# Properties of Gases | General Chemistry 2

## Gas Properties

**Properties of gases:**

- Variable volume and shape: Gases have a variable volume and shape, expanding to fill the entire container they occupy. Most of the volume is empty space due to the relative distance between particles.
- Effect of pressure and temperature: The volume of gas is significantly affected by changes in pressure and temperature.
- Low density: Gases have relatively low densities compared to other states of matter.
- Fluidity: Gases flow freely, allowing them to move unrestricted within a given space.
- Miscibility: Gases are miscible with each other, forming solutions in any proportions.
- Exert pressure: Gases exert pressure on the walls of any surface in contact with them.

**Gas pressure:**

Gas pressure is the force exerted by gas molecules when they collide with the walls of their container. It is defined as a force per unit area:

P = $\frac{\mathrm{F}}{\mathrm{A}}$

P = pressure (in N.m^{-2})

F = force (in N = kg.m.s^{-2})

A = area (in m^{2})

**Units of pressure:**

Pressure is measured in units such as pascals (Pa), atmospheres (atm), or millimeters of mercury (mmHg).

- atm and torr are most commonly used in gas law problems.
- Pa is used if SI units are needed.

Conversion factors:

- 1 atm = 1.013 x 10
^{5}Pa = 1.013 bar = 760 torr = 760 mmHg - 1 Pa = 1 N.m
^{-2}= 1 kg.m.s^{-2}

**Standard temperature and pressure (STP):**

The standard temperature and pressure for gases is defined as the conditions where the pressure is exerted by a column of mercury 760 mm high at sea level at 0°C. Thus, STP is:

- T = 0°C (273.15 K)
- P = 1 atm

## The Gas Laws

**Key variables in gas laws:**

4 important variables:

- Pressure (P): Measured in pascals (Pa).
- Temperature (T): Measured in kelvins (K).
- Volume (V): Measured in cubic meter (m
^{3}). - Number of moles (n): Measured in moles (mol).

**Boyle’s Law (P vs. V):**

Boyle’s Law states that the volume of a gas is inversely proportional to its pressure when the temperature and the number of moles of gas are held constant.

P_{1 }V_{1} = P_{2} V_{2}

(at constant T and n)

P = pressure (in Pa)

V = volume (in m^{3})

**Charles’s Law (V vs. T):**

Charles’s Law states that the volume of a gas is directly proportional to its temperature when the pressure and the number of moles of gas are held constant.

$\frac{{\mathrm{V}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{T}}_{2}}$

(at constant P and n)

V = volume (in m^{3})

T = temperature (in K)

**Gay-Lussac’s Law (P vs. T):**

Gay-Lussac’s Law states that the pressure of a gas is directly proportional to its temperature when the volume and the number of moles of gas are held constant.

$\frac{{\mathrm{P}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{P}}_{2}}{{\mathrm{T}}_{2}}$

(at constant V and n)

P = pressure (in Pa)

T = temperature (in K)

**Avogadro’s Law (V vs. n):**

Avogadro’s Law states that the volume of a gas is directly proportional to its number of moles when the temperature and the pressure are held constant.

$\frac{{\mathrm{V}}_{1}}{{\mathrm{n}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{n}}_{2}}$

(at constant T and P)

V = volume (in m^{3})

n = number of moles (in mol)

## The Ideal Gas Law

**Ideal Gas equation:**

PV = nRT

P = pressure (in Pa)

V = volume (in m^{3})

n = number of moles (in mol)

R = ideal gas constant = 8.314 (in J.K^{-1}.mol^{-1})

T = temperature (in K)

**Conditions for ideal behavior:**

The Ideal Gas Law applies to ideal gases, or gases that behave ideally. Conditions that favor ideal gas behavior include:

- Monoatomic gases
- Low pressure
- High temperature

**Molar volume, molar mass, and density in the Ideal Gas Law:**

- Molar Volume (V
_{m}) in the Ideal Gas Law:

