Unit 3.3 - Kinetic theory Flashcards
In which century were a great number of scientific discoveries being published which relate to this unit?
17th century
How were natural laws named in the 17th century?
After the scientist who first published them
Who built the equipment for the experiments of that proves the laws mentioned in this unit?
Robert Hooke
Who lends his name to one of the laws in this unit?
Robert Hooke’s mento, Robert Boyle
Who designed and published the experiments that prove the laws mentioned in this unit
Robert Boyle
What did Robert Hooke do?
Drew the first diagram observed through a microscope of a cell
Hooke’s law for a spring
What does Boyle’s law do?
Describes the behaviour of a gas (e.g - air) under special conditions
Boyle’s law (worded)
At a constant temperature, the product of the volume and pressure of a gas is constant
What is inversely proportional to what according to Boyle’ law?
Volume and pressure
Boyle’s law in symbol form
pV=k
What does it mean that volume and pressure are inversely proportional according to Boyle’s law?
When one increases, the other decreases
Unit of pressure
Pa
Volume unit
m^3
What is k in pV=k?
A constant
Equation for a change in volume of pressure and gas, following Boyle’s law?
p1v1 = p2v2
(Temperature is constant)
Equation to use for gases when the temperature is constant
p1v1 = p2v2
Is Boyle’s law a conservation law? Why?
No. It is a special case occurring at a constant temperature
What does thermodynamics deal with?
The processes that cause a change in energy due to he flow of heat into/out of a system and/or work done on/by a system
What is it possible to do for many thermodynamic systems?
Describe the state by noting 2 variables only
What are the 2 variables we use for gas to describe the state?
Pressures and volume
What is the basis of Boyle’s law?
For many thermodynamic systems,it is possible to describe the state by noting two variables only (for a gas, we use pressure and volume) - this is the basis of Boyle’s law
Gay lussac’s law
At constant volume
T1/p1 = T2/p2
Charles law
At constant pressure
T1/V1 = T2/V2
Equation to use with a constant volume
T1/P1 = T2/P2
Equation to use with a constant pressure
T1/V1 = T2/V2
How can we get the equation of the state of an ideal gas?
All of the different relationships (the different gas laws) can be combined with a constant of proportionality and the variable of the number of moles of gas present to give the equation of state of an ideal gas
Ideal gas equation
pV = nRT
R in pV=nRT
Molar gas constant
Other way of writing Pa for pressure
Nm^-2
What are usually constant in pV = nRT?
n and R
Avogadro’s gas law
Equal volumes of all gases, at the same temperatures and pressure, have the same number of molecules
How many of the three variables in the ideal gas equation is the state of the gas defined by?
2 of the 3
What does it mean due to the fact that the state of a gas is defined by two of the three variables in the ideal gas equation?
Regardless of what has happened to the system, if the volume and pressure return to the same values, then the temperature must also be the same
Combined gas law
P1V1/T2 = P2V2/T2
What is the unit of Mr (relative molar mass)?
Grams
What unit to we need molar mass in in physics?
Kg
What do we do if were given relative molar mass?
Make sure we convert this value from grams to kg for out molar mass (that isn’t relative)
Number of molecules
Number of moles x Avogadro’s constant
How do we go from cm^3 to m^3?
X10^-6
Equation to calculate the mean separation of molecules (explain this)
Cube root of volume/molecules
(Because it’s the number of molecules per m^3, then cube root to get the distance)
Compare the distance between molecules compared to the size of atoms
The distance between molecules is much higher than the size of the atom, which becomes basically negligible
What is meant by an “ideal gas”?
One that obeys the ideal gas equation (PV=nRT) under all conditions (e.g - under all values of pressure, volume and temperature)
What can be encapsulated by the ideal gas equation?
The macroscopic behaviour of an ideal gas (i.e- the features of the whole mass of the gas)
What does the kinetic theory of gases attempt to do?
Give an explanation of the macroscopic behaviour of gases based on the microscopic behaviour (the behaviour of the individual molecules)
What is the kinetic theory based upon?
The understanding of the motion of molecules that comprise the gas
Brownian motion
The random motion of particles suspended in a medium
Random motion
Moving in all directions without a preference
What does Brownian motion give evidence of and how?
The constant random motion of the molecules that form a gas
It gives evidence of the constant random motion of particles in gases and liquids
What does the kinetic theory make use of?
Brownian motion and adds to it by conserving the average behaviour of the molecules
What is the kinetic theory only valid for and why?
A large number of molecules only
Since the average behaviour is considered
Example of a situation that the kinetic theory can be applied to
A gas in a flask at room temperature and atmospheric pressure
Describe the type of molecules that the kinetic theory is based on
A vast number of molecules
All beaching as point masses
That bounce elastically off each other and off the walls of the container
Randomly
Why are the assumptions of the kinetic theory important?
