Ideal Gases-Microscope DefinitionWe define an ideal gas to have the following properties:

1 -There are no atomic interactions among the molecules or atoms comprising the gas, therefore, there is no internal potential

energy P.E resulting from such interactions, only kinetic energy 2 -The sizes of the atoms or molecules is extremely small

compared with their separations.3 -when the basic particles collide, they do so in a perfectly

elastic way

T constant (isothermal) P constant (isobaric) V constant (isochoric)

parabolas Straight line Straight line

Combinig these three relationships we get

PV=nRT or PV=RT

n= numper of molesN=numper of molecules

NA= Avogadro cnstantm= mass of gasMW= molar weight of gasR= gas constant

AN

Nn

WM

mn

Kinetic Theory of Gases

Kinetic theory of gases was built on several assumption which are:

Calculate the pressure of an ideal gas from Kinetic Theory

We consider a gas in a cubical vessel whose walls are perfectly elastic. Let each edge be of length L. call the faces normal to the x-axis A1 and A2 each of area L2.

Consider a molecule which has a velocity v which resolve into components vx, vy,

and vz.

If particle collides with A1, it rebound with its x- component of velocity

reversed.

∆P= mvx-(-mvx) =2mvx

Assuming no collisions in between, the round trip will take a time

The number of collisions per unit time is , so the change in momentum per unit time is =

Average force ).....( 223

22

21 xnxxx vvvv

L

mF ,Pressure P =

).....( 223

22

213 xnxxxx vvvv

L

mP ).....(

.22

322

212 xnxxxx vvvv

LL

mP

).....( 223

22

213 ynyyyy vvvv

L

mP ).....( 22

322

213 znzzzz vvvv

L

mP

)]....(

).....().....[(33

223

22

21

223

22

21

223

22

213

znzzz

ynyyyxnxxxzyx

vvvv

vvvvvvvvL

mPPPP

n

vvvvC n

223

22

212 ....

Pressure of gas equal in all direction

The root-mean square speed of the molecules is

).....(3

223

22

213 nvvvv

L

mP

The volume V=L3

M ( mass of gas)= m n ,

Units of pressure

1 -dyne/cm2 if ρ = g/cm3 and C = cm/s

2 -N/m2 if ρ = kg/m3 and C = m/s

3 -1 pascal (Pa) = 1 N/m2

4 -1 atom = ρ g h = 1.01ᵡ 105 N/m2

5 -Bar

The average kinetic energy (E) of molecules

𝜌=𝑀𝑉

𝜌=𝑀 If V=1 𝐸=12ρ𝐶2

𝑃=13𝑚𝑉𝑛𝐶

2

𝑃=23𝐸

The relationship between the kinetic energy and temperature:

𝑃=13𝜌𝐶2

𝑃𝑉=13𝑚𝑛𝐶2

,n=N 𝑃𝑉=13𝑚𝑁𝐶2

𝑃𝑉=𝑅𝑇 𝑅𝑇=13𝑚𝑁 𝐶2

𝑚=3𝑅𝑇𝑁 𝐶2

𝐸=12𝑚𝐶2 𝐸=3𝑅𝑇𝐶2

2𝑁𝐶2 𝐸=32𝑅𝑇𝑁

𝐾=𝑅𝑁 𝐸=

32𝐾 𝑇

𝐶2∝T𝐶1

𝐶2

=√ 𝑇1

𝑇2

The ideal gases laws from the kinetic energy theory of gases.

 1-Boyle's law

𝑃=13𝜌𝐶2¿ 1

3𝑚𝑉𝐶

2

2-charle's law

𝑃𝑉=13𝑚𝐶2

If T is constant ,C

also constant𝑃𝑉=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑃=13𝜌𝐶2¿ 1

3𝑀𝑉

𝐶2

𝑃𝑉=13𝑀𝐶2

M= m N

𝑃𝑉=13𝑚𝑁𝐶2

𝐸=32𝐾 𝑇𝑎𝑛𝑑𝐸=

12𝑚𝐶2 m𝐶2=3𝐾𝑇

𝑃𝑉=13

(3𝑘𝑇 )𝑁=𝑁𝑘𝑇 𝑁𝑘=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡V ∝T If P is constant

Van der Waals Equation

The Ideal Gas Law is based on the kinetic molecular theory assumptions that gases are composed of point masses that undergo perfectly elastic collisions, gas particles are much smaller than distance between particles, therefor the volume of a gas is mostly empty space and the volume of the gas molecules themselves is negligible, and there is no force of attraction between gas particles or between the particles and the walls of the container. However, real gases deviate from those assumptions at low temperatures or high pressures.The van der Waals equation is:

 

where P is the pressure, V is the volume, R is the universal gas constant, n is number of moles and T is the absolute temperature.

The constants a and b represent the magnitude of intermolecular attraction and excluded volume respectively, and are specific to a particular gas.

1-To calculate volume of real gas:

Approximate V → nR/TP

+nb

2-To calculate pressure of real gas:

Diviation are greater if:1 -intermolecular attractive forces are greater.

2 -mass and subsequently volume of gas molecules is greater

High T and low P Low T and high P

Conditions are (Ideal) at

Conditions are (Real) at

3 -Calculation the temperature of Boyle TB:

𝑃𝑉 +𝑎𝑉− Pb−

ab

𝑉 2 =RT

very small can negligible

𝑃𝑉 +𝑎𝑉− Pb=RT

𝑉=𝑅𝑇𝑃

𝑃𝑉=𝑅𝑇 +𝑃𝑏−𝑎𝑃𝑅𝑇

𝑃𝑉=𝑅𝑇 +𝑃 (𝑏−𝑎𝑅𝑇

)

At T=TB𝜕𝑃𝑉𝜕𝑃

=0

𝜕𝑃𝑉𝜕𝑃

=0+b−a

𝑅𝑇 𝐵

=0 𝑏=𝑎

𝑅𝑇 𝐵𝑇 𝐵=

𝑎𝑅𝑏

If T< TB b≈ 0

(𝑃+𝑎𝑉 2 )V=RT 𝑃𝑉=𝑅𝑇 −

𝑎𝑉

PV< RTDecreases the

pressure increases

If T> TBV smal ≈ 0

𝑃 (𝑉 −b)=RT 𝑃𝑉=RT+PbPV> RT

And increases with pressure