V_{m} = $\frac{\mathrm{V}}{\mathrm{n}}$ = $\frac{\mathrm{RT}}{\mathrm{P}}$

V_{m} = molar volume (in m^{3}.mol^{-1})

R = 8.314 J.mol^{-1}.K^{-1}

T = temperature (in K)

P = pressure (in Pa)

At standard temperature and pressure (STP): V_{m} = 22.41 L.mol^{-1}

- Molar Mass (M) in the Ideal Gas Law:

M = $\frac{\mathrm{m}}{\mathrm{n}}$ = $\frac{\mathrm{mRT}}{\mathrm{PV}}$

M = molar mass (in g.mol^{-1})

m = mass (in g)

R = 8.314 J.mol^{-1}.K^{-1}

T = temperature (in K)

P = pressure (in Pa)

V = volume (in m^{3})

- Density (ρ) in the Ideal Gas Law:

ρ = $\frac{\mathrm{m}}{\mathrm{V}}$ = $\frac{\mathrm{PM}}{\mathrm{RT}}$

ρ = density (in g.m^{-3})

P = pressure (in Pa)

M = molar mass (in g.mol^{-1})

R = 8.314 J.mol^{-1}.K^{-1}

T = temperature (in K)

## Gas Mixtures

**Dalton's Law of partial pressures:**

Dalton’s Law of partial pressures states that the total pressure exerted by a mixture of non-reacting gases at constant V and T is equal to the sum of the partial pressures of the individual gases.

P_{tot} = P_{1} + P_{2} + P_{3} + ... = $\sum _{}$ P_{i}

(at constant V and T)

P_{tot} = total pressure

P_{i} = partial pressure of gas i

**Partial pressure:**

The pressure exerted by an individual gas in a mixture is known as its partial pressure.

P_{i} = n_{i }$\left(\frac{\mathrm{RT}}{\mathrm{V}}\right)$

P_{i} = partial pressure of gas i

n_{i} = number of moles of gas i (in mol)

R = ideal gas constant = 8.314 (in J.K^{-1}.mol^{-1})

T = temperature (in K)

V = volume (in m^{3})

**Mole fraction ( X):**

The mole fraction of a component in a mixture is the ratio of the number of moles of that component to the total number of moles of all components in the mixture.

*X*_{i} = $\frac{{\mathrm{n}}_{\mathrm{i}}}{{\mathrm{n}}_{\mathrm{total}}}$

*X*_{i} = mole fraction of component i

n_{i} = number of moles of component i (in mol)

n_{total} = total number of moles of all components (in mol)

Consequence on the partial pressure: The expression for the partial pressure of a component i can be modified by replacing "n_{i}" with "*Xi* . n_{total}". Therefore, the partial pressure can be expressed as:

P_{i} = *X*_{i} P_{total}

P_{i} = partial pressure of gas i

*X*_{i} = mole fraction of component i

P_{total} = total pressure

## The Kinetic Molecular Theory of Gases

**Postulates of the Kinetic Molecular Theory:**

The kinetic molecular theory of gases is a set of principles that describe the behavior of gases based on the motion of their individual particles. The key postulates include:

- Constant, random motion: Gas particles are in continuous, random motion, colliding with each other and the walls of their container.
- Negligible volume: The volume occupied by gas particles is negligible compared to the total volume of the gas. Most of the space in a gas is empty.
- Elastic collisions: Collisions between gas particles are perfectly elastic, meaning there is no net loss of kinetic energy during collisions.
- No intermolecular forces: Gas particles do not exert attractive or repulsive forces on each other; they only interact during collisions and move in straight lines between collisions.
- Proportionality between temperature and kinetic energy: The mean kinetic energy ($\overline{{\mathrm{E}}_{\mathrm{k}}}$) of the gas molecules is proportional to the temperature. As the temperature increases, the average kinetic energy of the particles also increases.

