# Properties of Gases | General Chemistry 2

## Gas Properties

**Properties:**

- 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.
- The volume of gas is significantly affected by changes in pressure and temperature.
- Gases have relatively low densities compared to other states of matter.
- Gases flow freely, allowing them to move unrestricted within a given space.
- Gases are miscible with each other, forming solutions in any proportions.
- Gases exert pressure on the walls of any surface in contact with them.

**Pressure:**

Pressure is defined as a force per unit area:

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

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

A = area (in m^{2})

Pressure is measured in a variety of units:

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

Conversion factor: 1 atm = 760 torr = 1.013 bar = 1.013 x 10^{5} Pa

## Gas Laws

4 important variables:

- Pressure P (in Pa)
- Temperature T (in K)
- Volume V (in L)
- Number of moles n (in mol)

Ideal gas: a gas that exhibits linear relationships between these variables.

**Boyle’s Law (P vs. V):**

At constant temperature: P and V are inversely proportional

P_{1 }V_{1} = P_{2} V_{2} (at T = cst)

**Charles’s Law (V vs T):**

At constant pressure: V and T are directly proportional

$\frac{{\mathrm{V}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{T}}_{2}}$ (at P = cst)

**Gay-Lussac’s Law (P vs T):**

At constant volume: P and T are directly proportional

$\frac{{\mathrm{P}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{P}}_{2}}{{\mathrm{T}}_{2}}$ (at V = cst)

**Avogadro’s Law (V vs n):**

At constant temperature and pressure: V and n are directly proportional

$\frac{{\mathrm{V}}_{1}}{{\mathrm{n}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{n}}_{2}}$ (at T and P = cst)

## 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)

This equation applies to ideal gases, or gases that behave ideally.

Better conditions for a gas to behave like an ideal gas: monoatomic gas, low pressure, high temperature.

## Ideal Gas Law – Molar Volume, Molar Mass and Density

**Molar Volume V _{m} (in L.mol^{-1}) in the Ideal Gas Law:**

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

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 g.mol ^{-1}) in the Ideal Gas Law:**

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

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 g.m ^{-3}) in the Ideal Gas Law:**

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

P = pressure (in Pa)

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

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

T = temperature (in K)

## Dalton’s Law of Partial Pressures

Dalton’s Law: at low pressures, the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of all the gases in the mixture.

P_{tot} = P_{A} + P_{B} + P_{C} + … + P_{i} = $\sum _{}{P}_{i}$

## Kinetic Theory and Speed Distribution

**Kinetic energy of a gas molecule (in J):**

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

m = mass of the molecule (in kg)

v = velocity (in m.s^{-1})

**Kinetic energy of a mole of gas molecules (in J.mol ^{-1}):**

E_{k} = $\frac{1}{2}$M_{kg}v^{2}

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

v = velocity (in m.s^{-1})

**Kinetic theory of gases**

The kinetic 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 of the kinetic theory of gases include:

- Gases are composed of particles (atoms or molecules) that are in constant, random motion.
- The volume occupied by gas particles is negligible compared to the total volume of the gas ⇒ most of the space is empty.
- Collisions are perfectly elastic, meaning there is no net loss of kinetic energy.
- Gas particles do not exert attractive or repulsive forces on each other, they only interact during collisions ⇒ between collisions they move in straight lines.
- The mean kinetic energy of the gas molecules
*E*is proportional to the temperature. As temperature increases, the average kinetic energy of the particles also increases._{k}

**Mean kinetic energy of the gas molecules E_{k} (in J):**

$\overline{){E}_{k}}$ = $\frac{1}{2}$M_{kg}$\overline{{\mathrm{v}}^{2}}$ = αT

A detailed analysis of the collisions shows that:

α = $\frac{3}{2}$R ⇒ $\overline{{\mathrm{v}}^{2}}$ = $\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}$

**Root-mean-square speed v _{rms} (in m.s^{-1}):**

Square root of the mean of the square of the molecular speeds

v_{rms }= $\sqrt{\left(\overline{{\mathrm{v}}^{2}}\right)}$ = $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}}$

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

T = temperature (in K)

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

## Graham’s Law of Effusion

Diffusion: gradual dispersal of one gas through another (equal pressure process).

Effusion: escape of a gas through one or more small holes from an area of high pressure to an area of lower pressure.

**Graham’s Law:**

For 2 gases at the same temperature and pressure, the effusion rate is directly proportional to v_{rms}:

rate = α v_{rms} = α $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}}$

α = proportionality coefficient

The formula mass of a gas can be determined by using effusion:

$\frac{{\mathrm{rate}}_{\mathrm{A}}}{{\mathrm{rate}}_{\mathrm{B}}}$ = $\sqrt{\frac{{\mathrm{M}}_{\mathrm{B}}}{{\mathrm{M}}_{\mathrm{A}}}}$

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

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

## Mean Free Path

**Mean Free Path l (in m):**

Average distance a molecule travels between collisions.

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

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}):**

Number of collisions that a molecule undergoes per second.

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

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

l = mean free path (in m)

## Van der Waals Equation

Van der Waals equation is an equation for non-ideal gas ⇒ it addresses deviations from Gas Ideality. Within the ideal gas equation, adjustments are made to better represent non-ideal gas behavior:

- V is substituted by V – nb, where b = cst
- P is substituted by P + a , where a = cst

These modifications introduce two critical van der Waals constants, a and b, unique to each gas.

**Van der Waals equation:**

(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.