Molecular Kinetic Theory Model

1. Brownian motion:

• Brownian motion is the random movement of particles suspended in a fluid (such as a gas or liquid) due to collisions with surrounding fluid molecules. This phenomenon was first observed by Robert Brown in 1827 and later explained by Albert Einstein in 1905.
• 
  Figure 1 Brownian motion (random movement of particles)
• Kinetic Molecular Theory
• The kinetic molecular theory posits that:
  1. Matter is composed of tiny particles (atoms or molecules) in constant motion.
  2. The motion is due to thermal energy, which increases with temperature.
  3. Particles collide and transfer momentum, leading to random motion.
• Einstein’s Explanation:
• Einstein’s mathematical model of Brownian motion assumed:
  1. The fluid is composed of tiny molecules in constant motion.
  2. Suspended particles are bombarded by surrounding fluid molecules, causing random motion.
  3. The motion is characterized by rapid changes in direction and speed.
• Mathematical Derivation:
• Einstein derived the following equation to describe Brownian motion:
• [math] x = \sqrt{2Dt} [/math]
• where:
  – x is the displacement of the particle
  – D is the diffusion coefficient
  – t is time
• Predictions and Confirmation:
• Einstein’s theory predicted:
  1. The mean squared displacement of particles would increase linearly with time.
  2. The distribution of particle displacements would follow a Gaussian distribution.
  3. The speed and frequency of particle movement would increase with temperature.
• Experiments confirmed these predictions, providing strong evidence for the kinetic molecular theory and the existence of atoms.
• Implications:
• Brownian motion:
  1. Provided direct visual evidence for the existence of atoms and molecules.
  2. Confirmed the kinetic molecular theory, establishing the foundation for modern physics and chemistry.
  3. Demonstrated the power of statistical mechanics in predicting physical phenomena.
• Brownian motion, explained by Einstein’s mathematical model, offers compelling evidence for the existence of atoms.
• The kinetic molecular theory, supported by experimental confirmation, forms the basis of our understanding of the physical world.

2. Pressure, volume and temperature of a simple molecular model:

• One should not undervalue the significance of ideal gas characteristics and Brownian motion.
• A microscopic molecular model may be used to explain the macroscopic gas law quantities and, consequently, the gas laws themselves, thanks to Einstein’s theory and the observation of the random motion of fluid particles.

• ⇒ Pressure:

• A force operating across a certain region is used to describe macroscopic pressure.
• The force is caused by the molecules colliding with the container walls, as demonstrated by the kinetic theory model of an ideal gas.
• The molecules have a mean average velocity and are travelling in random directions.
• Due to the elastic nature of all impacts, the particles strike the container walls and bounce back at the same speed. This results in a change in momentum.
• Because the motion is random, the force per unit area created by all the particles impacting throughout the container’s whole interior surface area per second exerts pressure in all directions.

• ⇒ Volume:

• Molecules moving within a container travel in an unpredictable path.
• As a result, the molecules will disperse across the container, filling its capacity, with no preferred orientation.
• Gases absorb the volume of the space they occupy.
• The molecules’ mobility will adapt to any changes in the container’s size and will keep filling the available capacity.
• At low pressures, actual gases behave most as ideal gases do, staying well outside of their phase boundary where they transform into liquids.

• ⇒ Temperature:

• Since there are no intermolecular forces for an ideal gas, increasing the temperature of the working fluid means that you can only increase or change kinetic energy to do work.
• This will increase the average velocity of those particles.
• Those particles are still just moving in various random ways and fill up the container. This will increase the pressure, due to a greater average speed of particles and therefore an increased change in momentum during the particle-wall collision pressures push which cause p = F / A is higher where increasing number of particles hitting against walls also. This results in greater forces and thus pressure.
• For constant pressure the volume will need to change.

• ⇒ Relationship:

1. P-V Relationship: At constant temperature (T), increasing volume (V) reduces pressure (P), as molecules have more space to move and collide less frequently. Decreasing volume increases pressure, as collisions become more frequent.
2. P-T Relationship: At constant volume (V), increasing temperature (T) increases pressure (P), as molecules move faster and collide more energetically. Decreasing temperature reduces pressure.
3. V-T Relationship: At constant pressure (P), increasing temperature (T) increases volume (V), as molecules expand and occupy more space. Decreasing temperature reduces volume.
• These relationships form the basis of the ideal gas law:
• [math] PV = nRT [/math]
• where n is the number of moles and R is the gas constant.
• This simple molecular model helps us understand the relationships between pressure, volume, and temperature, laying the foundation for more advanced topics in physics and chemistry.