**Kinetic energy of gases:**

- Kinetic energy of a molecule (E
_{k}):

E_{k} = $\frac{1}{2}$ mu^{2}

E_{k} = kinetic energy (in J)

m = mass of the molecule (in kg)

u = speed of a gas molecule (in m.s^{-1})

- Average kinetic energy of one molecule ($\overline{{\mathrm{E}}_{\mathrm{k}}}$)

$\overline{{\mathrm{E}}_{\mathrm{k}}}$ = $\frac{1}{2}$ m$\overline{{\mathrm{u}}^{2}}$

$\overline{{\mathrm{E}}_{\mathrm{k}}}$ = average kinetic energy of one molecule (in J)

m = mass of one molecule (in kg)

$\overline{\mathrm{u}}$ = average of the speed of one molecule (in m.s^{-1})

- Total kinetic energy of a mole of gas:

According to the kinetic molecular theory, the total kinetic energy of a mole of gas is:

E_{k,tot} = $\frac{3}{2}$ RT

E_{k,tot} = total kinetic energy of a mole of gas (in J)

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

**Molecular speed:**

- Relationship between average kinetic energy of one molecule and total kinetic energy of a mole of gas:

E_{k,tot} = N_{A}$\overline{{\mathrm{E}}_{\mathrm{k}}}$

E_{k,tot} = total kinetic energy of a mole of gas (in J)

N_{A} = Avogadro's number

$\overline{{\mathrm{E}}_{\mathrm{k}}}$ = average kinetic energy of one molecule (in J)

$\frac{3}{2}$ RT = N_{A}$\frac{1}{2}$ m$\overline{{\mathrm{u}}^{2}}$

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

N_{A} = Avogadro's number

m = mass of one molecule (in kg)

$\overline{\mathrm{u}}$ = average of the speed of one molecule (in m.s^{-1})

- Root-mean-square (rms) speed (u
_{rms}):

Because the molar mass M is equal to m x N_{A}, we can rearrange the preceding equation as follows:

$\overline{{\mathrm{u}}^{2}}$ = $\frac{3\mathrm{RT}}{\mathrm{M}}$

$\overline{\mathrm{u}}$ = average of the speed of one molecule (in m.s^{-1})

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

M = molar mass (in kg.mol^{-1})

Therefore,

u_{rms }= $\sqrt{\overline{{\mathrm{u}}^{2}}}$ = $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{}}}$

u_{rms} = root-mean-square speed (in m.s^{-1})

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

M = molar mass (in kg.mol^{-1})

## Gas Diffusion and Effusion

**Diffusion:**

Diffusion is the process by which gas molecules spread out in response to a concentration gradient.

- Gas molecules move from an area of higher concentration to an area of lower concentration until uniform distribution is achieved.
- The rate of diffusion is influenced by the mass of the gas molecules and the temperature of the environment. Lighter molecules diffuse faster than heavier ones.

**Effusion:**

Effusion is the process by which gas molecules escape through a tiny hole into a vacuum.

- Effusion depends on the size of the hole, the pressure of the gas, and the temperature.
- Like diffusion, gas molecules move from an area of higher concentration to an area of lower concentration until uniform distribution is achieved.

**Graham’s Law:**

Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

rate of effusion = α $\frac{1}{\sqrt{\mathrm{M}}}$

α = proportionality coefficient

m = molar mass (in kg.mol^{-1})

Therefore, the ratio of the rate of effusion of 2 gases at the same temperature and pressure is given by:

$\frac{{\mathrm{rate}}_{\mathrm{1}}}{{\mathrm{rate}}_{\mathrm{2}}}$ = $\sqrt{\frac{{\mathrm{M}}_{\mathrm{2}}}{{\mathrm{M}}_{\mathrm{1}}}}$

M_{1} = molar mass of the gas 1 (in kg.mol^{-1})

M_{2} = molar mass of the gas B (in kg.mol^{-1})

