Microscopic model
Square Container
Area A
a
Pressure (P) is caused by
the collisions of molecules
with walls
Face B
Our task is to show that
Face A
vy
vx
Gas atoms
Temperature (T) is related
to average speed of
molecules
a
a
Fig.1.15: The gas molecules in the container are in random motion.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Definition: P (pressure) = force per unit area
Connecting the pressure with the average speed of gas molecules
Step 1. Change in Momentum of a Molecule
∆p = 2mvx
N
∆p = change in momentum,
m = mass of the molecule,
vx = velocity in the x direction
Step2. Force produced by unique molecule
∆p 2mv x mv x
=
=
F=
∆t 2 a / v x
a
2
Step3. Calculating pressure
P=
Total _ force
=
2
a
mv x1 + mv x2 + ... + mv x N
2
2
a3
Step4. Averaging speed
v x + v y + v z = v 2 = 3v x
2
2
2
2
2
2
mN v x
mN v x1 + v x2 + ... + v x N
= 3
=
a
N
V
2
2
2
Nmv 2 1 2
= ρv
P=
3V
3
Ν = total number of molecules
ρ = density of gas
v = velocity of molecules
Pressure ÙTemperature
Ú
Ú
Volume
Ideal Gas Equation
⎛ N ⎞
⎟⎟ RT
PV = ⎜⎜
⎝ NA ⎠
P is the pressure, i.e. force per unit area
N is the total number of molecules in volume V
R is the gas constant (8.3144 J mol-1 K-1)
NA is the Avogadro’s number (6.022 ×1023)
Experimental law – macroscopic point of view
Gas Pressure in the Kinetic Theory
2 ⎞
⎛
m
v
1
2
2
⎟
P = ρ v = N⎜
3
3 ⎜⎝ 2 ⎟⎠
Ideal Gas Equation
⎛ N ⎞
⎟⎟ RT
PV = ⎜⎜
⎝ NA ⎠
P = gas pressure, N = number of molecules,
m = mass of the gas molecule, v = velocity,
V = volume, ρ = density.
Mean Kinetic Energy per Atom
3
1 2
kT
KE = mv =
2
2
k =R/NA Boltzmann constant,
T = absolute temperature
0 K ≈ -273 0C
Gas constant
Boltzmann’s constant
R = 8.3144
J mol-1 K-1
k = 8.61 × 10-5
eV K-1
R = k × N0
R – macroscopic (technical, thermodynamical etc.) calculations
k – microscopic (atomic) calculations
Mole (a gram molecule) = a quantity of a substance
equal to the molecular weight of a substance
expressed in grams
Example:
Carbon has atomic mass of 12 ⇒ a mole of carbon is 12 grams
Mole contains 6.0220 × 1023 species (atoms, molecules, etc.)
N0
= 6.0220 × 1023 - an Avogadro's number
Internal Energy per Mole for a Monatomic Gas
KE =
1 2
3
mv =
kT
2
2
⎛ 1 2⎞ 3
U = N A ⎜ mv ⎟ = N A kT
⎝2
⎠ 2
U = total internal energy per mole, NA = Avogadro’s number, m =
mass of the gas molecule, k = Boltzmann constant, T = temperature
Molar Heat Capacity at Constant Volume
3
3
dU
R
N Ak =
=
Cm =
2
2
dT
Cm = specific heat per mole at constant volume (J K-1 mole-1), U =
total internal energy per mole, R = gas constant
Definition: Heat capacity is the rise of internal energy per unit temperature