Ideal Gases Flashcards
Boyle’s Law
Pressure exerted by a fixed mass is indirectly proportional to its volume, provided that temperature is constant
p ∝ 1/v
p1v1 = p2V2
Charles Law
V is directly proportional to T
- pressure is constant
Pressure Law
P is directly proportional to T
- volume is constant
Combining all the equations
pV = NT
N is the number of molecules
the greater number of molecules, the greater the pressure
The two equations of Ideal Gases
pV = NkT
pV = nRT
N = ???
n = ???
N = number of molecules
n = number of moles
Avogardo Constant (1 mole)
the number of atoms of carbon-12 = 6.02x10^23 molecules
Equation linking N, n and Na (Avogardo’s constant)
n = N / Na
number of moles = number of molecules / Avogardo constant
Assumptions of Gases
R - random motion
A - attraction of gas molecules is none
V - volume of gas negligible
E - elastic collision // Ek is conserved
D - duration of collision is negligible
Kinetic theory of gases equation
P = 1/3Nm<c^2>
Derivation of Kinetic theory of gases equation
Consider a box of gas:
x = length
y = width
z = height
c = speed of the molecule
t = time taken for the particle to travel forward AND backward
P = F/A
F = 2mc / t
t = 2x / c
F = 2mc / (2x / c )
P = (mc^2 / x ) / xyz
P = mc^2 / V —— (1)
Since it’s a box, gas movements are in 3D. The (1) equation only considers 2D. Therefore we use the 3D Pythagoras Theorem to calculate the mean speed.
c^2 = cx^2 + cy^2 + cz^2
<cx^2> = <cy^2> = <cz^2>
<cx^2> = 1/3 <c^2> —— (2)
Substitute (2) to (1):
P = 1/3Nm<c^2> / V
V = xyz
Final equation:
pV = 1/3Nm<c^2>
Definition of mole
the SI base unit of an amount of substance. it is the amount containing as many particles as there are atoms in 12g of carbon-12
Definition of molar mass
the mass of 1 mole
Explain what happens to the pressure inside a box with moving gas particles
- a gas exerts pressure on any surface with which it comes into contact
- when an air molecule collides with the surface of the box, it exerts a small force on the box
- the pressure inside the box is a result of the forces exerted by a vast number of molecules in the box
Factors that affect the pressure that gas exerts
- number of molecules that hit the box in one second
- the force with which a molecule collides with the wall
The average speed of molecules
400 ms^-1
Explain pressure when gas molecules hit a wall in relation to momentum
If a molecule of mass m hits the wall with speed v, it will rebound with speed v at the opposite direction
Hence change in momentum = 2mv
Since force is equal to rate of change of momentum, the higher the speed of the molecule, the greater the force it exerts when colliding the wall; the pressure on the wall increases.
Properties of gases to measure
- pressure
- temperature
- volume
- mass
Changing degrees Celsius to Kelvin
θ°C + 273.15
Pressure in ideal gases
The frequency of collisions of the gas molecules per unit area of the container
Mass in ideal gases
the amount of gas measured in moles
Mass of a proton / neutron (1u)
1.66 x 10^-23
According to Boyle’s law, if gas is compressed at constant temperature…
… its pressure increases, and volume decreases.
volume decrease because there are more particles per unit volume and more collisions per second of the particles with unit area of the wall
constant temperature = speed doesn’t change
each collision with the wall involves the same change in momentum, but with more collisions per second on unit area of the wall there is a greater rate of change of momentum and, therefore, a larger pressure on the wall.
How does Charles Law prove absolute zero?
- graph of Charles Law
- gradient is constant
- the graph does show that there is a temperature at which the volume of a gas does, in principle, shrink to zero.
- Looking at the lower temperature scale on the graph, where temperatures are shown in kelvin (K), we can see that this temperature is 0 K, or absolute zero.
R (constant of proportionality) value
R = 8.31 J mol−1 K−1
What does the equation pV = 1/3Nm<c^2> suggest?
- pressure is proportional to the number of molecules
- the greater the mass of the molecule, the greater force it will exert in collision
- pressure is proportional to the average value speed squared
- pressure is inversely proportional to volume (Boyles law)
Combining equations pV = 1/3Nm<c^2> and pV = nRT
nRT = 1/3Nm<c^2>
3nRT/N = m<c^2>
m<c^2> = 3RT / Na
KE = 1/2mv^2
KE = 1/2m<c^2>
1/2M<c^2> = 3RT / Na
k = R / Na (Boltzmann constant)
1/2 m<c^2> = 3/2kT
KE = 3/2kT -> the kinetic energy of only one molecule of gas
What does KE = 3/2kT suggest
the mean kinetic energy of an ideal gas molecule is proportional to its thermodynamic temperature
definition of translational kinetic energy
the energy a molecule has as it moves from one point to another
Calculating root mean squared speed
Imagine three molecules with speeds 10, 20 and 30 m s−1
mean speed <c> = 10+ 20 + 30 / 3 = 20ms^-1</c>
mean square speeds = 10^2 + 10^2 + 30^2 / 3 = 467 ms^-1
√ <c^2> = √ 467 = 22ms^-1