## Mean Free Path

**Mean Free Path (l):**

The mean free path is the average distance a molecule travels between collisions. It is given by:

I = $\frac{\mathrm{RT}}{\mathrm{\pi}\sqrt{2}{\mathrm{d}}^{2}{\mathrm{N}}_{\mathrm{A}}\mathrm{P}}$

I = mean free path (in m)

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

d = diameter of the gas molecule (in m)

N_{A} = Avogadro’s number = 6.022 x 10^{23}

P = pressure (in Pa)

**Collision frequency (z) (in collisions.s ^{-1}):**

The collision frequency is the number of collisions that a molecule undergoes per second. It is given by:

z = $\frac{{\mathrm{u}}_{\mathrm{rms}}}{\mathrm{I}}$

z = collision frequency (in s^{-1})

u_{rms} = root-mean-square speed (in m.s^{-1})

l = mean free path (in m)

## Real Gases

**Deviations from ideal behavior:**

Real gases deviate from ideal gas behavior due to intermolecular forces and the finite volume of gas molecules:

- Intermolecular forces: In real gases, attractive and repulsive forces between molecules cause deviations from ideal behavior. At high pressures and low temperatures, these forces become significant.
- Finite molecular volume: Gas molecules occupy a finite volume, which is not negligible at high pressures. The volume available for gas molecules to move is less than the container volume.

**Van der Waals equation:**

The Van der Waals equation accounts for the deviations from ideal gas behavior by introducing 2 correction factors unique to each gas: a and b. Within the ideal gas equation:

- P is substituted by P + a $\frac{{\mathrm{n}}^{2}}{{\mathrm{V}}^{2}}$ , where a is a constant accounting for intermolecular attractions.
- V is substituted by V - nb, where b is a constante accounting for the finite volume of gas molecules.

(P + a $\frac{{\mathrm{n}}^{2}}{{\mathrm{V}}^{2}}$)(V - nb) = nRT

P = pressure (in Pa)

V = volume (in m^{3})

n = number of moles (in mol)

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

### Check your knowledge about this Chapter

Gases are one of the three classical states of matter, defined by their ability to fill a container of any shape and volume.

Unlike solids that have a fixed shape and volume and liquids that have a fixed volume but no fixed shape, gases have neither fixed volume nor shape, and they expand to fill the volume and take the shape of the container they are in. Additionally, gas molecules are in constant, random motion and are much farther apart compared to the molecules in liquids and solids, which allows for the compressibility and expandability of gases.

Real gases differ from ideal gases primarily in how they conform to the assumptions of the Ideal Gas Law. Ideal gases are assumed to have no intermolecular forces and occupy no volume. Real gases, on the other hand, do have intermolecular attractive and repulsive forces, and their molecules occupy space. These differences become significant at high pressures and low temperatures, where deviations from ideal behavior are more pronounced. The Van der Waals equation is often used to correct the Ideal Gas Law for these deviations by introducing constants that adjust for the volume of the gas molecules (b) and the intermolecular forces (a).

The Ideal Gas Law is an equation that relates the pressure, volume, temperature, and number of moles of a gas through the relationship:

PV = nRT

P = pressure (in Pa)

V = volume (in m^{3})

n = number of moles (in mol)

R = ideal gas constant = 8.314 (in J.K^{-1}.mol^{-1})

T = temperature (in K)

This law assumes that gases are composed of a large number of particles that move randomly and do not interact with each other, which is an idealized model. The Ideal Gas Law is very useful in predicting the behavior of gases under various conditions, although it becomes less accurate at high pressures and low temperatures where real gases start to deviate from ideal behavior.

According to Gay-Lussac's law, for a given quantity of gas at constant volume, the pressure of the gas is directly proportional to its absolute temperature (measured in Kelvin). This means that as the temperature of the gas increases, so does the pressure. Conversely, as the temperature decreases, the pressure also decreases.