They’re the things that make an ideal gas obey PV=nRT
Assumptions of the kinetic theory
- The gas is formed of atoms or molecules which behave as identical small spheres
- A gas is for me from a large number of these spheres
- The volume of the spheres is negligible compares to the volume of the container vessel
- There are no forces of attraction or repulsion between the molecules
- All the collision between the molecules and between the molecules and the vessel wall are perfectly elastic
- The time during a collision is negligible compares to the time between the collisions
- The energy is distributed randomly amongst the gas molecules
Why are gas molecules assumed to behave as identical small spheres?
So that they bounce off of each other in the same way each time
Why is it true that the volume of the spheres (kinetic theory) is negligible compares to the volume of the container vessel?
It you picture a cubic metre of a gas in a box, the volume of particles in the box is negligible compared to the box itself
What does the fact that there are no forces of attraction or repulsion between the molecules in the kinetic theory mean?
There’s no potential energy (all kinetic energy)
“Perfectly elastic” meaning
No loss in kinetic energy
Momentum and kinetic energy are conserved
What does the fact that the energy is assumed to be distributed randomly amongst all the gas molecules in the kinetic theory mean?
There’s no preferred direction of molecules
Statistical treatment
Why is it not necessarily always true that gases can be assumed to behave as identical small spheres?
It’s less true for more complex molecules (e.g - methane)
Why is it important that the gas is formed of a large number of spheres in the kinetic theory?
At low densities, there’s only a few particles.
This would five a less smooth Boltzmann distribution curve.
When may it not be true that the time during a collision is negligible compares to the time between the collisions?
Smaller volumes
High temperatures
High pressure
How can we link microscopic and macroscopic properties for the kinetic theory?
By using the assumptions and Newton’s laws
Microscopic properties of the kinetic theory
Mass and speed of molecules
Macroscopic properties of the kinetic theory
Pressure
Volume
Expression for the pressures of a gas in terms of its microscopic preopeorties
P = 1/2pcbar^2
How does gas give pressure?
Consider one molecule of gas that is moving towards a wall of a cubic container
The momentum of the molecule before the collision will be equal the magnitude of the momentum after the collision, but in the opposite direction
The molecule that returns along its previous path must have experienced a force
From the third law —> every force has an equal an opposite reaction force
In this case, the force will push the wall of the vessel outwards
In our model of the gas, there are billions of these molecules moving randomly, all of them exerting a force on the container wall
What do we need to consider about the molecules of gas that give pressure?
The number present (N)
The volume of the container (V)
The mass of each molecule (m)
The average speed of the molecules (cbar^2)
Pressure equation
M/V or Nm(mass of the number of molecules)/V
Equation to link the properties of the number of molecules present, the volume of the container, the mass of each molecule and the average speed to the molecules and how it can be adapted
P = 1/3N/^mcbar^2
We can substitute p = m/v = Nm/V to give
P = 1/3pcbar^2
What’s the assumption that leads to 1/3 being included in the equation p = 1/3pcbar^2
Random motion so that there are an equal number of molecules in all 3 dimensions
Number of dimensions of molecules
3
Why is p = 1/3pcbar^2 an important equation?
It links the macroscopic features (pressure and density) with the microscopic features (mean square speed) of the molecules
Describe the pressure with more molecules and explain
More pressure (more collisions per second)
Describe the pressure with less molecules and explain
Less pressure
More travel time between collisions
Avogadro’s constant (NA)
The number of particles per mole
The mole
SI unit for an “amount of substance”
The amount containing as many particles (e.g - molecules) as there area atoms in 12g of carbon-12
Number of moles symbol and meaning
n
The total number of moles of a gas present in a given mass or volume of the gas
Two ways of working out n
Total mass of gas (M)/Molar mass (Mm)
Or
Total number of particles (N)/Avogadro’s constant (NA)
Number of molecules symbol and meaning
N
The total number of molecules present in a sample of gas
How do you work out N (the number of molecules)
n x NA
Molar mass symbol and meaning
Mm
Mass in kg of a mole of a substance
Explain how we work out molar mass (Mm)
If you are dealing with a molecule (e.g - O2), it will be equal to the mass number expressed in kg
Which mass is in kg and which is in g?
Molar mass (Mm) = kg
Relative molecular mass (Mr) = g
Relative molecular mass symbol and meaning
Mr
The mass ratio to 1/2th of the mass of a carbon-12 (12C) atom. The same value as the molar mass but in grams.
Molecular mass symbol and meaning
m
The mass of a single molecule of a mass, and can be found from the molar mass
Two ways of working out m, molecular mass
Mm/NA or M/N
Total mass symbol and meaning
M
The bulk mass of a given volume of gas
Ways of working out M, total mass
Mr x n
Or
N x m
Deriving an equation for the bulk energy/total kinetic energy of the gas
We have an expression that has a microscopic quantity (the mean square speed of individual atoms/molecules, and a macroscopic quantity (the density of the mass, p)
(Be aware that big P is for pressure here and small p is for density)
P = 1/3pcbar^2
This can be rewritten as
P = 1/3M/Vcbar^2 (since P = M/V)
M=nM, so
P = 1/3mN/Vcbar^2
Move V to the left side
PV = 1/3mNcbar^2
Looking at the ideal gas equation
PV = nRT
We see that the LHS is identical to the expression above, so
1/4mNcbar^2 = nRT
Multiply both sides by 3/2
3/2 x 1/3mNcbar^2 = 3/2nRT
1/2nMcbar^2 = nRT
Why do we multiply both sides by 3/2 when driving the equation for bulk energy?
