3.4.A.1 Ideal Gas Law: PV = nRT:

1. Macroscopic Properties of Gases:

i. Ideal Gas Law

The relationships between these four properties are encapsulated in the Ideal Gas Law:

$$PV = nRT$$

Where:

• P = pressure
• V = volume
• n = number of moles
• R = the ideal gas constant (8.314 J/mol·K or 0.0821 L·atm/mol·K)
• T = temperature in Kelvin

This equation shows that the pressure, volume, and temperature of a gas are all interdependent, and can be manipulated based on changes in the number of moles.

2. Gas Laws:

i. Boyle’s Law (P₁V₁ = P₂V₂) – At constant temperature, the volume of a gas is inversely proportional to its pressure.

ii. Charles’s Law (V₁/T₁ = V₂/T₂) – At constant pressure, the volume of a gas is directly proportional to its temperature.

iii. Avogadro’s Law (V₁/n₁ = V₂/n₂) – At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.

3. Ideal Gas Law:

i. Derivation of the Ideal Gas Law:

$PV = nRT$

The Ideal Gas Law relates pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas through a constant

$R$ (the ideal gas constant). Here’s the step-by-step derivation and the assumptions that go into it:

a. Boyle’s Law (at constant temperature):

Boyle’s Law states that for a fixed amount of gas at constant temperature, the pressure (P) and volume (V) are inversely proportional:

$$P \propto \frac{1}{V} \quad \text{or} \quad PV = \text{constant}$$

This relationship shows that when the volume of the gas decreases, the pressure increases, assuming the temperature is constant.

b. Charles’s Law (at constant pressure):

Charles’s Law states that for a fixed amount of gas at constant pressure, the volume (V) is directly proportional to its temperature (T) in Kelvin:

$$V \propto T \quad \text{or} \quad \frac{V}{T} = \text{constant}$$

This means that when the temperature increases, the volume also increases, as long as the pressure is kept constant.

c. Avogadro’s Law (at constant temperature and pressure):

Avogadro’s Law states that at constant temperature and pressure, the volume (V) of a gas is directly proportional to the number of moles (n):

$$V \propto n \quad \text{or} \quad \frac{V}{n} = \text{constant}$$

This implies that as more gas molecules are added to a fixed volume, the volume increases.

Combining the Three Laws:

To combine these three relationships into one, we consider each of them individually:

• Boyle’s Law tells us that
  $P \propto \frac{1}{V}$

  or

  $PV = \text{constant}$
• Charles’s Law tells us that
  $V \propto T$

  or

  $\frac{V}{T} = \text{constant}$
• Avogadro’s Law tells us that
  $V \propto n$

  or

  $\frac{V}{n} = \text{constant}$

If we combine these proportionalities, we get the general form of the ideal gas law:

$$\frac{PV}{nT} = \text{constant}$$

This constant is the ideal gas constant,

$R$ so we get the equation:

$$PV = nRT$$

where

$R$ is the ideal gas constant, whose value is:

• R = 0.0821 L·atm/mol·K
• R = 8.314 J/mol·K (this is the SI unit value)
• R = 1.987 cal/mol·K

ii. Application of the Ideal Gas Law

The Ideal Gas Law

$PV = nRT$

is incredibly useful for calculating one of the four macroscopic properties (P, V, T, or n) of an ideal gas if the other three are known.

iii. Assumptions of the Ideal Gas Law

a. Ideal Gas Behavior: The gas must behave ideally, meaning the gas molecules have no volume and experience no intermolecular forces (except elastic collisions).

b. High Temperature & Low Pressure: Real gases approximate ideal behavior at high temperatures and low pressures, where interactions between gas molecules are minimal.

4. Kinetic Molecular Theory:

The Kinetic Molecular Theory (KMT) provides a molecular-level explanation for the behavior of gases. It helps explain why gases follow the gas laws (like the Ideal Gas Law) and describes the motion and interactions of gas molecules. Here’s a summary of the key postulates of KMT and how they explain gas behavior:

i. Key Postulates of the Kinetic Molecular Theory:

a. Gas molecules are in constant, random motion.