Boyle's Law states that for a given mass of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means if the pressure on the gas increases, the volume decreases, and vice versa as long as the temperature remains unchanged. It can be expressed mathematically as:

PV = k (at T = cst)

P = pressure

V = volume

k = constant for the given mass of gas at a specific temperature

The Ideal Gas Law, expressed as PV = nRT, defines the relationship between pressure (P), volume (V), number of moles (n), temperature (T), and the ideal gas constant (R). At Standard Temperature and Pressure (STP), which is 0°C (273.15 K) and 1 atmosphere, one mole of an ideal gas occupies 22.414 liters. By using the Ideal Gas Law, you can plug these standard conditions into the equation to solve for the volume (V) that one mole of gas would occupy.

For 1 mole of gas at STP, the calculation would be:

V = $\raisebox{1ex}{$\mathrm{nRT}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{P}$}\right.$

where n = 1 mole

R = 0.0821 L.atm.K^{-1}.mol^{-1}

T = 273.15 K

P = 1 atm

This will yield a volume of approximately 22.4 liters.

The molar mass of a gas can be determined using the Ideal Gas Equation, PV = nRT, where P is pressure, V is volume, n is the number of moles of the gas, R is the ideal gas constant, and T is temperature. By rearranging the equation to solve for n, and knowing the mass of the gas, you can calculate its molar mass (M) using the relationship M = $\raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.=\raisebox{1ex}{$\mathrm{mRT}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{PV}$}\right.$. This allows you to find the molar mass by measuring the mass of a known volume of the gas at a certain temperature and pressure.

To calculate the total pressure of a mixture of non-reacting gases, you can use Dalton’s Law of Partial Pressures. Dalton’s Law states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual gas in the mixture. The partial pressure of each gas in a mixture can be found by multiplying the mole fraction of the gas by the total pressure of the mixture. Mathematically, if you have a mixture of n gases, the total pressure P_{total} is given by P_{total} = P_{1} + P_{2} + ... + P_{n}, where P_{i} is the partial pressure of each gas.

The kinetic molecular theory provides a microscopic explanation for the macroscopic behavior of gases. It postulates that gases are composed of particles in constant, random motion, and that the pressure exerted by a gas results from collisions of these particles with the walls of the container. This theory is significant because it provides the basis for understanding the relationships between pressure, volume, temperature, and the amount of gas that are quantitatively described by the gas laws, such as Boyle's law, Charles's law, and the ideal gas law.

Furthermore, kinetic molecular theory explains the elastic nature of gas particle collisions, and the negligible volume occupied by gas particles themselves in comparison to the overall volume of the gas. These insights help to rationalize the behavior of real gases under varying conditions and also lead to the concept of the mean free path, providing a foundational understanding of gas diffusion and effusion rates as described by Graham's Law.

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. In other words, lighter gases effuse more quickly than heavier gases because they move at higher speeds, a consequence of their smaller mass under the same temperature conditions. Mathematically, it is expressed as $\raisebox{1ex}{${\mathrm{r}}_{1}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{r}}_{2}$}\right.$ = $\sqrt{\raisebox{1ex}{${\mathrm{M}}_{2}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{M}}_{1}$}\right.}$, where r represents the rate of effusion and M represents the molar mass of the gases being compared.

The mean free path is the average distance a gas molecule travels between collisions with other molecules. In conditions where the gas density is low or the temperature is high, the mean free path increases because molecules move faster and are more spread out, thereby encountering fewer collisions. Conversely, at higher densities or lower temperatures, the mean free path decreases due to more frequent collisions.

The Van der Waals Equation modifies the Ideal Gas Law to account for real gas behavior. It includes two correction factors:

- one for intermolecular attractions, represented by 'a', which lowers the pressure from the ideal prediction
- another for the finite size of gas molecules, represented by 'b', which effectively reduces the volume available for gas particle movement

By considering these factors, the equation provides a more accurate description of gas behavior under various conditions, especially those at high pressures and low temperatures where deviations from ideal behavior are more significant.