3. The kinetic theory model:

• Empirical Gas Laws:
• The gas laws (Boyle’s Law, Charles’ Law, Avogadro’s Law, and Gay-Lussac’s Law) are empirical, meaning they’re derived from experimental observations and data. These laws describe the relationships between pressure, volume, temperature, and amount of gas, but they don’t provide a theoretical explanation for why these relationships exist.
• Kinetic Theory:
• In contrast, the kinetic theory (also known as the kinetic molecular theory) is a theoretical framework that attempts to explain the behavior of gases at a molecular level. It posits that gases consist of tiny molecules in constant motion, and their behavior is governed by statistical mechanics. The kinetic theory provides a conceptual understanding of gas behavior, but it’s a theoretical model, not a direct empirical observation.
  1. Origin: Gas laws come from experimentation, while kinetic theory arises from theoretical reasoning.
  2. Purpose: Gas laws describe relationships, whereas kinetic theory explains the underlying mechanisms.
  3. Scope: Gas laws apply specifically to ideal gases, whereas kinetic theory attempts to explain gas behavior in general.
  4. Assumptions: Gas laws don’t require assumptions about molecular structure or behavior, whereas kinetic theory relies on assumptions about molecular motion and interactions.
• By recognizing the distinction between empirical gas laws and theoretical kinetic theory, students can better appreciate the strengths and limitations of each approach and develop a deeper understanding of gas behavior.
• Let us consider a particle that moves with a velocity C₁ parallel to the x-axis. The shaded wall shows where the particle collides.

  
  Figure 2 particle in a box

• In the ideal gas theory, it is assumed that this collision is perfectly elastic; since so the particle bounces back of wall with a velocity -C₁.
• The net change in momentum of the particle during this collision thus equals 2mc₁, regardless of which analysis you prefer.
• If the assumption of totally elastic collision is false, then during collisions particles will lose some amount energy and with this their average velocity in the box decreases so that reduces overall gas pressure. From experimental evidence, we know that this is not the case
• The particle then bounces back to the other wall at velocity C₁ and collides with lace as seen in time interval [math] \Delta t = \frac{2L}{c_1} [/math] , a force exerted by the particles stops it when after somepoint reacheswalls. That is, If the particle satisfies Newton’s Second Law of motion

• [math] \text{Force} = \frac{\text{change in momentum}}{\text{time for change}} \\ F = \frac{2mc_1}{\left(\frac{2L}{c_1}\right)} \\
  F = 2mc_1 * \frac{ c_1}{2L} \\
  F = \frac{m c_1^2}{L} [/math]
• The shaded wall has an area,[math] A=L^2 [/math] , so the pressure exerted by the one particle is
• [math] p = \frac{F}{A} = \frac{m c_1^2}{L^3} [/math]
•  There are N particles in the box, and if they were all travelling parallel to the a-axis: total pressure on the shaded wall would be
• [math] p = \frac{m}{L^3} (C_1^2 + C_2^2 + C_3^2 + \cdots + C_N^2) [/math]

In actuality, however, particles move randomly, with a velocity, (C), made up of elements in the x, y, and z directions [math](C_x ,C_z, \text{ and } C_z) [/math] that are at right angles to one another. The three-dimensional theorem is used, [math] C^2 = C_x^2 + C_y^2 + C_z^2 [/math]  but on average, [math] C_x^2 = C_y^2 = C_z^2, \text{ so } C_x^2 = \frac{1}{3} C^2 [/math]. As there are N particles in the box, the pressure, p, parallel to the x-axis is therefore

[math] p = \frac{1}{3} * \frac{m}{L^3} (C_1^2 + C_2^2 + C_3^2 + \cdots + C_N^2) [/math]

We now define a quantity called the root mean square velocity, (C_rms),
(the square root of the average of the square velocities) where:

[math] C_{\text{rms}} = \sqrt{\frac{C_1^2 + C_2^2 + C_3^2 + \cdots + C_N^2}{N}} \, \text{so} \, N (C_{\text{rms}})^2 = C_1^2 + C_2^2 + C_3^2 + \cdots + C_N^2 [/math]