1. • Gas molecules move in all directions and with various speeds, colliding with one another and with the walls of the container. These collisions are elastic, meaning that no energy is lost in the collisions.
   • The molecules’ kinetic energy (energy of motion) is directly related to the temperature of the gas.

b. The volume of the gas molecules is negligible.

• • Compared to the space between gas molecules, the size of the molecules themselves is very small. This assumption allows us to treat gases as if they have no volume.
  • The volume of the gas is determined primarily by the space between the molecules (not their individual sizes).

c. Gas molecules do not exert attractive or repulsive forces on each other.

• • In an ideal gas, molecules do not interact with each other except when they collide. There are no intermolecular forces (like van der Waals forces or hydrogen bonds) acting between them.
  • This assumption makes gases behave differently from liquids and solids, where intermolecular forces play a significant role in their behavior.

d. The average kinetic energy of gas molecules is proportional to the temperature.

• The kinetic energy of gas molecules is directly related to the temperature in Kelvin. The higher the temperature, the faster the molecules move and the greater their kinetic energy.
• The equation for the average kinetic energy of a gas molecule is:

$$\text{KE} = \frac{3}{2} k_B T$$

where

$k_B$ is Boltzmann’s constant ($1.38\times 10^{-23}\text{J/K}$) and

$T$ is the temperature in Kelvin.

e. Gas molecules undergo perfectly elastic collisions.

• • When gas molecules collide with each other or with the walls of the container, they do so in a way that the total kinetic energy of the system is conserved.
  • In other words, the energy lost by one molecule during a collision is exactly gained by the other molecule(s) involved in the collision.

ii. Molecular Explanation of Gas Behavior:

Using the postulates of the Kinetic Molecular Theory, we can explain the following behaviors of gases:

a. Pressure:

• Pressure arises from the collisions of gas molecules with the walls of the container. Each time a molecule hits the wall, it exerts a small force. The total pressure is the sum of the forces from all the collisions per unit area.
• Since gas molecules are in constant motion, the more frequent and forceful the collisions, the higher the pressure.

b. Volume and Temperature Relationship (Charles’s Law):

• As temperature increases, the kinetic energy of the gas molecules also increases. This results in the molecules moving faster and colliding with the walls of the container more forcefully, causing the gas to expand (increase in volume).
• In a rigid container, if the temperature is increased, the molecules speed up, and their kinetic energy increases, which could lead to increased pressure if the volume is held constant.

c. Pressure and Volume Relationship (Boyle’s Law):

• As the volume of a gas decreases, the gas molecules have less space to move. This means that the number of collisions with the container walls increases, and the pressure increases, assuming the temperature and number of molecules are constant.

d. Number of Moles and Volume Relationship (Avogadro’s Law):

• If the number of gas molecules increases while the temperature and pressure remain constant, the volume increases because there are more molecules colliding with the container walls. This corresponds to an increase in the frequency of collisions and the overall volume the gas occupies.

e. Gas Distribution and Speed of Molecules:

• Not all molecules in a gas travel at the same speed. The distribution of molecular speeds is described by the Maxwell-Boltzmann distribution.
• At higher temperatures, the distribution shifts, meaning more molecules have higher speeds. The average kinetic energy of the molecules increases with temperature, as described earlier.

5. Real Gases vs. Ideal Gases:

While Ideal Gases are a theoretical concept based on the assumptions of the Kinetic Molecular Theory (KMT), Real Gases are actual gases that exhibit behaviors influenced by intermolecular forces and the finite volume of gas molecules. The Ideal Gas Law is a good approximation for many gases under standard conditions, but real gases often deviate from this behavior under certain circumstances.

i. Assumptions of Ideal Gases (KMT):

The Ideal Gas Law assumes:

• Gas molecules have no volume: They are considered point particles with negligible size.
• No intermolecular forces: Gas molecules are assumed to not attract or repel each other.
• Elastic collisions: Molecules collide without loss of kinetic energy.
• Constant motion: Molecules are in continuous, random motion, and their average kinetic energy is proportional to the temperature.

