Molecular Kinetic Theory Model 2

Test yourself

Average molecular kinetic energy

The average molecular energy can be used to find the pressure of radiation.

Kinetic energy

We assume that a gas is homogeneous (the same everywhere) and isotropic (has the same value when measured in different directions).
This means that the average speed of a molecule in the x-direction must be the same as the y and the z-directions. So we can write:
- $\langle{c_x}\rangle=\langle{c_y}\rangle=\langle{c_z}\rangle$
This means that the average kinetic energy in each direction is the same.

Momentum

We consider a cube of length l which contains a gas.
If we only look at the x-direction, we can say that the time taken for a particle to travel from one end to the other is:
- $t=\frac{l}{\langle{c_x}\rangle}$
If a particle collides with the wall of the box and is absorbed, we can write the change in momentum as:
- Change in momentum = mass x velocity
- $P=m\langle{c_x}\rangle$

Force

We can write the force exerted on the wall when it absorbs a particle as:
- Force = change in momentum ÷ change in time
- $F=\frac{P}{t}$
We can substitute in the equations for momentum and time to get:
- $F=\frac{m\langle{c_x}^2\rangle}{l}$

Pressure

The pressure on the wall of the box when it absorbs one particle is given by the equation:
- Pressure = force ÷ area
- $p=\frac{F}{A}$
We can use the value for force and that the area is length squared to get:
- $p=\frac{m\langle{c_x}^2\rangle}{l^3}$
Since volume is length cubed we get:
- $p=\frac{m\langle{c_x}^2\rangle}{V}$

Pressure 2

We have found the pressure for one particle if N particles hit the wall the pressure would be:
- Total pressure = number of particles x pressure from one particle.
- ${p_t}=Np$
We can substitute the pressure in to give:
- ${p_t}=N\frac{m\langle{c_x}^2\rangle}{V}$

Kinetic equation

Recalling that the velocity is the same in all directions, we can write an equation for the total velocity squared as:
- $\langle{c}^2\rangle= \langle{c_x}^2\rangle+\langle{c_y}^2\rangle+\langle{c_z}^2\rangle$
Because the velocities are equal, we can simplify to give:
- $\langle{c_x}^2\rangle=\frac{1}{3}\langle{c_t}^2\rangle$
So the equation for pressure can be written as:
- $pV=\frac{1}{3}Nm\langle c^2\rangle$

Kinetic Energy of a Molecule

Using the results we obtained from the molecular kinetic theory derivation, we can find the temperature dependence of the kinetic energy of a molecule.

Ideal gas law

Recall that the ideal gas law in terms of the Boltzmann constant, $k$ is:
- $pV=NkT$
And the equation of the molecular kinetic theory is:
- $pV = \frac{1}{3} Nm\langle c^2\rangle$
Equating the two gives:
- $NkT=\frac{1}{3}Nm\langle c^2\rangle$

Kinetic energy of a molecule

We can cancel a factor of $N$ from each side to give:
- $kT=\frac{1}{3}m\langle c^2\rangle$
Finally, multiplying each side by $\frac{3}{2}$ , we obtain an expression for the average kinetic energy for a molecule:
- $\frac{1}{2}m\langle c^2\rangle = \frac{3}{2}kT$
The average kinetic energy for a molecule is proportional to the temperature.

Molecular Kinetic Theory - Developments Over Time

Ideas about the underlying structure of materials have changed considerably over time.

Early ideas

The idea that the atom, or atoms that are too small to directly view, move around is ancient. This dates from Lucretius in approx 50 BCE.
This idea was largely ignored because of the predominance of Aristotelian ideas about elements such as fire, earth, air and water.

Particle nature of matter

The modern theory of the particle nature of matter was attributed to Bernoulli.
This was prior to the ideas of conservation of energy and how collisions between particles could be elastic.
The model did not predict anything in itself.

