Ideal Gases

#### 3.3.1 State the macroscopic gas laws relating pressure, volume and temperature.

Boyle’s Law – For a gas at a constant temperature the volume and the pressure are inversely proportional:

(1)
\begin{align} p \propto {1 \over V} \end{align}
 Plotting pressure vs. volume produces the following graph: Plotting pressure vs. inverse volume produces a straight line graph:

If a thermodynamic system changes, but its temperature remains constant then the following the initial pressure and volume are related to the final pressure and volume by the following equation:

(2)
$$P_1 V_1 = P_2 V_2$$

Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the gas is directly proportional to its absolute temperature.

(3)
\begin{align} V \propto T \end{align}

If a thermodynamic system changes, but its mass and pressure are held constant the initial volume and temperature are related to the final volume and temperature by the following equation:

(4)
\begin{align} {V_1 \over T_1} = {V_2 \over T_2} \end{align}

Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute pressure of the gas is directly proportional to its absolute temperature.

(5)
\begin{align} P \propto T \end{align}

If a thermodynamic system changes, but the volume are held constant the initial pressure and temperature are related to the final pressure and temperature by the following equation:

(6)
\begin{align} {P_1 \over T_1} = {P_2 \over T_2} \end{align}

All of the above mentioned laws required special circumstances that are not often satisfied. A more general relationship between the volume, pressure and temperature can be found:

(7)
\begin{align} {P_1 V_1 \over T_1} = {P_2 V_2 \over T_2} \end{align}

This still does not take into account a change in mass. If the temperature and pressure of a gas is held constant and more gas is added (mass increases) the volume must increase. If the volume and temperature of a gas are held constant while more gas is added the pressure must increase. If we combine these experimental observations we can write an even more general relation:

(8)
\begin{align} {P_1 V_1 \over m_1 T_1} = {P_2 V_2 \over m_2 T_2} \end{align}

#### 3.3.2 Define the terms mole and molar mass

In general we use relatively small amount of gas or just a few numbers of atoms or molecules. It is possible to determine the mass of individual atoms or molecules, but the numbers are messy, a gram is an incredibly large mass compared to the mass of an atom. So we define a new mass unit, called an atomic mass unit or amu.

An atomic mass unit is defined as 1/12th the mass of a carbon-12 atom. Basically the mass of a proton or neutron, or a proton or neutron have mass 1 amu. Note: Neutrons and protons do not have exactly the same mass, but its close enough. Thus one carbon-12 atom has mass 12 amu.

Helium-4 has an mass 4 amu or approximately and fluorine-19 has a mass of 19 amu or approximately $1.99 \times 10^{-23} g$. Which numbers would you rather use?

It would be a rare find to find a scale that measures in amu. So how do we relate amu to grams? Here we define the concept of a mole. A mole (mol) is the number of particles in a sample such that the mass in grams of the sample is equal to the mass in amu of a single particle. So 1 mol of carbon-12 atoms has a mass of 12 g, 1 mol of helium-4 has a mass of 4 g.

From this we can define the concept of molar mass, this is simply the grams per mole. Carbon-12’s molar mass is $12g \cdot mol^-1$.

#### 3.3.3 Define the Avogadro constant

One day some body got bored and wondered exactly how many particles are in a mole, this guy’s name was Avogadro (written in symbol form NA). I don’t think he actually made a good measurement of his number, but hey he got a symbol named after him, better than I could do.

(9)
\begin{align} N_A = {Number of Particles \over Number of Moles} \end{align}

This number is called Avogadro’s number of Avogadro’s constant. The accepted value is:

(10)
\begin{align} N_A = 6.023 \times 10^23 \end{align}

#### 3.3.4 State that the equation of state of an ideal gas is

The number of moles of gas to the mass of gas is related by the molecular mass of the gas. Therefore we can write the mass as:

(11)
$$m = nM$$

Where m is the mass, n is the number of moles and M is the molecular mass. If we substitute this expression in to the general gas law:

#### 3.3.1 State the macroscopic gas laws relating pressure, volume and temperature.

Boyle’s Law – For a gas at a constant temperature the volume and the pressure are inversely proportional:

(12)
\begin{align} p \propto {1 \over V} \end{align}
 Plotting pressure vs. volume produces the following graph: Plotting pressure vs. inverse volume produces a straight line graph:

If a thermodynamic system changes, but its temperature remains constant then the following the initial pressure and volume are related to the final pressure and volume by the following equation:

(13)
$$P_1 V_1 = P_2 V_2$$

Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the gas is directly proportional to its absolute temperature.

