Kinetic Theory of Gases and Radiation - Introduction and Gas Laws

Foundation and Recalled Concepts

Before delving into the Kinetic Theory of Gases and Radiation, it is essential to establish the foundational knowledge required for understanding the behavior of matter at macroscopic and microscopic levels.

• States of Matter: Matter exists in three common states: solid, liquid, and gas.
• Distinguishing States: The solid, liquid, and gaseous states are distinguished by the arrangement, energy, and bonding of their constituent particles.
• Gas Laws: These are empirical laws that describe how the pressure, volume, and temperature of a gas relate to one another.
• Absolute Zero Temperature: The theoretical lowest temperature possible where all molecular motion ceases ($0\,K$ or $-273.15\,^{\circ}C$).
• Avogadro Number ($N_A$): The number of constituent particles (usually atoms or molecules) in one mole of a substance.
• The Mole: A unit of measurement for the amount of substance; one mole of a gas contains a number of molecules equal to the Avogadro number.
• Ideal Gas Equation Derivation: Combining the individual gas laws (Boyle’s, Charles’, and Gay-Lussac’s) results in the unified ideal gas equation.
• Ideal vs. Real Gases: Ideal gases are theoretical constructs that perfectly follow gas laws under all conditions, whereas real gases exhibit deviations, especially at high pressures and low temperatures.
• Elastic Collision: A collision in which there is no net loss in kinetic energy in the system as a result of the collision.
• Dalton's Law of Partial Pressures: The total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of individual gases.

Introduction to Gas Laws

The behavior of gases is fundamentally described by three primary laws, applicable to a fixed mass ($m$) of an enclosed gas:

• Boyle's Law: The volume ($V$) of a gas is inversely proportional to the pressure ($P$) when the temperature ($T$) is held constant.     $V \propto \frac{1}{P} \text{ at constant T}$

• Charles' Law: The volume ($V$) of a gas is directly proportional to its temperature ($T$) when the pressure ($P$) is held constant.     $V \propto T \text{ at constant P}$

• Gay-Lussac's Law: The pressure ($P$) of a gas is directly proportional to its temperature ($T$) when the volume ($V$) is held constant.     $P \propto T \text{ at constant V}$

The Ideal Gas Equation and State Variables

Combining the three laws for a fixed mass of gas yields a single relation known as the ideal gas equation:

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

For comparing two states of the same gas: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

Molar Representation

When expressing the fixed mass in terms of the number of moles ($n$), the relationship becomes: $PV = nRT$

Where:

• $n$ (Number of Moles): Defined by the ratio of the mass of the gas ($M$) to its molar mass ($M$), or the number of molecules ($N$) to the Avogadro number ($N_A$).     $n = \frac{M}{M} = \frac{N}{N_A}$
• Molar Mass: This is defined as the mass of one mole of gas.
• $R$ (Universal Gas Constant): A constant applicable to all gases, with a value of $8.314\,J\,mol^{-1}\,K^{-1}$.
• $N_A$ (Avogadro Number): The number of molecules contained in one mole of gas.

Molecular Representation

Alternatively, the ideal gas equation can be written in terms of the total number of molecules ($N$): $PV = Nk_B T$

Where:

• $k_B$ (Boltzmann Constant): A constant related to the universal gas constant and Avogadro number by the following relation:     $R = N_A k_B$

Real Gases vs. Ideal Gases

• Validity of Gas Laws: The laws established by Boyle, Charles, and Gay-Lussac are strictly valid for real gases only under specific conditions: the pressure must not be too high, and the temperature must not be close to the liquefaction temperature of the gas.
• Definition of an Ideal Gas: A gas that obeys the equation of state $PV = nRT$ at all pressures and all temperatures is considered an ideal gas.
• Equation of State: For any gas, the state is specified by physical quantities such as pressure ($P$), temperature ($T$), volume ($V$), and internal energy ($E$). The mathematical relationship between these quantities is known as the equation of state.

Macroscopic vs. Microscopic Behavior of Gases

Understanding gas behavior requires distinguishing between different scales of description:

• The Macroscopic Description: A gas enclosed in a container is characterized by bulk properties such as pressure, volume, and temperature.
• The Microscopic Description: At the particle level, gas molecules are in constant motion. While the motion of a single macroscopic object, like a stone thrown upwards, can be described by Newton’s laws of motion to predict its height and return to ground, describing a gas via individual particle motion is practically impossible.
• Scale and Complexity: The number of particles in a gas is extremely large, approximately $10^{23}\,\text{particles per m}^3$. Because of this scale, attempting to relate macroscopic parameters ($P, V, T, E$) to the individual motion of every single particle is futile. Instead, the behavior is understood through averages and statistical approaches.

Questions & Discussion

1. What are different states of matter? Common states include solid, liquid, and gas. (Other states like plasma exist but are not the primary focus here).

2. How do you distinguish between solid, liquid and gaseous states? Distinction is made based on molecular spacing, intermolecular forces, and whether the substance has a definite shape or volume.

3. What are gas laws? They are the mathematical relationships (Boyle's Law, Charles' Law, Gay-Lussac's Law) that describe the behavior of gas properties ($P, V, T, n$).

4. What is absolute zero temperature? The zero point on the Kelvin scale ($0\,K$), equal to $-273.15\,^{\circ}C$.

5. What is Avogadro number? What is a mole? Avogadro number ($N_A$) is the count of particles in one mole. A mole is the unit for amount of substance.

6. How do you get ideal gas equation from the gas laws? By combining the proportionalities of Boyle's Law ($V \propto \frac{1}{P}$) and Charles' Law ($V \propto T$) into a single expression $PV \propto T$, and introducing the gas constant.

7. How is ideal gas different from real gases? An ideal gas follows $PV = nRT$ perfectly under all conditions, whereas real gases only approximate this behavior at low pressures and high temperatures.

8. What is elastic collision of particles? A collision where total kinetic energy and momentum are conserved.

9. What is Dalton's law of partial pressures? It states that the total pressure of a gas mixture is the sum of the pressures each gas would exert if it were alone in the container.