Thermal Physics3.2 Modelling a gas

Understanding Pressure Equation of state for an ideal gas Kinetic model of an ideal gas Mole, molar mass, and the Avogadro

constant Differences between real and ideal gases

Applications and Skills Equation of state for an ideal gas Kinetic model of an ideal gas Boltzmann equation Mole, molar mass, and the Avogadro

constant Differences between real and ideal gases

Equations Pressure p = F/A # moles of a gas as

ratio of # molecules to Avogadro’s constant n = N/NA

Equation of state for an ideal gas pV=nRT

Pressure and mean square velocity of an ideal gas

cp 2

31

Understand the proof for the formula

cp 2

31

Equations Mean kinetic energy of ideal gas molecules Ek(mean) = (3/2)kBT = (3/2)(R/NA)T

The Gas Laws (1) Developed independently & experimentally between mid 17th and

start of 19th centuries Ideal gases defined as those which obey the gas laws under all

conditions i.e. no intermolecular interactions between molecules and only exert forces when colliding. Real gases only approximate ideal gases as long as pressures are slightly greater than normal atmospheric pressure

Boyle showed that p α 1/V or pV = k(at const temp) pV graphs aka isothermal curves Charles, around 1787 confirmed that all gases expanded by equal

amounts when subjected to equal pressure. The volume changed by 1/273 of the volume at zero. At -273 °C volume becomes zero.

For a fixed mass at constant pressure, volume directly proportional to absolute temperature V α T (const pressure)

V/T = constant (at constant pressure)

The Gas laws (2) Third gas law, for a gas of fixed mass and

volume, the pressure is directly proportional to the absolute temperature

p α T (const volume) p/T = constant (at const volume)

Avogadro stated that the number of particles in a gas at const temp and pressure is directly proportional to the volume of the gas

n α V n/V = constant

Gas laws (3) Combining the four equations and four

constants gives pV/nT = R or pV = nRT R = 8.31 JK-1mol-1 when p in pascals , V in

m3, n = # moles of gas

The mole and Avogadro’s constant

The mole (mol) measures the amount of substance something has and is one of he seven base SI units.

Defined as the amount of substance having the same number of particles as there are neutral atoms in 12 grams of carbon – 12

One mole of gas contains 6.02 x 1023 atoms or molecules (Avogadro’s constant NA ). So 2 moles of oxygen gas contains 12.04 x 1023 molecules.

Molar mass Since diatomic gases have two atoms per molecule a

mole of a diatomic gas will have 6.02 x 1023 molecules but 12.04 x 1023 atoms.

One mole of oxygen atoms has approximate mass 16.0 g so a mole of oxygen molecules will have mass 32.0 g i.e. its molar mass.

Consider one mole of CO2 (g) which contains one mole of carbon atoms has mass 12.0 g and one mole of oxygen molecules 32.0 g. The molar mass of CO2 is 44.0 g mol-1

Example 1

Molar mass of Oxygen is 32 x10-3 kg mol-1 If I have 20g of Oxygen, how many moles do I

have and how many molecules? Molar mass of Oxygen gas is 20 x 10-3 kg / 32 x10-3 kg mol-1 0.625 mol 0.625 mol x 6.02 x 1023 molecules 3.7625 x 1023 molecules

Example 2 Calculate the percentage change in the

volume of a fixed mass of an ideal gas when its pressure is increased by a factor of 3 and its temperature increases from 40.° C to 100.° C.

n is constant so (p1V1/T1) = (p2V2/T2) V2/V1 = [(p1T2 )/(p2T1) = 373/(3 x 313) ≈0.40 i.e. a 60% reduction in volume of gas

Kinetic Model of Ideal gases- Key assumptions Gas consists of large number of identical tiny particles- molecules in

constant random motion Statistical averages of this number can be made Each molecule’s volume is negligible when compared to the volume of the

whole gas At any instant as many molecules are moving in one direction as any other

direction Molecules undergo perfectly elastic collisions between each other and with

the walls of the container; momentum is reversed during collision No intermolecular forces between molecules between collisions i.e. energy

is completely kinetic Duration of collision negligible compared with the time between collisions Each molecule produces a force on the wall of the container The forces of individual molecules will average out to produce a uniform

pressure throughout the gas- ignoring the effect of gravity

Developing a relation between pressure and density (1) Refer to text p. 107 Figure 7 Consider one molecule which has

momentum and collides elastically with the right side of the box of length L

∆p = -2mcx F= ∆p/∆t and ∆t =2L/cx so F=(-mcx

2/L) is force of box on molecule Newton’s III states molecule exerts an equal and opposite force F=mcx

2/L on box

Developing a relation between pressure and density (2)

N molecules exert a total force Fx= (m/L)(cx1

2 + cx22+ cx3

2+ … cxN2)

The forces average out giving a constant force with so many molecules

The mean value of the square of the velocities (cx1

2 + cx22+ cx3

2+ … cxN2)/N

The total force on right hand wall is Fx= (m/L)

2c

2c

Developing a relation between pressure and density (3) From Pythagoras,

2

223

2xx

2

2222

2222

31

mass totalis Nmbut 31

31

&AF p since and

31

now31or 3

sodirection any in movingmolecule of likelihood equalan is there

averageon and

cp

cVNmc

LNmp

LAcLNmF

cccc

cccc

x

xx

zyx

Molecular interpretation of temperature

cp 2

31 2

3c

VNmp

222

21)

323(

23

3mcNcNmpVcNmpV

22

21

23 so

nNbut

21

23 cm

NRTNcm

NnRT

AA

Boltzmann Constant

21

23

,/ as defined isconstant new a If

2cmTk

thenNRk

B

AB

123123

11

1038.11002.6

3.8

JKxmolxKJmol

NRkA

B

Linking temperature with energy The kinetic theory now links temperature with

the microscopic energies of the gas molecules The equation resembles the kinetic energy

formula. Adjusting for N molecules gives 3/2 NkBT This represents the total internal energy of an

ideal gas (only considering translational motion of molecules of monoatomic gases)

Alternative equation of state for ideal gas

231002.6 then , 1 if

and , but ,

xNNmoln

NknRTNknRTsoTNkpVnRTpV

A

BB

B

Real Gases vs Ideal Gases (1) Since an ideal gas obeys the ideal gas laws under

all conditions, ideal gases cannot be liquefied In 1863, Andrews experiments showed a deviation

from Boyles’ pV curves for CO2 at high pressures and low temperatures

Later experimentation showed that real gases do not behave like ideal gases and that all gases can be liquefied at high pressures and low temperatures

Real Gases vs Ideal gases (2) The ideal gas law describes the behaviour of all gases at

relatively low pressures and high temperatures. The ideal gas law fails when the main assumptions of

the Kinetic Theory are invalid i.e molecular volumes and intermolecular forces are negligible.

Real gases can only be compressed so far indicating that molecules occupy a non negligible volume and weak attractive forces exist between the molecules of real gases just as those between molecules of liquids

Deviation of a real gas from an ideal gas Real Gases vs Ideal gases (3)

Source Google images

Real Gases vs Ideal gases (4) Short range repulsive forces act between gas molecules

when they approach each other reducing the distance they can effectively move i.e. reducing gas volume below the V used to develop ideal gas law.

At slightly greater distances molecules attract each other slightly forming small groups and reducing the effective number of particles. This slightly reduces the pressure.

Enter van der Waals who modified the ideal gas law introducing two new constants.

No single simple equation of state applying to all gases has been found to this date