Chapter 12 Notes Dr. Floyd Beckford

Recall that matter exists in three common physical states: solids, liquids and gases. Solids tend to be rigid and have a definite shape and a fixed volume. Solids are not very compressible and their volumes do not depend on temperature and pressure. Liquids are confined to a particular volume. They flow and assume the shape of their containers. Liquids are a bit more compressible than solids. Gases have particles that are independent of each other. As such they take up all the space of their containers and are readily compressible. They consist primarily of empty space.

Common properties of gases
1. Gases can be compressed i.e. their densities can be increased by applying pressure.
2. Gases exert pressure on their surroundings.
3. Gases expand without limits and so gas samples completely and uniformly occupy the volume of any container.
4. Gases diffuse into each other and so samples of gases placed in the same container, mix completely.
5. The amounts and properties of gases are described in terms of temperature, pressure, volume and number of moles of gas.

PRESSURE
Pressure is defined as force per unit area e.g. lbs/in², pounds per square inch (psi). There are many different units of pressure. The pressure of the atmosphere, called atmospheric pressure, is measured by an instrument called the barometer that measures pressure as a difference in height of a mercury column. Therefore the most widely used unit of pressure is mmHg (millimeters of mercury). Another unit of pressure is the torr.

• V is proportional to 1/P or V = k x (1/P) (n, T constant) {n = number of moles}
• k is called a proportionality constant

One can carry out two different experiments on a gas sample

	Experiment 1: P1V1 = k
	Experiment 2: P2V2 = k
	That is, P1V1 = k = P2V2

This is the common way to mathematically expressed Boyle’s Law: P₁V₁ = P₂V₂ (at constant n, T)

CHARLES’ LAW: Volume/temperature relationship

When experiments involving changing temperature and volume are carried out, there is a temperature where the volume of the gas, in principle, is zero. This temperature is called absolute zero. The temperature scale is therefore called the absolute zero temperature scale or more commonly as the Kelvin temperature scale.

On the Celcius scale absolute zero occurs at –273.15 °C. The Kelvin temperature, K, is given by:

As seen from the graph above,

STANDARD TEMPERATURE AND PRESSURE (STP)

• Temperature: 0 °C (273.15 K)
• Pressure: 1 atm (760 torr)

If we combine Boyle’s Law and Charles’, we get the combined gas law equation. P₁V₁/T₁ = P₂V₂/T₂ (at constant n)

AVOGADRO’S LAW
At the same temperature and pressure, equal volumes of all gases contain the same number of molecules. This can be expressed as Avogadro’s Law:

• At constant temperature and pressure, the volume occupied by a gas sample is directly proportional to the number of moles of gas. So, V is directly proportional to n or V = kn
• For two different samples of gas at the same temperature and pressure, V₁n₁ = V₂n₂ (same T, P)

The density of a gas is proportional to its molar mass. So,

THE IDEAL GAS EQUATION
The mathematical expressions describing Boyle’s Law, Charles’ Law and Avogadro’s Law can be used to describe the behavior of an ideal gas. An ideal gas is one that obeys these laws strictly.

• V is directly proportional to (1/P),
• V is directly proportional to T,
• and V is directly proportional to n

• V is proportional to (1/P)xTxn i.e. V is proportional to nT/P

• PV = nRT

R = 8.314 J/mol.K = 0.0826 L.atm/mol.K = 8.314 kPa.dm³/mol.K

DALTON’S LAW (OF PARTIAL PRESSURE)

The product contains a mixture of gases. The total pressure of the gaseous product mixture is the sum of the pressures of the individual gas pressures; their so-called partial pressures. Obviously the total number of moles is equal to the sum of the number of moles of the individual gases.
That is, n_total = n_a + n_b + n_c + ….
So

	
	PtotalV = ntotalRT

		= (na + nb + nc + ….)RT

	
	Ptotal = [(na + nb + nc + ….)/V]RT

		= naRT/V + nbRT/V + ncRT/V + ….

		= Pa + Pb + Pc + ….

But n_A = P_AV/RT

So, X_A = P_A/P_total

Likewise X_B = P_B/P_total
Therefore
X_A + X_B + X_C + … = 1

THE KINETIC THEORY
The ideas used to explain the gas laws are collectively referred to as the kinetic molecular theory. Main points of the kinetic molecular theory:
1. Gases consist of molecules whose separation is much larger than the molecules themselves.
2. Molecules of a gas are in continuous random motion.
3. Gas molecules collide with each other and with the walls of their container, but they do so without loss of energy. These collisions are said to be perfectly elastic.
4. Between collisions the molecules exerts no attractive or repulsive force on one another.
5. The average kinetic energy of a gas is directly proportional to the absolute temperature of the gas. All gas molecules at the same temperature, regardless of mass, have the same kinetic energy.

Average KE is proportional to T (T is the absolute temperature)

Boyle’s Law: KMT interpretation
Recall that P is proportional to 1/V
According to KMT, gas pressure is a force that arises from collisions of molecules with a surface. This force (pressure) depends on
1. the number of collisions with the surface and
2. the average force per collision

If T and n is fixed, more collisions will occur at ˝V, leading to twice as many collisions per unit time and therefore doubling the pressure.

Charles’ Law: KMT interpretation
Recall that V is proportional to T
Kinetic energy is directly proportional to absolute temperature. If temperature is increases then the force of the collisions with the wall (pressure) increases. If pressure is held constant, the volume of the gas has to increase. Likewise, decreasing the temperature causes a decrease in volume.

Dalton’s Law: KMT interpretation
In a gas, the molecules behave as independent entities. Therefore each gas in a mixture creates its own “pressure” despite the presence of other molecules and so the total pressure is the sum of the individual “pressures”.

REAL GASES – DEVIATIONS FROM IDEALITY
Recall that an ideal gas obeys the gas laws strictly. In ideal gases the identity of the gases does not matter. Under most conditions real gases do behave ideally. However deviations from ideality occur some under conditions, which implies that under these conditions, the gases are not behaving as stated in the kinetic molecular theory.

Non-ideal behavior occurs near or under conditions that the gas liquefies. High pressures and/or low temperatures typify such conditions.

(a) According to the KMT gases are mostly empty space (V_ideal).
At high pressures, the volume of the molecules becomes a significant fraction of the total volume occupied by the gas.

(b) The KMT also assumes no attractive forces between the molecules. At low temperature and high pressure, real gases do attract each other. This attraction causes the energy of the collisions to be less than for an ideal gas.

Applying the corrections to the ideal gas equation results in the van der Waals equation:

Here P, V, T and n are measured values just as in the ideal gas equation. Values of a and b depends on the forces of attraction between the molecules. So a is small for the noble gases and nonpolar molecules because only weak attractive forces exist between them. These weak forces are called London forces. As seen earlier, larger molecules have greater values for b.