A minimum of

A minimum of

Thermodynamics and of

Kinetic Theory of Gases

GACHON UNIVERSITY

2.1 The motion of Molecules

• A gas is a collection of particles (molecules)

• This particles interacts with other particles through elastic collision that conserve both energy and momentum.

• If they move along three-dimension only, degree of freedom would be three. (+ vibration, spin)

• At a given moment, some molecules have large kinetic energy, while others have little. Over a sufficiently long period of time, each have same “average kinetic energy” < W_mol > – principle of equipartition of energy (에너 지 등분배 원리)

• Average kinetic energy is same in a collection of molecules even with different masses. (But velocity is different.)

• Instantaneous velocity (순간속도) have Maxwellian distribution.

2.2 Temperature

• Two component of kinetic energy

• Temperature is measure of <Wmol, linear>: It is defined by

, k is Boltzmann’s constant ( k = 1.38* 10

  W



T k

3 2











 W

W

W

• Average energy of linear motion in 3d gas:

• Per degree of freedom :

• Average total molecular energy:

k T W

3 2

,



 k T W

deg

2

/

,





) :

(

2 kT , v 자유도 W

 v



2.3 The Perfect-Gas Law: Relation between pressure and temperature

• Upon impact, single molecular deposits a momentum of 2mv (mv + mv) on the wall.

• The number of molecules passing through a unit area in unit time (i.e. flux):

• Pressure by the gas: ( n : number of molecular / volume)

• Kinetic energy = ½ mv² (Newton’s dynamics) = 1/2kT, p = nkT

• Pressure is proportional to temperature and concentration (농도, i.e. number molecular/volume)

•

• Perfect-Gas (ideal gas): do neither attract nor repel one another, just collide

2

/ 1

*

2 mv nv nmv

p  

2 nv

1

GACHON UNIVERSITY

2.3 The Perfect-Gas Law: Relation between pressure and temperature

• Pressure, p = nkT

• 1 kilomole: - Avogadro’s number ( Like 담배 1보루 = 200 개비)

• u is number of kilomole, N₀ is number of molecule, v is volume

• Pressure with gas constant R:

1023

022 .

6 

V N n  

/

V RT V

kT p   N

 

1 1

26 23

0

 1 . 38  10

 6 . 022  10  8314

 kN JK kmole

R

i.e. PV = nRT

2.4 Internal Energy

• Total internal energy U: sum of the energy of all molecules

• Total internal energy U: depends only on the temperature T and degree of freedom

v RT v kT

N W

N W

U

i

mol mol_i

2

0

2

0

 

    



 

2.5 Specific Heat (비열) at Constant Volume

• At constant volume (gas in the rigid container), the rate of change of internal energy with a change of temperature (per kilomole of gas)

• specitic heat of gas is proportional to degree of freedom ν

• If temperature of gas change, internal energy change:





 

v R dT

c

dU

2

1 

  ^Jkmole

^K

“specific heat” is the amount of heat per unit mass required to raise the

temperature by one degree Celsius.

GACHON UNIVERSITY

2.6 The First law of Thermodynamics

• In the piston, Internal energy U, heat energy Q, external work W:

ΔU = ΔQ −ΔW, dU = dQ −dW ~ mathematical expression of the first law of TMD

• Heat is added at constant volume,

dt dQ dt

c

 dU 



2.7 The pressure-Volume work

• The force on the piston is pA (p pressure, A area, because F=p/A)

• If the piston moves “dx”, it does an amount of work: dW = pAdx

• The volume of the cylinder is changed by: dV = Adx

• Thus dW = pdV ,

^{W }  ^pdv

2.8 Specific Heat at Constant Pressure

• If the pressure is kept constant, in order increase the temperature by the same 1 K, more energy is needed. The extra energy is required because in addition to

increaseing the internal energy, heat must also do work lifting the piston.

• Frictionless Cylinder

• The ratio of the two specific heats is

p is

constant