Pressure and KMT
•
I’m going to quickly show the derivation pressure, you
aren’t responsible for the derivation – it’s shown to
give you an idea of…
•
… where this model comes from
•
… how even simple models can give us deep insights
into a system
•
(the power of p-chem!!)
•
Pressure is force per area
•
Force is change in momentum per time
•
For a particle (atom/molecule) traveling along
x
, the
change in momentum in an elastic collision will be
𝐶ℎ???? 𝑖? ???????? ??? ????𝑖??? = 2 ? ?
?
8

Pressure and KMT, pt2
•
How many particles will hit the wall?
•
If a particle is traveling at a speed of
?
?
for some time (
Δ?
), then only particles
within a distance of
Δ?
will collide with the wall.
Δ? = ?
?
Δ?
•
If
A
is the total area of the wall, then only particles in a volume of
𝐴 ∙ Δ?
can collide with the wall
?????? ?? ????𝑖???? ?ℎ?? ??? ℎ𝑖? ?ℎ? ???? = 𝐴 ?
?
Δ?
•
The density of particles in the volume is:
𝑃???𝑖??? ????𝑖?? =
?
𝑉
=
? ?
𝐴
𝑉
9

Pressure and KMT, pt3
•
How many particles will hit the wall (continued)?
•
Velocity is assumed to be random, so only half of the particles have velocity of
?
?
the other half
travel at
−?
?
and won’t hit the wall
?????? ?? ????𝑖???? ℎ𝑖??𝑖?? ?ℎ? ???? ????? ? =
?????? ?? ????𝑖???? ?ℎ?? ??? ????? ?????ℎ ?? ℎ𝑖? ?ℎ? ????
×
????𝑖??? ????𝑖??
×
1
2
=
?∙?
𝐴
∙𝐴∙𝑣
𝑥
∙Δ?
2𝑉
•
Total change in momentum during time
Δ
t:
?ℎ???? 𝑖? ???????? =
(?????? ?? ????𝑖????) × (???????? ?ℎ???? ??? ????𝑖???) =
? ∙ ?
𝐴
∙ 𝐴 ∙ ?
?
∙ Δ?
2?
2??
?
10

Pressure and KMT, pt4
(almost there)
•
Force is change in momentum per time
•
(note)N
A
∙m
= M, the molar mass
•
Pressure is force per area
•
And we expect to have a distribution of speeds, so we’ll use the average
square velocity in the
x
-direction:
<v
x
2
>
11
? Δ? =
? ∙ ?
𝐴
∙ 𝐴 ∙ ?
?
∙ Δ?
2?
2??
?
=
? ∙ ? ∙ ?
𝐴
∙ 𝐴 ∙ ?
?
2
∙ Δ?
?
? =
? ∙ ? ∙ 𝐴 ∙ ?
?
2
?
? =
?
𝐴
=
? ∙ ? ∙ 𝐴 ∙ ?
?
2
𝐴 ∙ ?
=
? ∙ ? ∙ ?
?
2
?
? =
? ∙ ? ∙
?
?
2
?

Pressure and KMT, pt5
(phew, and we did it!)
•
We want total velocity, not just along the
x
-direction
•
Which we get from a Pythagorean identity!
•
No one direction is favored, so we expect to have the
same average square velocity along all directions
•
Lastly, as we will see in the next part (distribution of
speeds) the square-root of the average square velocity is
called the: root-mean-square velocity (
v
rms
)
12
? =
? ∙ ? ∙ ?
???
2
3?
?
2
= ?
?
2
+?
?
2
+?
?
2
?
?
2
=
?
?
2
=
?
?
2
?
2
= 3
?
?
2
?? =
? ∙ ? ∙ ?
???
2
3