Module 5: C15 - Ideal Gases Flashcards
Equation for Pressure
Pressure = Force/Area
How can we express the number of atoms or molecules in a given volume of air
We can express the number of atoms or molecules in a given volume of gas, using moles (mol), the SI unit of measurement for the amount of a substance.
What is 1 mol of any substance
1 mol of any substance contains 6.02x10^23 individual atoms or molecules. Therefore, the total number of atoms or molecules in a substance, N, is given by the equation
N = n x Na
(n is the number of moles in the substance)
What equation gives the total number of atoms or molecules in a substance
N = n x Na
(n is the number of moles in the substance)
Equation for molar mass
m = n x M
What is the kinetic theory of matter
The kinetic theory of matter is a model used to describe the behaviour of the atoms or molecules in an ideal gas. Real gases have complex behaviour, so to keep the model simple, a number of assumptions are made about the atoms or molecules in an ideal gas.
What are the assumptions made in the kinetic model for an ideal gas
- The gas contains a very large number if atoms or molecules moving in random directions with random speeds
- The atoms or molecules of the gas occupy a negligible volume compared with the volume of the gas.
- The collisions of atoms or molecules with each other and the container walls are perfectly elastic (no kinetic energy is lost)
- The time of collisions between the atoms or molecules is negligible compared to the time between collisions.
- Electrostatic forces between atoms or molecules are negligible except during collisions.
6 assumptions made about ideal gases and kinetic theory
- Molecules are points- the volume of the molecules
is insignificant compared to the volume of the ideal
gas. - Molecules do not attract each other– if they did
then the pressure exerted by the gas on its container
would be reduced. - Molecules move in constant random motion.
- All collisions between gas molecules and their
container are elastic– there is no loss of kinetic
energy. - The time taken for a collision is much shorter
than the time between collisions - Any sample of an ideal gas contains a very large
number of molecules.
What is the Avogadro Constant
The Avagadro constant NA is equal to the number of atoms in exactly 12g of the isotope carbon 12.
To 3sf. : NA = 6.02 x 10^23
How does the pressure and volume relate to each other in ideal gases
If the temperature and mass of a gas remain constant, then the pressure of an ideal gas is inversely proportional to its volume V. This can be expressed as
p ∝ 1/V
or
PVA = constant
What is the equation to show When a gas changes
pressure from p1 to p2 while undergoing a volume change from V1 to V2
P1 x V1 = P2 x V2
How does the pressure and temperature relate to each other in ideal gases
If the volume and mass of the gas remain constant the pressure p of an ideal gas is directly proportional to its absolute (thermodynamic) temperature T in kelvin. This relationship can be expressed as
p ∝ T
or
p/T = constant
What is an ideal gas defined as
An ideal gas is defined as a gas that obeys
Boyle’s law at all pressures.
An ideal gas is defined as a gas that obeys
Boyle’s law at all pressures. Real gases do not obey Boyle’s law at very high pressures or when they are cooled to near their condensation point.
What is the equation of state of an ideal gas
pV/T = nR
or
pV = nRT
Boyle’s law question:
A gas has an initial volume of 300 m^3 at standard
temperature and pressure (100 kPa). Calculate the final volume of this gas if its pressure is increased by 400 kPa at a constant temperature.
P1 = 100kOa = 100x10^3 Pa
V1 = 300m^3
P2 = 500kPa =500x10^3 Pa
V2 = ?
P1V1 = P2V2
100x10^3 = 300 = 500x10^3 V2
V2 = 60m^3
Pressure law question:
A gas has an initial pressure of 100kPa at a temperature of 27°C. Calculate the final pressure if its temperature is increased by 300°C at a constant volume.
P1 = 100kPa = 100x10^3 Pa
T1 = 27°C = 30pK
P2 = ?