(14)
\begin{align} V \propto T \end{align}

If a thermodynamic system changes, but its mass and pressure are held constant the initial volume and temperature are related to the final volume and temperature by the following equation:

(15)
\begin{align} {V_1 \over T_1} = {V_2 \over T_2} \end{align}

Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute pressure of the gas is directly proportional to its absolute temperature.

(16)
\begin{align} P \propto T \end{align}

If a thermodynamic system changes, but the volume are held constant the initial pressure and temperature are related to the final pressure and temperature by the following equation:

(17)
\begin{align} {P_1 \over T_1} = {P_2 \over T_2} \end{align}

All of the above mentioned laws required special circumstances that are not often satisfied. A more general relationship between the volume, pressure and temperature can be found:

(18)
\begin{align} {P_1 V_1 \over T_1} = {P_2 V_2 \over T_2} \end{align}

This still does not take into account a change in mass. If the temperature and pressure of a gas is held constant and more gas is added (mass increases) the volume must increase. If the volume and temperature of a gas are held constant while more gas is added the pressure must increase. If we combine these experimental observations we can write an even more general relation:

(19)
\begin{align} {P_1 V_1 \over n_1 M T_1} = {P_2 V_2 \over n_2 M T_2} \end{align}

Assuming the molecular mass is constant:

(20)
\begin{align} {P_1 V_1 \over n_1 T_1} = {P_2 V_2 \over n_2 T_2} \end{align}

This can be written yet again as:

(21)
\begin{align} {P_1 V_1 \over n_1 T_1} =R \end{align}

Or as the ideal gas law:

(22)
$$pV = nRT$$

An ideal gas being a gas that obeys the ideal gas law at all pressures, volumes and temperatures. No real gas is ideal, but most gases obey the ideal gas law for low pressure and high temperatures (above their freezing point).

R is called the universal gas constant, the value of R is dependent on the choice of units:

(23)
\begin{align} R = 8.314 J \cdot mol^-1 \cdot K^-1 \end{align}

The universal gas constant is given in the IB formula booklet.

#### 3.3.5 Describe the concept of absolute zero and the Kelvin scale

If the volume of a gas is measured at constant pressure but at different temperatures a linear relationship is found. If the line is extended to the left it will eventually intercept the temperature axis, if the line went farther than this it would represent a negative volume which does not make physical sense. If lines are plotted for several different gases it can be found that they have different slopes, but intercept the temperature axis at the same point. This temperature is in theory a minimum temperature, or absolute zero. Absolute zero does not mean there is no internal energy, but only that internal energy is at a minimum.

It should be noticed also that the equations presented above would provide nonsensical answers for negative temperatures, i.e. negative volume or negative pressure. This is simply a mathematical issue. Due to this, a new scale the Kelvin scale was developed where 0 K is absolute zero. One degree Kelvin is the same change in temperature as one degree Celsius. 0 K is defined as -273.16°C, for the purposes of the IB -273°C is sufficient.

#### 3.3.8 Explain the macroscopic behavior of an ideal gas in terms of molecular model

Straight from Physics 2nd Edition, Kerr, Kerr, Ruth pages 220-221:

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 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

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.

The pressure that the molecules exert is due to their collisions with the sides of the container. 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. When a force is applied to a piston in a cylinder containing a volume of gas, the molecules take up a smaller volume and hence collisions are more frequent leading to an increase in pressure.

Because the collisions are perfectly elastic there is no loss in kinetic energy as a result of the collisions. Temperature is a measure of the average kinetic energy per molecule.