More complex ideas

Clausius developed a much more complex set of ideas.
The observation of Brownian motion is the first direct evidence of the existence of particles.
The kinetic theory of gases became universally accepted because of Einstein and Smoluchowski’s theoretical model, which made specific predictions about Brownian motion and diffusion.

1Physical Quantities & Units

1.1Physical Quantities & Units

1.1.1Use of SI Units

1.1.2SI Prefixes, Standard Form & Converting Units

1.1.3End of Topic Test - Units & Prefixes

1.1.4Avogadro's Constant

1.1.5Scalars & Vectors

2Measurement Techniques

2.1Measurement, Errors & Uncertainty

2.1.1Measuring Length, Time & Volume

2.1.2Measuring Temperature

2.1.3Circuit Measurements

2.1.4Limitation of Physical Measurements

2.1.5Uncertainty

2.1.6Estimation

2.1.7Displaying Data

2.1.8End of Topic Test - Measurements & Errors

3Kinematics

3.1Equations of Motion

3.1.1Motion in a Straight Line

3.1.2Graphs of Motion

3.1.3Bouncing Ball Example

3.1.4End of Topic Test - Motion in a Straight Line

3.1.5Acceleration Due to Gravity

4Dynamics

4.1Momentum & Newton's Laws of Motion

4.1.1Mass & Inertia

4.1.2Newton's Laws

4.1.3Momentum

4.1.4End of Topic Test - Newton's Laws & Momentum

4.1.5A-A* (AO3/4) - Newton's Third Law

4.2Non-Uniform Motion

4.2.1Projectile Motion

4.3Linear Momentum & Conservation

4.3.1Conservation of Momentum

4.4Force, Density & Pressure

4.4.1Fields

4.4.2Force in Uniform Fields

4.4.3Friction

4.4.4Buoyancy

4.4.5Terminal Speed

4.4.6End of Topic Test - Acceleration Due to Gravity

4.4.7Centre of Mass

4.4.8Forces & Equilibrium

4.4.9End of Topic Test - Scalars & Vectors

4.4.10Moments

4.4.11End of Topic Test - Moments & Centre of Mass

4.4.12Density

4.4.13Pressure

4.5Work, Energy & Power

4.5.1Work & Energy

4.5.2Power & Efficiency

4.5.3Conservation of Energy

4.5.4End of Topic Test - Work, Energy & Power

4.6Circular Motion

4.6.1Circular Motion

4.6.2Circular Motion 2

4.6.3End of Topic Test - Circular Motion

5Gravitational Fields

5.1Gravitational Fields (A2 only)