T2 = 327°C = 600K
P1/T1 = P2/T2
P2 = P1T2/T1
100x10^3 / 300 = p2 / 600
600 (100x10^3) / 300 = P2
P2 = 200MPa
What is Charles’ Law (equation relating volume and temperature in an ideal gas)
For a fixed mass of gas at a constant pressure:
V/T = Constant
When a gas changes volume from V1 to V2 while undergoing a temperature change from T1 to T2 :
V1/T1 = V2/T2
Example Question:
A gas has an initial volume of 50 m^3 at a temperature of 127°C. Calculate the final temperature required in 0°C to decrease the volume to 20m^3 at a constant pressure.
V1/T1 = V2/T2
T2 = V2T1/V1
T2 = 20x400 / 50
0K = -273°C
T2 = 160K
160-273 = -113°C
T2 = -113°C
What is the Ideal Gas equation and how do you get it
Combining all three gas laws for a constant mass of gas gives:
pV/T = a constant
the constant = nR and so:
Therefore:
pV = nRT
(n = number of moles of the gas
R = molar gas constant = 8.31 J K^-1 mol^-1)
Equation for Number of Molecules involving Number of Moles and Avogadros Constant
N = nNa
(Number of molecules = number of moles x avogadros constant)
Equation for mass of a gas involving the number of moles and molar mass
m = nM
Mass of Gas = number of moles x Molar Mass
Calculate the volume of 1 mole of an ideal gas at 0°C and 101kPa using pV=nRT
pV = nRT
V = nRT/p
V = 1x8.31x273 / 101x10^3
V = 0.0225m^3
V = 2.25x10^-2 m^3
Example Question:
A fixed mass of gas has its pressure increased from 101 kPa to 303 kPa, its volume increased by 5 m3 to 1 m3 whilst its temperature is raised from 200C. Calculate its final temperature.
p1V1/T1 = p2V2/T2
T2= P2V2T1/P1V1
T2 = 303x10^3 x 6 x 293 / 101x10^3 x 1
T2 = 5274 K
or
T2 = 5001°C
Example Question:
A container of volume 2.0x10^-3 m^3,
temperature 20°C, contains 60g of oxygen of
molar mass 32g. Calculate its pressure.
pV = nRT
becomes: p = nRT/V
But n = m/M
T = 20°C = 293K
n = 60/32
n = 1.875
p = nRT/V
p = 1.875x8.31x293 / 2x10^-3
p = 2.28x10^6 Pa
p = 2.28 MPa
What is the Boltzmann Constant, and how it is found
The Boltzmann constant, k = R/Na
k = 1.38x10^-23 JK^-1
Example Question:
Estimate the number of air molecules in this room.
[Typical values: room volume = 300m^3]
room temperature = 20°C = 293K atmospheric pressure = 101 kPa = 101x10^3 Pa
pV = nRT
N = pV/kT
N = 101x10^3 x 300 / 1.38x10^-23 x 293
N = 7.49x10^27
N = 7.50x10^27
How can pV = nRT become pV = NkT
n = N/Na
pV = nRT
pV = N/Na x RT
pV = R/Ra x NT
k = R/Na
And so the ideal gas equation can also be stated as:
pV = NkT
What is the Kinetic Theory of Gases
The kinetic theory of gases states that a gas consists of point molecules moving about in random motion.
According to the kinetic theory, what happens when the Volume of a container is decreased
– There will be a greater number of molecules hitting the inside of the container per second
– A greater force will be exerted
– Pressure will increase
According to the kinetic theory, what happens when the Temperature of a container is decreased
If the temperature of a container is increased:
– Molecules will be moving at greater speeds.
– More molecules will be hitting the inside of the container per second and they will each exert a greater force.
