Kinetic Theory of Gases (KTG) and Thermodynamics

Introduction to KTG and Thermodynamics

• Temperature must be in Kelvin (K) for all calculations.
• Understanding molecular behavior is crucial.
• Important to know formulas for problem-solving.
• Important for those aiming to achieve success.
• Covers basics, important formulas, and completes the theory.
• Useful for those with backlogs or needing crisp content.

Kinetic Theory of Gases (KTG)

• Developed by Clark Maxwell and Boltzmann.
• Helps understand gas behavior and gives molecular interpretation of pressure and temperature.
• Pressure and Temperature are Microscopic Variables.
• Explains Avogadro's law, specific heat capacity of gases.
• A successful model for ideal gases.

Ideal vs. Real Gases

• Ideal gases follow a specific equation of state.
• Real gases behave closely to ideal gases under low pressure and high temperature.

Gas Constant (R) and Boltzmann Constant (k)

• Gas constant R is related to Boltzmann constant k: R = Avogadro's number * k.
• Physics uses R = 8.31 Joules per mole per Kelvin (SI unit).
• Avogadro's number: 6.02 * 10^23 particles per mole.

Assumptions of KTG

• All gas molecules are identical.
• Molecules are rigid, perfectly elastic spheres with negligible volume.
• Molecules move randomly with varying speeds at any temperature above 0 Kelvin.
• No net force acts on molecules except during collisions.
• Molecules move in straight lines between collisions.
• Molecules have kinetic energy but no potential energy.
• Collision time is very small compared to the time between collisions.
• Average number of molecules per unit volume remains constant.
• Molecular distribution is uniform within the vessel.
• Gravity has negligible effect on molecular motion.

Pressure Derivation

• Consider a cube of side L containing an ideal gas.
• A molecule has velocities v_x, v_y, v_z in x, y, z directions.
• Molecule collides with a wall (yz-plane), velocity in x direction reverses due to elastic collision.
• Velocity before collision: v_x i + v_y j + v_z k.
• Velocity after collision: -v_x i + v_y j + v_z k.

Collision Mechanics

• Consider collision with wall in yz-plane.
• Change in momentum magnitude is 2m*v_x.
• Time between successive collisions with the same wall: delta_t = 2L / v_x.
• Each molecule collides once with all six surfaces in this time.
• The force exerted on the wall is the rate of change of momentum.

Relating Force to Number of Molecules

• Force F = N * (delta_p / delta_t), where N is total molecules.
• Capital N = number of moles * Avogadro's number.
• Substitute values to get F = M * (v_x)^2 / L, where M is total mass.
• Pressure P = F / Area = rho * (v_x)^2, where rho is density.

Equipartition Theorem

• (0.5)m(v_x)^2 = (0.5)m(v_y)^2 = (0.5)m(v_z)^2 = (0.5)kT.
• Velocity components are equal: (v_x)^2 = (v^2) / 3.
• Substitute into pressure equation: P = (1/3) * rho * (v^2).

Maxwell's Velocity Distribution

• It addresses what v should be used
• Instead of random velocities, use specific ones based on Maxwell curve,
  • Such as most probable, average, or root mean square (RMS) velocity.
• Use v_RMS in P = (1/3) * rho * (v^2).

RMS, Average, and most probable

• Formulas for different velocities:
  • v_RMS = sqrt(3RT / M_0)
  • v_average = sqrt(8RT / (pi * M_0))
  • v_most_probable = sqrt(2RT / M_0).
• Relationship: v_RMS > v_average > v_most_probable.
• Ratio: v_RMS : v_average : v_most_probable = sqrt(3) : sqrt(8/pi) : sqrt(2).

Replacing R

• R = Avogadro's number * k.
• M_0 = Avogadro's number * m , m is the molecular mass
• The formulas for velocities
• After substitution include RMS, Average, most probable

Key Temperature Error

• Always use Kelvin for temperature in thermodynamics and KTG.

Impact of Temperature

• Higher temperature broadens the distribution curve.

Relation b/w Pressure and KE

• Average kinetic energy is (3/2)PV.
• Pressure is (2/3) of kinetic energy per unit volume.
• Translational Kinetic Energy (KE) of a molecule is (3/2)kT.
• KE of one mole: (3/2)RT.

Mean Free Path

• Average distance between successive collisions.
• The Mean free path is given by symbol Lambda
• Lambda = 1 / (sqrt(2) * pi * d^2 * n'), where d is molecular diameter and n' is number of molecules per unit volume.
• n' = P / (kT), Lambda = kT / (sqrt(2) * pi * d^2 * P).
• Mean Free Time: time spent is (Lambda / v_average).
• Collision Frequency is mean free time
• Then given directly using parameters from free path
• If Volume is constant
  • Do NOT depend on Pressure and Temp
• Volume can Change
  • Lambda will be directly porportional to Temp
  • And inversely porportional to Pressure
• Graphs are mass versus mean path and Density versus mean path

Impact of Ideal Equation

• Boyles Charles etc etc can be solved using P V = n R T

Degrees of Freedom

• Independent motions a molecule can have.
• Types: translational, rotational, vibrational (or combinations).
• Translational: 3 degrees (x, y, z).
• Rotational: max 3 degrees.
• Vibrational: max 2 degrees (kinetic + potential energy).
• Monoatomic gas has 3 translational degrees of freedom.
• Diatomic has:
  • 3 transitional degrees
  • 2 rotational dof (Rotation possible about 2 axes)
  • Not about the axis with the molecules
• High temperature has additional dof but that depends.

