627-1691) discussed that the pressure of a
gas at constant temperature is proportional to its density. He also investigated how the pressure is related to the volume for a fixed mass of gas at constant temperature. Boyle’s Law relates pressure and volume for a gas at fixed temperature.
Boyle’s Law for gases states that the pressure of a fixed mass of gas is inversely proportional to its volume at constant temperature.
P α 1 __
V ⇔ PV = constant
When the conditions are changed, with the temperature still constant
P 1 V 1 = P 2 V 2
The readings of P and V must be taken slowly to maintain constant temperature because when air is compressed, it warms up slightly.
When a pressure versus volume graph is drawn for the collected data a hyperbola shape is obtained, and when pressure is plotted against the reciprocal of volume a straight line (direct proportionality) is obtained. See Figure 327. pressure, P mm Hg pressure, P mm Hg PV volume, V cm3 1 V cm –3 P
Figure 327 (a), (b) and)c) pressure-volume graphs.
The pressure that the molecules exert is due to their collisions with the sides of the container. When the volume of the container is decreased, the frequency of the particle collisions with the walls of the container increases. This means that there is a greater force in a smaller area leading to an increase in pressure. The pressure increase has nothing to do with the collisions of the particles with each other.
In 1787 Jacques Charles (1746–1823) performed experiments to investigate how the volume of a gas changed with temperature. Gay-Lussac (1778–1850) published more accurate investigations in 1802.
A very simple apparatus to investigate Charles’ Law is shown in Figure 328. A sample of dry air is trapped in a capillary tube by a bead of concentrated sulfuric acid. The capillary tube is heated in a water bath and the water is constantly stirred to ensure that the whole air column is at the same temperature.
thermometer
water bath
bead of acid (e.g. sulfuric acid)
air column capillary tube
H E A T
Figure 328 Apparatus for Charles’ law.
The investigation should be carried out slowly to allow thermal energy to pass into or out of the thick glass walls of the capillary tube. When the volume and temperature measurements are plotted, a graph similar to Figure 328 is obtained. V c m 3 –273 100 T °C 0 2730 373 T K
Figure 329 Variation of volume with temperature.
Note that from the extrapolation of the straight line that the volume of gases would be theoretically zero at –273 °C called absolute zero. The scale chosen is called the Kelvin scale K.
The Charles (Gay-Lussac) Law of gases states that:
The volume of a fixed mass of gas at constant pressure is directly proportional to its absolute (Kelvin) temperature.
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This can also be stated as:
The volume of a fixed mass of gas increases by 1/273 of its volume at 0 °C for every degree Celsius rise in temperature provided the pressure is constant.
V α T ⇒ V = kt so that V ___ 1 T 1 = k Therefore, V ___ 1 T 1 = V 2 ___ T 2
As the temperature of a gas is increased, the average kinetic energy per molecule increases. The increase in velocity of the molecules leads to a greater rate of collisions, and each collision involves greater impulse. Hence the volume of the gas increases as the collisions with the sides of the container increase.
Experiments were similarly carried out to investigate the relationship between the pressure and temperature of a fixed mass of various gases.
The essential parts of the apparatus shown in Figure 330 are a metal sphere or round bottomed flask, and a Bourdon pressure gauge. The sphere/flask and bourdon gauge are connected by a short column of metal tubing/ capillary tube to ensure that as little air as possible is at a different temperature from the main body of enclosed gas. The apparatus in Figure 330 allows the pressure of a fixed volume of gas to be determined as the gas is heated.
Bourdon gauge counter-balance metal stem thermometer retort stand air enclosed in a metal sphere
Figure 330 Pressure law apparatus.
The variation in pressure as the temperature is changed is measured and graphed. A typical graph is shown in Figure 331. –273 100 T °C Pr es su re , P kP a 0 2730 373 T K
Figure 331 Variation of pressure with temperature.
The Pressure (Admonton) Law of Gases states that:
The pressure of a fixed mass of gas at constant volume is directly proportional to its absolute (Kelvin) temperature. P ∝T⇔P kT P1 T1 --- ∴ k = = Therefore, P1 T1 --- P2 T2 --- =
As the temperature of a gas is increased, the average kinetic energy per molecule increases. The increase in velocity of the molecules leads to a greater rate of collisions, and each collision involves greater impulse. Hence the pressure of the gas increases as the collisions with the sides of the container increase.