Under these idealized conditions, gases obey the Ideal Gas Law perfectly:

$$PV = nRT$$

However, these assumptions don’t hold perfectly in reality, especially under extreme conditions.

ii. Real Gases and Deviations:

Real gases deviate from ideal behavior mainly because of intermolecular forces and the finite size of the gas molecules. These deviations are most noticeable at high pressure and low temperature.

a) Deviations at High Pressure:

• High Pressure means gas molecules are forced closer together, and the volume of the gas molecules themselves becomes significant (i.e., the volume of the molecules is no longer negligible compared to the container’s volume).

• Intermolecular Forces also become more important at high pressure. At closer distances, the attractive forces (e.g., van der Waals forces) between gas molecules cause the gas to behave less ideally.

  Effect of High Pressure:

  • At high pressure, the volume of the gas is less than predicted by the Ideal Gas Law. This happens because the molecules are packed together, and the space they occupy is more significant.
  • Attractive forces between molecules cause a reduction in the overall pressure because the molecules don’t collide with the walls of the container as frequently (due to the attractive forces slowing them down).

b) Deviations at Low Temperature:

• Low Temperature means gas molecules move more slowly, and intermolecular forces become more significant.

• As molecules slow down, attractive forces (like van der Waals forces) cause the gas to condense, and the gas no longer behaves ideally.

  Effect of Low Temperature:

  • At low temperatures, real gases tend to condense into a liquid because intermolecular forces are more prominent. The Ideal Gas Law assumes that there’s no attraction between gas molecules, but in reality, attractive forces become stronger as the molecules move slower at lower temperatures.
  • The pressure of a real gas is lower than predicted by the Ideal Gas Law because the molecules are not moving as fast, and attractive forces decrease the frequency of collisions with the container walls.

iii. Van der Waals Equation: A Correction to the Ideal Gas Law:

To account for the deviations of real gases from ideal behavior, we use the Van der Waals equation, which includes two correction factors:

• Correction for intermolecular attractions: Attracting forces between gas molecules reduce the pressure, so the pressure term is adjusted.
• Correction for molecular volume: The finite volume of gas molecules is accounted for, as real gas molecules occupy space.

The Van der Waals equation is:

$$\left(P + \frac{a}{V^2}\right)(V – b) = nRT$$

Where:

P = Pressure

$\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;">V\; =\; Volume\; of\; the\; gas</span>}$ $n$

n = Number of moles

T = Temperature

R = Ideal gas constant

a = A constant that accounts for the attractive forces between molecules

$\mathrm{<span\; style="font-size:\; 16px;">b\; <span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">=\; A\; constant\; that\; accounts\; for\; the\; finite\; volume\; of\; gas\; molecules</span></span>}$

iv. Why Do Real Gases Deviate?

• Intermolecular forces: Real gas molecules attract or repel each other. For example, gases like oxygen (O₂) or nitrogen (N₂) exhibit weak London dispersion forces, while water vapor (H₂O) experiences hydrogen bonding, which leads to more significant deviations from ideal gas behavior.

• Molecular size: The molecules of real gases have a finite size, meaning that the volume occupied by the molecules themselves must be accounted for, especially at high pressures.

• Temperature: At low temperatures, the attractive forces become more significant as the gas molecules move more slowly. This causes real gases to condense, deviating from ideal behavior.

v. Conditions Under Which Gases Behave Ideally:

Real gases behave more ideally when:

• High temperature: Molecules move faster, and intermolecular forces have less impact.
• Low pressure: Molecules are far apart, so their volume and intermolecular forces become negligible.
• Nonpolar gases: Gases with weak intermolecular forces (e.g., noble gases like helium, neon) behave more ideally than those with strong intermolecular forces (e.g., hydrogen bonding in water vapor).