5.1.1Newton's Law

5.1.2Gravitational Field Strength

5.1.3Gravitational Potential

6Deformation of Solids

6.1Materials

6.1.1Bulk Properties of Solids

6.1.2Energy in Materials

6.1.3Young Modulus

6.1.4End of Topic Test - Materials

7Thermal Physics

7.1Thermal Physics

7.1.1Temperature

7.1.2Measuring Temperature

7.1.3Ideal Gas Law

7.1.4Ideal Gases

7.1.5Boyle's Law & Charles' Law

7.1.6Molecular Kinetic Theory Model

7.1.7Molecular Kinetic Theory Model 2

7.1.8Thermal Energy Transfer

7.1.9Thermal Energy Transfer Experiments

7.1.10End of Topic Test - Thermal Energy & Ideal Gases

7.1.11First Law of Thermodynamics

8Oscillations

8.1Simple Harmonic Motion

8.1.1Simple Harmonic Motion

8.1.2Simple Harmonic Systems

8.1.3Energy in Simple Harmonic Motion

8.1.4Resonance

8.1.5End of Topic Test - Simple Harmonic Motion

8.1.6A-A* (AO3/4) - Simple Harmonic Motion

8.1.7End of Topic Test - Gravitational Fields

8.2Waves

8.2.1Progressive Waves

8.2.2Intensity of Waves

8.2.3Wave Speed & Phase Difference

8.2.4Longitudinal & Transverse Waves

8.2.5End of Topic Test - Progressive Waves

8.2.6Electromagnetic Waves

8.2.7Doppler Effect

8.2.8Sound Waves

8.2.9Measuring Sound Waves

8.2.10End of Topic Test - Waves

8.2.11Ultrasound Imaging

8.2.12Ultrasound Imaging 2

8.3Superposition

8.3.1Stationary Waves

8.3.2Stationary Waves 2

8.3.3End of Topic Test - Stationary Waves

8.3.4A-A* (AO3/4) - Stationary Waves

8.3.5Interference

8.3.6Interference 2

8.3.7End of Topic Test - Interference

8.3.8Diffraction

8.3.9Diffraction Gratings

8.3.10End of Topic Test - Diffraction

9Communication

9.1Communication Channels

9.1.1Communication Channels

9.1.2Modulation

9.1.3Attenuation

9.2Digital Communication

9.2.1Digital & Analogue Signals

9.2.2Representing Sound

10Electric Fields

10.1Electric Fields

10.1.1Coulomb's Law

10.1.2Electric Field Strength

10.1.3Electric Field Strength 2

10.1.4Electric Potential

10.1.5End of Topic Test - Electric Fields

10.1.6A-A* (AO3/4) - Electric and Gravitational Field

10.2Capacitance

10.2.1Capacitance

10.2.2Parallel Plate Capacitor

10.2.3Energy Stored by a Capacitor

10.2.4Capacitor Discharge

10.2.5Capacitor Charge

11Current Electricity

11.1Current Electricity

11.1.1Basics of Electricity

11.1.2Mean Drift Velocity

11.1.3Current-Voltage Characteristics

11.1.4End of Topic Test - Basics of Electricity

11.1.5Resistivity

11.1.6End of Topic Test - Resistivity & Superconductors

11.1.7Power and Conservation

11.1.8Microphones

11.1.9Components

11.1.10Relays

11.1.11Strain Gauges

11.2DC Circuits

11.2.1Circuits

11.2.2Potential Divider

11.2.3Emf & Internal Resistance

11.2.4End of Topic Test - Power & Potential

12Magnetic Fields

12.1Magnetic Fields

12.1.1Magnetic Flux Density

12.1.2End of Topic Test - Capacitance & Flux Density

12.1.3Moving Charges in a Magnetic Field

12.1.4Magnetic Flux & Flux Linkage

12.1.5Electromagnetic Induction

12.1.6Electromagnetic Induction 2

12.1.7Magnetic Flux Density

12.1.8Magnetic Resonance Scanning

12.2Alternating Currents

12.2.1Alternating Currents

12.2.2Operation of a Transformer

12.2.3End of Topic Test - Electromagnetic Induction & AC

12.2.4Rectification

13Modern Physics

13.1Quantum Physics

13.1.1The Photoelectric Effect

13.1.2The Photoelectric Effect Explanation

13.1.3End of Topic Test - The Photoelectric Effect

13.1.4Collisions of Electrons with Atoms

13.1.5Energy Levels & Photon Emission

13.1.6Wave-Particle Duality

13.1.7End of Topic Test - Absorption & Emission

13.1.8Band Theory

13.1.9Diagnostic X-Rays

13.1.10X-Ray Image Processing

13.1.11Absorption of X-Rays

13.1.12CT Scanners

13.2Nuclear Physics

13.2.1Rutherford Scattering

13.2.2Atomic Model

13.2.3Isotopes

13.2.4Stable & Unstable Nuclei

13.2.5A-A* (AO3/4) - Stable & Unstable Nuclei

13.2.6Alpha & Beta Radiation

13.2.7Gamma Radiation

13.2.8Particles, Antiparticles & Photons

13.2.9Quarks & Antiquarks

13.2.10Particle Interactions

13.2.11Radioactive Decay

13.2.12Half Life

13.2.13End of Topic Test - Radioactivity

13.2.14Nuclear Instability

13.2.15Mass & Energy

13.2.16Binding Energy

13.2.17A-A* (AO3/4) - Nuclear Fusion

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