We need 1/2 for kinetic energy
Derive the equation for the individual kinetic energy of gas molecules
Looking at the LHS of the equation for bulk energy, 1/2mcbar^2 is equal to the mean kinetic energy of a single molecule of gas, and since it is multiplied by NA, we have an expression for the total kinetic energy of the gas (remember that all of the energy is kinetic - there are no forces between the molecules)
To calculate the mean kinetic energy of a single molecule, we move the NA to the right hand side to obtain…
1/2mcbar^2 = 3/2R/NA T
This is simplified by the use of the Boltzmann constant (K = R/NA) to get
1/2mcbar^2 = 3/2kT
What is 1/2mNcbar^2 = 3/2nRT the equation for? Why?
The total/bulk kinetic energy of the gas
Since it’s equal to the mean kinetic energy of a single molecule, then multiplied by NA
What is all of the energy in gas molecules and why?
Kinetic
There are no forces between the molecules
What is 1/2mcbar^2 = 3/2kT the equation for?
The mean kinetic energy of a single gas molecule
Relationship between temperature and internal energy of a gas
Proportional
What does the equation for the energy of gas molecules do?
Directly links the mean kinetic entry of the individual molecules (microscopic) with the thermodynamic temperature (microscopic)
What is temperature a measure of and what does this mean?
The internal energy of a gas
They’re proportional
Cbar^2 meaning
Route mean square velocity
Route mean square velocity symbol
Cbar^2
Route mean square velocity equation and explanation
Root of the sum of velocities squared/number of molecules
So, square each speed, and square root the whole thing
Why is route mean square velocity used instead of just the speed of a single molecule?
Different molecules will have different speeds at different times
The route mean square velocity gives a mean value of all the molecules since the theory is a statistical one
Why is route mean square velocity used instead of the normal mean velocity?
Since it’s slightly skewed
What will the kinetic energy of different types of gases by in a mixture and why?
The same kinetic energy since it’s dependent only on temperature
(If they have a higher mass, they’ll just move slower)
What is the only thing kinetic energy is dependent on?
Temperature
Distinction between gas laws and kinetic theory
Gas laws = describe the empirical relationships between the variables (can observe the effect of changing the variables)
Kinetic theory = statistical model, so gives a value for pressure based on statistical information about the gas
The property of an ideal gas that’s equal to its internal energy
The sum of all the individual kinetic energies of the molecules
What does U equal from the equation U = 3/2nRT = 3/2nKT in the data book?
U = N1/2mcbar^2
How do we use U = N1/2mcbar^2 for one molecule?
U/N = one molecule
What is pressure (in terms of gas in a container)?
The force per unit are on the walls of the container
Describe how particles move
In straight lines
What is the change in momentum per collision that a particle has with the wall of a container vessel? Explain this
-2mu
(-mu)-(+mu)
=-2mu
If some is doubled and it’s squared, what happens?
x4
How many particles does 1 mole of anything have?
Avogadro’s constant number of moles
Why does the equation m = Mm/NA work?
Think of the units
kgmol-1 over mol-1 leaves kg
Describe an ideal gas in terms of its molecules and their motion
A large number of small identical spheres that are moving randomly and colliding elastically with each other and the walls of the container.
There are no forces of attraction or repulsion between the molecules.
If the mean kinetic energy of molecules increases by a factor of 4, to what factor does the root mean square speed of the molecules Increase by?
A factor of 2 (sqrt of 4)
What do we need to remember to input into any equation used to work out the rms speed of molecules?
When working out m, remember it’s Mm/NA
What do we need to do to get the final answer with rms speed?
Square root it
Force of a gas on a piston
Pressure x area
If using p= 1/3N/Vmc2, what do we need to remember to do?
Do the N/V first before moving things over
What do we need to mention when explaining how a gas gives pressure?
That all collisions are perfectly elastic (no loss in KE)
Why is pressure higher with a higher temperature?
Each collision gives more pressure and there’s more collisions per second
Using 1/2mc^2 = 3/2kT, explain what happens when T is doubled
c2 increases by a factor of sqrt2
(Multiply sqrt2 by original speed for new speed)
What do we put into the PV = nRT equation if we’re asked for something “per m^3”?
V = 1
How do we work out the number per m^3 of atom x?
Density divided by the atomic mass of x
Heat
Energy entering a system by virtue of a temperature difference
Why does a positive change in internal energy at a constant pressure mean that the temperature is increasing?
It’s the gas that’s expanding
What does a clockwise energy loop imply?
Net work done done is positive
What does an anticlockwise energy loop imply?
Net work is negative (work done on gas)
How do we know that the net work is positive in an energy loop?
It’s clockwise
How do we know that the net work done is negative in an energy loop?
Anti-clockwise loop (work done on gas)
How do we go from relative molecular mass to kg?
multiply by u (1.66x10^-27)