Substituting and replacing [math] L^3 = V [/math] , gives:

[math] pV = \frac{1}{3} N m (C_{\text{rms}})^2[/math]

Nm is the total mass of the gas inside the box so the density of the gas inside the box is given by:

[math] \text{Density} = \frac{\text{total mass}}{\text{Volume}} \\
\rho = \frac{Nm}{V} [/math]

So,

[math] p = \frac{1}{3} \frac{Nm}{V} (C_{\text{rms}})^2 [/math]
[math] p = \frac{1}{3} \rho (C_{\text{rms}})^2 [/math]

4. A simple algebraic in Conservation of momentum:

• Consider a container with a gas and a piston. Assume the piston moves a distance dx, and the gas molecules collide with the piston, transferring momentum.

  
  Figure 3 A container with a gas
  and a piston

• Momentum Transfer:
  The momentum transfer (change in momentum (dp)) is equal to the force (F) times the time (change in time (dt)):
• [math] dp = F dt \qquad  (1) [/math]
• Conservation of Momentum:
  The momentum transfer is equal to the change in momentum of the gas molecules (dmv):
• [math] dp = dmv  \qquad (2) [/math]
• Force and Pressure:
  The force (F) is equal to the pressure (P) times the area (A):
• [math] F = PA \qquad (3) [/math]
• Velocity and Distance:
  The velocity (v) is equal to the distance (dx) divided by the time (dt):
• [math] v = \frac{dx}{dt} [/math]
• Substitution:
  Substitute the expressions for F and v into the momentum transfer equation:
• [math] dp = PA \, dt = dmv [/math]
• Algebraic Manipulation:
  Rearrange and simplify the equation:
• [math] P = \frac{dm}{dt} * \frac{v}{A} [/math]
• Conservation of Mass:
  Assume the mass (m) is constant, so dm/dt = 0:
• [math] P = m *  \frac{v}{A} [/math]
• Velocity and Pressure:
  Solve for v:
• [math] v = \frac{P}{\frac{m}{A}} [/math]
• This derivation shows that pressure (P) is proportional to velocity (v). The constant of proportionality is the mass per unit area (m/A).
• This algebraic approach using conservation of momentum provides a simple and intuitive understanding of the relationship between pressure and velocity.

5. Ideal gas internal energy:

• For an ideal gas, the internal energy (U) is indeed solely comprised of the kinetic energy of the atoms or molecules. This is because ideal gases are assumed to have no intermolecular interactions, potential energy, or other forms of internal energy.
• Kinetic Energy:
  The kinetic energy of the atoms or molecules is due to their translational motion. In an ideal gas, the atoms or molecules are free to move and collide with each other, and their kinetic energy is directly proportional to the temperature (T) of the gas.
• Internal Energy:
  The internal energy (U) of an ideal gas is therefore equal to the total kinetic energy of all the atoms or molecules:
• [math] U = \frac{1}{2} mv^2 [/math]
• where m is the mass and v is the velocity of the atoms or molecules.
• Ideal Gas Law:
• This relationship can be combined with the ideal gas law:
  
• [math] PV = nRT [/math]
• where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.
• Equating Expressions:
• Equating the two expressions for internal energy, we get:
  
• [math] \frac{1}{2} mv^2  = nRT [/math]
• This shows that the internal energy of an ideal gas is directly proportional to the temperature, and that the kinetic energy of the atoms or molecules is the sole contributor to the internal energy.
• This appreciation is fundamental to understanding the behavior of ideal gases and the underlying principles of thermodynamics.

6. Comparing two models of the behavior of gases:

• Two theories that explain a gas’s behavior are now available to us.
• The first model, known as the ideal gas equation, explains the macroscopic, experimental behavior of the gas:
• [math] pV = nRT [/math]
• The second model, which uses the kinetic theory model, uses a tiny, mechanical model of the particles to theoretically explain the behavior.
• [math] p = \frac{1}{3} \rho (C_{rms})^2 \\
  p = \frac{1}{3} \frac{N m}{V} (C_{rms})^2 [/math]
• If the average molecular kinetic energy is (the bar above the quantity means ‘mean average’) the [math] E_k = \frac{1}{m} m (C_{rms})^2 [/math] and so
• [math] p = \frac{(2)1}{(2)3}  \frac{N m}{V} (C_{rms})^2 \\
  p = \frac{2}{3} \frac{N}{V} * \left(\frac{1}{2} m (C_{rms})^2 \right) \\
  pV = \frac{2}{3} N * \overline{E_k} [/math]
• If this equation is compared to the ideal gas equation
  