– A greater overall force will be exerted
– Pressure will increase
What is the equation for the Root Mean Square Speed
Crms = √(C1^2 + C2^2 + C3^2 + C4^2 / N)
_
C^2 = Crms^2
Where:
_
C^2 is the mean square speed
Example Question:
Calculate the RMS speed of four molecules having speeds 300, 340, 350 and 380 ms^-1
√(300^2 + 340^2 + 350^2 + 380^2 / 4) = 75√21
= 344ms^-1
What is the Kinetic Theory Equation
For an ideal gas containing N identical
molecules, each of mass, m in a container
of volume, V, the pressure, p of the gas is
given by:
pV = 1/3 Nmc̅^2
Example Question:
A container of volume 0.05m^3 has 0.4kg of an ideal gas at a pressure of 2.0x10^7 Pa. Calculate the RMS speed of the gas molecules.
pV = 1/3 Nmc̅^2
c̅^2 = 3pV / Nm
Nm = mass of the gas = 0.4kg
c̅^2 = 3 x 2x10^7 x 0.05 / 0.4
c̅^2 = 7.5x10^6 ms^-1
Crms = √7.5x10^6 ms^-1
Crms = 2740ms^-1
How do you work out Average Molecular Kinetic Energy
If 𝒑𝑽 =𝟏/𝟑𝑵𝒎c̅^𝟐 and 𝒑𝑽 = 𝑵𝒌𝑻 then:
𝟏/𝟑𝑵𝒎c̅^𝟐 = 𝑵𝒌𝑻
𝟏/𝟑𝒎c̅^𝟐 = 𝒌𝑻
x 3/2 on both sides
3/2 x 1/3mc̅^2 = 3/2kT
1/2mc̅^2 = 3/2 kT
1/2mc̅^2 is the average translational kinetic energy of a molecule in the gas, so:
Ek = 3/2 kT
The kinetic energy of a molecule is directly proportional to its absolute temperature
How can the equation pV = 1/3Nmc̅^2 be derived
It can be derived by considering how the movement of atoms or molecules of gas inside a sealed box gives rise to pressure.
Consider a single gas particle making repeated collisions with a container wall. The container is a cube with sides L. The gas particle has mass m and velocity c. It hits the surface of the wall at right angles.
The elastic collision results in a change in momentum of magnitude 2mc. The time t between collisions is the total distance covered by the particle divided by its speed. Therefore t = 2L/c. According to Newton’s second and third laws, the force exerted by the particle on the wall is
Force = Δp/Δt = 2mc x c/2L = mc^2/L
If there are N particles in the container moving randomly, the average force exerted by each particle must be mc̅^2/L, where c̅^2 is the mean square speed of the particles.
On average, because if the random motion of the gas particles, about 1/3 of the particles will be moving between two opposite faces of the container. Consequently, the total force on one container wall of cross-sectional area L^2 due to collisions from all particles must be
Force = mc̅^2/L x N x 1/3 = Nmc̅^2/3L
Finally, the pressure p exerted by the gas must be equal to the total force exerted by all the particles divided by the cross-sectional area of the wall. Therefore,
p = Nmc̅^2/3L x 1/L^2 = Nmc̅^2 /3L^3 = Nmc̅^2/3V
where V is the volume of the container. Therefore pV = 1/3Nmc̅^2
Example Question:
Calculate the mean Ek of air molecules at 0°C
0°C = 273K
Ek = 3/2kT
Ek = 3/2 x 1.38x10^-23 x 27
Ek = 5.65x10^-21 J
Work out the RMS speed of Oxygen-16 where Ek = 5.65x10^-21 J
Ek = 3/2kT
Ek = 1/2mc̅^2
3/2kT = 1/2mc̅^2
c̅^2 = 2Ek/m
And Ek = 5.65x10^-21
Mass of O2 molecule
m = M/Na
m = 0.032/6,02x10^23
m = 5.3x10^-26 kg
c̅^2 = 2Ek/m
c̅^2 = (2 x 5.65x10^-21)/5.3x10^-26
c̅^2 = 2.13x10^5 m^2 s^-2
rms speed = √c̅^2
rms speed = 462ms^-1
What is the Nm in the equation Pv = 1/3 Nmc̅^2
The Nm in the equation is the mass of the gas
