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Chemistry 431
Lecture 1
Ideal Gas Behavior
NC State University
Macroscopic
p variables P, T
Pressure is a force per unit area (P= F/A)
The force arises from the
change in momentum as
particles hit an object and
change direction.
Temperature derives from molecular
i molar
l
motion (3/2RT = 1/2M<u2>) M is
mass
Greater average
g velocity
y
results in a higher
temperature.
u is the velocity
Mass and molar mass
We can multiply the equation:
3 RT = 1 M <u 2>
2
2
by the number of moles, n, to obtain:
3 nRT = 1 nM <u 2>
2
2
If m is the mass and M is the molar of a
particle then we can also write:
p
nM = Nm (N is the number of particles)
Mass and molar mass
In other words nNA = N where NA is
Avagadro’s
Avagadro
s number
number.
3 nRT = 1 Nm <u 2>
2
2
Average properties
<u2> represents the average speed
Kinetic Model of Gases
Assumptions:
1. A gas consists of molecules that move randomly.
2 Th
2.
The size
i off th
the molecules
l
l iis negligible.
li ibl
3. There are no interactions between the gas molecules.
Because there are such large numbers of gas molecules
in any system we will interested in average quantities.
We have written average with an angle bracket.
For example, the average speed is:
s 12 + s 22 + s 32 + ... + s N2
<u > = c =
N
2
2
c=
s 1 + s 2 + s 3 + ... + s N
N
We use s for speed
and
d c ffor mean speed.
d
Velocity and Speed
When we considered the derivation of pressure using a
kinetic model we used the fact that the gas exchanges
momentum with the wall of the container. Therefore, the
vector (directional) quantity velocity was appropriate.
However, in the energy expression the velocity enters as
the square and so the sign of the velocity does not matter.
In essence it is the average speed that is relevant for the
energy Another way to say this is the energy is a scalar
energy.
scalar.
E = 1 m<u 2> = 1 mv 2 = 1 mc 2
2
2
2
p = mu = mv
All of these notations
mean the same thing.
The root-mean-square
q
speed
p
The ideal gas equation of state is consistent with an
p
of temperature
p
as p
proportional
p
to the kinetic
interpretation
energy of a gas.
1 M u 2 = RT
3
If we solve for <u2> we have the mean-square speed.
〈 u 2 〉 = 3RT
M
If we take the square root of both sides we have the r.m.s.
speed.
u
2 1/2
=
3RT
M
The mean speed
The mean value is more commonly used than the
root-mean-square of a value. The root-mean-square speed
Is equal to the root-mean-square velocity:
c 2 = 〈 u 2〉
The mean speed is:
c=
8 c2
3π
The r.m.s. speed of oxygen at 25 oC (298 K) is 482 m/s.
Note: M is converted to kg/mol!
〈u 〉
2 1/2
3 8.31
8 31 J/mol–K
J/mol K 298 K
=
0.032 kg/mol
= 481.8 m / s
The Maxwell Distribution
Not all molecules have the same speed. Maxwell assumed
that the distribution of speeds was Gaussian
Gaussian.
F(s) = 4π M
2πRT
3/2
2
Ms
s exp –
RT
2
As temperature
p
increases the r.m.s. speed
p
increases and
the width of the distribution increases. Moreover, the
functions is a normalized distribution. This just means
that the integral over the distribution function is equal to 1
1.
∞
F(s)ds = 1
0
See the MAPLE
worksheets for examples.
Molecular Collisions
Cross section
σ = πd2
Interation Volume
πd2<u>t
Center location of
target molecule
mean free path estimate =
<u>t
distance traveled
volume of interaction * number density
n/V = moles p
per unit volume ((molar density)
y)
N/V = molecules per unit volume (number density)
mean free path estimate =
<u>t
σ<u>t N/V
Refinement of mean free path
The analysis of molecular collisions assumed that the target
atom was stationary. If we include the fact that the target
atom
t
is
i moving
i we fifind
d th
thatt th
the relative
l ti velocity
l it iis:
< u > rel = 2 < u >
Th f
Therefore
<u>t
1
1
RT
=
=
=
λ=
2σ < u > t N / V
2σ N / V
2 σ N A n/V
2 σ NA P
As the pressure increases the number density increases
and the distance between collision ((mean free p
path))
becomes shorter.