• [math] pV = nRT [/math]
• then
• [math] nRT = \frac{2}{3} N \overline{E_k} \\ \overline{E_k} = \frac{3}{2} \frac{n}{N} RT [/math]
• Because n is the number of moles of the gas and N is the number of particles of the gas, then
• [math] N = n N_A \quad \text{or} \quad \frac{n}{N} = \frac{1}{N_A} [/math]
• Substituting this for n/N gives
• [math] \overline{E_k} = \frac{3}{2} \frac{R}{N_A} T [/math]
• But has already been defined as equal to k, the Boltzmann constant, which effectively is the gas constant per particle of gas. So
• [math] \overline{E_k} = \frac{3}{2} kT [/math]
• Where E_k represents the kinetic energy of a single gas particle. This is a really impressive conclusion. Starting with three readily measured, macroscopic parameters of a gas, we arrive to a straightforward equation that lets us determine the kinetic energy of a gas particle by only observing its temperature.

Examples:

Particles of neon in a bulb:
Neon particles are present in neon-filled light bulbs used in advertising signs at a temperature of 60°C and a pressure of 1.03*10⁵Pa. Neon has a molar mass of 20.2 g mol^-1. Determine the neon’s density within the lightbulb.

Solution:

First calculate the kinetic energy o the individual neon particles:

[math]
\overline{E_k} = \frac{3}{2} k T\\
\overline{E_k} = \frac{3}{2} (1.38 \times 10^{-23}) (273 + 60) \\
\overline{E_k} = 6.89 \times 10^{-21} \, \text{J}
[/math]

This can then be used to calculate the value of

[math]
\overline{E_k} = \frac{1}{2} m C_{rms}^2 \\
|C_{rms}|^2 = \frac{2E_k}{m} \\
|C_{rms}|^2 = \frac{2E_k}{M_m / N_A} \\
|C_{rms}|^2 = \frac{2E_k N_A}{M_m} \\
|C_{rms}|^2 = \frac{2(6.89 \times 10^{-21}) (6.02 \times 10^{23})}{20.2 \times 10^{-3}} \\
|C_{rms}|^2 = 408648 \, \text{m}^2 \, \text{s}^{-2} \approx 4.1 \times 10^5 \, \text{m}^2 \, \text{s}^{-2}
[/math]

Substituting into

[math]
p = \frac{1}{3} \rho |C_{rms}|^2 \\
\rho = \frac{3p}{|C_{rms}|^2} \\
\rho = \frac{3 \times 1.03 \times 10^5}{4.1 \times 10^5} \\
\rho = 0.76 \, \text{kg} \, \text{m}^{-3} [/math]

7. The behavior of a gas has changed over time:

• Let’s dive deeper into the evolution of our understanding of ideal gas behavior:
  1. Boyle’s Law (1662): Robert Boyle discovered the pressure-volume relationship, laying the foundation for ideal gas behavior.
  2. Charles’ Law (1787): Jacques Charles identified the temperature-volume relationship, further expanding our understanding.
  3. Avogadro’s Law (1811): Amedeo Avogadro revealed the connection between the number of molecules and volume, introducing the concept of an ideal gas.
  4. Kinetic Theory (late 19th century): Scientists like Maxwell, Boltzmann, and Einstein developed the kinetic theory, explaining ideal gas behavior using statistical mechanics.
  5. Ideal Gas Equation (1904): The combination of Boyle’s, Charles’, and Avogadro’s Laws led to the derivation of the ideal gas equation: .
  6. Quantum Mechanics (20th century): Refinements to the kinetic theory using quantum mechanics provided a more detailed understanding of ideal gas behavior at the atomic and subatomic level.
  7. Modern Understanding: Today, we have a comprehensive understanding of ideal gas behavior, including:
     – Equations of state (e.g., ideal gas equation)
     – Thermodynamic properties (e.g., internal energy, entropy)
     – Microscopic behavior (e.g., molecular motion, collisions)
• Our understanding of ideal gas behavior has transformed significantly over time, from empirical observations to a rigorous, molecular-level comprehension.