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8/22/2008
Chemistry 431
Macroscopic variables P, T
Pressure is a force per unit area (P= F/A)
Lecture 1
Ideal Gas Behavior
NC State University
The force arises from the
change in momentum as
particles hit an object and
change direction.
Temperature derives from molecular
motion (3/2RT = 1/2M<u2>) M is molar
mass
Greater average velocity
results in a higher
temperature.
Mass and molar mass
We can multiply the equation:
3 RT = 1 M <u 2>
2
2
by the number of moles
moles, n
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:
nM = Nm (N is the number of particles)
Kinetic Model of Gases
Assumptions:
1. A gas consists of molecules that move randomly.
2. The size of the molecules is negligible.
3. There are no interactions between the gas molecules.
Because there are such large numbers of gas molecules
i any system
in
t
we will
ill interested
i t
t d in
i average quantities.
titi
We have written average with an angle bracket.
For example, the average speed is:
<u 2> = c 2 =
c=
s 12 + s 22 + s 32 + ... + s N2
N
s 1 + s 2 + s 3 + ... + s N
N
We use s for speed
and c for mean speed.
u is the velocity
Mass and molar mass
In other words nNA = N where NA is
Avagadro’s number.
3 nRT = 1 Nm <u 2>
2
2
Average properties
<u2> represents the average speed
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.
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.
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The mean speed
The root-mean-square speed
The ideal gas equation of state is consistent with an
interpretation of temperature as proportional to the kinetic
energy of a gas.
1 M u 2 = RT
3
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:
If we solve for <u2> we have the mean-square speed.
c=
〈 u 2 〉 = 3RT
M
If we take the square root of both sides we have the r.m.s.
speed.
u2
1/2
The Maxwell Distribution
Not all molecules have the same speed. Maxwell assumed
that the distribution of speeds was Gaussian.
F(s) = 4π M
2πRT
3/2
2
s exp – Ms
RT
2
As temperature increases the r.m.s. speed 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.
∞
F(s)ds = 1
0
The r.m.s. speed of oxygen at 25 oC (298 K) is 482 m/s.
Note: M is converted to kg/mol!
3RT
M
=
〈 u 2 〉 1/2 =
As the pressure increases the number density increases
and the distance between collision (mean free path)
becomes shorter.
As the temperature increases at constant pressure the
number density must decrease and the mean free path
ill increase.
= 481.8 m / s
0.032 kg/mol
Molecular Collisions
Interation Volume
πd2<u>t
Center location of
target molecule
<u>t
distance traveled
mean free path estimate =
volume of interaction * number density
n/V = moles per unit volume (molar density)
N/V = molecules per unit volume (number density)
mean free path estimate =
<u>t
σ<u>t N/V
Mean free path
Collision frequency
Refinement of mean free path
< u > rel = 2 < u >
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
3 8.31 J/mol–K 298 K
Cross section
σ = πd2
See the MAPLE
worksheets for examples.
The analysis of molecular collisions assumed that the target
atom was stationary. If we include the fact that the target
atom is moving we find that the relative velocity is:
8 c2
3π
The mean free path, λ is the average distance that a molecule
travels between collisions.
The collision frequency, z is the average rate of collisions
made by one molecule.
The collision cross section
section, σ is target area presented by one
molecule to another.
When interpreted in the kinetic model it can be shown that:
λ=
2N Aσ 〈 u 2 〉 P
RT
, z=
, σ = πd 2
RT
2N AσP
The product of the mean free path and collision frequency is
equal to the room mean square speed.
〈 u 2 〉 = λz
2
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Units of Pressure
Force has units of Newtons
F = ma (kg m/s2)
Pressure has units of Newtons/meter2
P F/A = (kg
P=
(k m/s
/ 2/m
/ 2 = kg/s
k / 2/m)
/ )
These units are also called Pascals (Pa).
1 bar = 105 Pa = 105 N/m2.
1 atm = 1.01325 x 105 Pa
Thermal Energy
Units of Energy
Energy has units of Joules
1 J = 1 Nm
Work and energy have the same units.
Work is defined as the result of a force
acting through a distance.
We can also define chemical energy in
terms of the energy per mole.
1 kJ/mol
1 kcal/mol = 4.184 kJ/mol
Extensive and Intensive Variables
Thermal energy can be defined as RT.
Its magnitude depends 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, k.
R = NAk
NA = 6.022 x 1023 molecules/mol
Extensive variables are proportional to the
size of the system.
Extensive variables: volume, mass, energy
Equation of state relates P, V and T
Microsopic view of momentum
Intensive variables do not depend on the
size of the system.
Intensive variables: pressure, temperature,
density
c
The ideal gas equation of state is
PV = nRT
An equation of state relates macroscopic
properties which result from the average
behavior of a large number of particles.
b
area = bc
P
Macroscopic
ux
Microscopic
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.
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Transit time
Transit time
c
c
ux
b
area = bc
a
Round trip
distance is 2a
ux
b
area = bc
a
The time between collisions with one wall is
Δt = 2a/ux.
This is also the round trip time.
The time between collision is Δt = 2a/ux.
velocity = distance/time.
time = distance/velocity.
The pressure on the wall
Average properties
force = rate of change of momentum
Pressure does not result from a single
particle striking the wall but from many
particles. Thus, the velocity is the average
velocity times the number of particles.
Δ p 2mu x mu x2
=
= a
Δt 2a/u x
The pressure is the force per unit area
area.
The area is A = bc and the
volume of the box is V = abc
F=
mu 2x mu 2x
P= F =
=
V
bc abc
Average properties
There are three dimensions so the velocity along
the x-direction is 1/3 the total.
〈u 2x 〉= 13 〈u 2 〉
PV =
Nm〈 u 2 〉
3
From the kinetic theory of gases
1 Nm〈 u 2 〉 = 3 nRT
2
2
P=
Nm 〈u 2x 〉
V
PV = Nm 〈u 2x 〉
Putting the results together
When we combine of microscopic view of
pressure with the kinetic theory of gases
result we find the ideal gas law.
PV = nRT
RT
This approach assumes that the molecules
have no size (take up no space) and that
they have no interactions.
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