Energy and Degrees of Freedom

• Energy of 1 molecule is is (f * (1/2) * k * T).
  • Where if if number of degrees of freedom
• Energy per mole is f / 2 * R * T , R is the gas constant
• Energy of N moles is N * F / 2 * R * T
• Hence change in Internal Temp, for a N mole gas, is deltaU = N * F / 2 * R * deltaT.

Degrees of freedom for combinations

• f_equivalent = summation(n_i * f_i) / summation(n_i).

Beginning Thermodynamics

• Thermo means heat/temp, dynamics means state change (movement)

Relating System and Surroundings

• System versus Surroundings
• System exchanges energy through:
  • DeltaQ Heat energy - flow
  • Mc - Mechanical Energy through Work
• Total energy exchange in systen:
  • Depends on Temp Exchange
  • DeltaQ = NCdeltaT
  • C, molar specific heat
    • Heat depends (or changes based on other parameters: 1 / M * dQ over dT
      • Depends calories per degree

First Law

• Delta Q is + if heat comes from outside of sustem
• Otherwise Q goes down
• Energy conservation is expressed as
• DeltaQ = internal energy of system + Work
• U is only function of TEMP
• Functions are called PROCESSES - iso blah blah batric

Specific Processes

• Isotonic vs isochoric
• Constant P : Volume and Temp proportional
• Constant V: Pressure Temperature Proportionate
• Isothermal, where temp constant
• Slow Changes happen with Isothermal: - PV and volume is constant (boyles law) - DeltaU is zero as well

Adiabatic

• A different Process - no heat energy exchange is possible
• Occurs very rapidly
  • When you pull a tube out fast
  • Suddent changes compress and expand gases

Cycles

• Returns to orig P V T, volume
• Combination Of all the above = closed cycle -
• Heat Energy into WD
• Polytropic
  • thousands of other processes
• Depends (relation b/w volume change

Indicatior diagrams

• Shows all related graphs
• Pressure V graph
• Isobar - P constant
• Isochoric is Volume

Delta Q

• Defined when entering sustem
• Depends on state - hence the path it has
• C molar function changes depending on the process and can be tricky
• Isoteric function P = constant in C P where P is onstant pressure
• Q where v is constant

Where the Specific process is Zero

• Adiabatic - that means it depends on energy

Where C is undiefned

• Isothermal process.

State Function

• DEltaU - it means it does not defend on oath only temp
• Hence: No matter the oath :
  • Delta u = N * f/r * delta T where f is degree

To relate

• Isometric Q Delta U
• That relates to V does not change
• Then C V degrees of FREEDOM

Meyers

• C p - C V = R

Relating Work Volume Pressure

• INtegrating PDeltaV to get energy
• Energy is path dependent

If Gas expands

• Positive work, pushing away surroundings
• NEG if goes in
• P to point 2 (area gives value of work)
• With Volume on X
• INdicator = tells type of process happening

First and Second law of thermo

• Based on energy
• No difference b/w work and heat with laws (not always)
• Only changes depend on Temp changes but NOT THE PROCESS
• First - direction of heat
  • Heat into mechanical

Free expansion

• Volume high work zero
• Volume increase but WORK is ZERO
• IN vaccume - push does nothing (no force)
• Rapid: Q and delta u also zero
• Hence intiial = final values

Recap

Cyclic Proicess (returns, cycles back) = back to PVT values (cycle, loops

How

• Constant volume at same space
• Constant pressure
• Or using heat energy to change
• Hence all internal parts become the same

Hence

• Summation delta U is zero
• With combined functions
• The work and delta u combine to a net zero
• If process reverses - same will happen

Heats

• Can only happen if works happens otherwise loops cant be completeed

Second law

Must release some heat - cant take take take - or vice versa Energy to lower side

Process and Steps For Loop

• If loops dont have more neg values not eng

Efficiency - output / input.

Recap.

Heat Engines Work

• Heat turns mechanical
• Happens thru clockwise.

Types of Engines

• To relate engines- is deltaHeat / over / total Heat

Carntot

• highest possible

In reverse IsoThermal -

A

• The Addaiabatic with the process where states of gases change

Recap Points on temp

T1 is heat of source or furnace

• Where heat taken T2 is sinv. T1 is a greater > than High /

Efficiency - one mint temp

• Only kevelin