As the temperature increases at constant pressure the
number density must decrease and the mean free path
ill increase.
Mean free path
Collision frequency
The mean free path
path, λ is the average distance that a molecule
travels between collisions.
The collision frequency, z is the average rate of collisions
made
d b
by one molecule.
l
l
The collision cross section, σ is target area presented by one
molecule to another.
When interpreted in the kinetic model it can be shown that:
2
2
N
σ
〈
u
〉P
RT
A
λ=
, z=
, σ = πd 2
RT
2 N AσP
The product of the mean free path and collision frequency is
equall tto the
th room mean square speed.
d
〈 u 2 〉 = λz
Units of Pressure
Force has units of Newtons
F = ma (kg m/s2)
Pressure has units of Newtons/meter2
P= F/A = (kg m/s2/m2 = kg/s2/m)
These units are also called Pascals (Pa).
1 bar = 105 Pa = 105 N/m2.
1 atm = 1.01325 x 105 Pa
Units of Energy
gy
Energy has units of Joules
1 J = 1 Nm
N
Work and energy have the same units.
Work is defined as the result of a force
g through
g a distance.
acting
We can also define chemical energy in
terms of the energy per mole
mole.
1 kJ/mol
1 kcal/mol
k l/ l = 4.184
4 184 kJ/mol
kJ/ l
Thermal Energy
gy
Thermal energy can be defined as RT.
I magnitude
Its
i d d
depends
d on temperature.
R = 8.31 J/mol-K or 1.98 cal/mol-K
At 298 K, RT = 2476 J/mol (2.476 kJ/mol)
Thermal energy can also be expressed on a
per molecule basis. The molecular
equivalent of R is the Boltzmann constant
constant, kk.
R = NAk
NA = 6.022
6 022 x 1023 molecules/mol
l
l / l
Extensive and Intensive Variables
Extensive variables are proportional to the
size of the system.
Extensive variables: volume,, mass,, energy
gy
Intensive variables do not depend on the
size of the system.
I t
Intensive
i variables:
i bl
pressure, ttemperature,
t
density
Equation of state relates P, V and T
The ideal gas equation of state is
PV = nRT
RT
An equation of state relates macroscopic
properties which result from the average
behavior of a large number of particles.
P
Macroscopic
Microscopic
Microsopic
p view of momentum
c
ux
b
area = bc
a
A particle with velocity ux strikes a wall.
The momentum of the particle changes from mux
to –mux. The momentum change is Δp = 2mux.
Transit time
c
ux
b
area = bc
a
The time between collisions with one wall is
Δt = 2a/ux.
This is also the round trip time.
Transit time
c
Round trip
distance is 2a
area = bc
ux
b
a
The time between collision is Δt = 2a/ux.
velocityy = distance/time.
time = distance/velocity.
The p
pressure on the wall
force = rate of change of momentum
Δ p 2mu x mu x2
F=
=
= a
Δt 2a/u
a/u x
The pressure is the force per unit area.
The area is A = bc and the
volume of the box is V = abc
2
2
mu
mu
x
x
F
P=
=
=
V
bc abc
Average
g p
properties
p
Pressure does not result from a single
particle
ti l striking
t iki th
the wallll b
butt ffrom many
particles. Thus, the velocity is the average
velocity
l it titimes th
the number
b off particles.
ti l
Nm 〈u 2x 〉
P=
V
PV = Nm 〈
2
ux
〉
Average
g p
properties
p
There are three dimensions so the velocity along
the xx-direction
direction is 1/3 the total
total.
〈
u 2x
1
2
=
u
〉 3〈 〉
Nm 〈 u 2 〉
PV =
3
From the kinetic theory of gases
1 Nm〈 u 2 〉 = 3 nRT
2
2
Putting
g the results together
g
When we combine of microscopic view of
pressure with
ith th
the ki
kinetic
ti th
theory off gases
result we find the ideal gas law.
PV = nRT
This approach assumes that the molecules
have no size (take up no space) and that
they have no interactions
interactions.