Ideal gases are gases of low pressure, or gases that have a low density.

• We use ideal gases to approximate the more complex relationships that can exist.

• Relationships between the pressure, volume and temperature are called equations of state.

• These equations can be very complicated but relationship between pressure , volume and temperature can be found for ideal gases and is much simpler than the more complex general equations of state.

Kinetic Theory of GasesKinetic Theory of Gases• A theory that attempts to explain the behavior of an ideal gas.

•An ideal gas can be defined as: The gas that obeys the assumptions of the kinetic theory of gases.

The assumptions Kinetic Theory

• Gas consists of large number of tiny particles (atoms or molecules) with freedom of movement

• There exist no external forces (density constant) and no forces between particles except when they collide.

• Particles, on average, separated by distances large compared to their diameters

• Particles make elastic collisions with each other and with walls of container

• The average kinetic energy of the gas particles is directly proportional to the Kelvin temperature of the gas

KINETIC THEORY

From Newton’s LawThe internal energy of monatomic ideal gas

For a single molecule, the average force is:

For N molecules, the average force is:

root-mean-squarespeed

V: is the volume

Hence, The pressure is given by:

We can relate the state equation of the ideal gas with its average kinetic energy as

The internal energy of monatomic ideal gas

kB = R/NA & / An N NBut

The Equation of State of Ideal Gases

P – pressure [N/m2]V – volume [m3]n – number of moles of gasT – the temperature in Kelvins [K]

R – a universal constant

nRTPV

KmolJR

31.8

The ideal gas equation of state:

An equation that relates macroscopic variables (e.g., P, V, and T) for a given substance in thermodynamic equilibrium.In equilibrium ( no macroscopic motion), just a few macroscopic parameters are required to describe the state of a system.

f (P,V,T) = 0

Geometrical representation of the

equation of state:

P

VT

an equilibriumstate

the equation-of-state surface

R= 0.08214 L atm/mol K

Mm – Molar mass [g/mol]N – Number of molecules [molecules]NA – Avogadro’s Number of molecules per mole [molecules/mol]

mMmn

ANNn

Where, TNkPV B

the Boltzmann constant

kB = R/NA 1.3810-23 J/K(introduced by Planck in 1899)

Avogadro’s Law: equal volumes of different gases at the same P and T contain the same amount of molecules.

Ideal gas, constant mass (fixed quantity of gas)

1 1 2 2

1 2

PV P VT T

Boyle's Law (constant temperature) P = constant / V

Charles Law (constant pressure) V = constant T

Gay-Lussac’s Law (constant volume) P = constant T

Important laws

Isothermals pV = constant

0

20

40

60

80

100

120

140

160

180

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

volume V (m3)

pres

sure

p (k

Pa)

100 K200 K300 K400 K

n RTpV

Energy in terms of Temperature

At T = 300K

32 BKE k T

The kinetic energy is proportional to the temperature, and the Boltzmann constant kB is the coefficient of proportionality that provides one-to-one correspondence between the units of energy and temperature.

If the temperature is measured in Kelvins, and the energy – in Joules: kB = 1.38×10-23 J/K

23

19

1.38 10 300 261.6 10Bk T meV

In General

The internal energy of an ideal gas with f degrees of freedom:

TNkfU B2

f 3 (monatomic), 5 (diatomic), 6 (polyatomic)

How does the internal energy of air in this (not-air-tight) room change with T if the external P = const?

2 2Bf fU Nk T PV

does not change at all, an increase of the kinetic energy of individual molecules with T is compensated by a decrease of their number.

Real Gases General Observations

• Deviations from ideal gas law are particularly important at high pressures and low temperatures

• Real gases differ from ideal gases in that there can be interactions between molecules in the gas state– Repulsive forces important only when molecules are

nearly in contact, i.e. very high pressures• Gases at high pressures , gases less compressible

– Attractive forces operate at relatively long range (several molecular diameters)

• Gases at moderate pressures are more compressible since attractive forces dominate

– At low pressures, neither repulsive or attractive forces dominate → ideal behavior

Real Gases Real Gases Deviations from IdealityDeviations from Ideality

J. van der Waals, 1837-1923, J. van der Waals, 1837-1923, Professor of Physics, Amsterdam. Professor of Physics, Amsterdam. Nobel Prize 1910.Nobel Prize 1910.

. The Ideal Gas Law ignores both the volume occupied by the molecules of a gas and all interactions between molecules, whether attractive or repulsive

In reality, all gases have a volume and the molecules of real gases interact with one another.

For an ideal gas, a plot of PV/nRT versus P gives a horizontal line with an intercept of 1 on the PV/nRT axis.

The reasons for the deviations from ideality are:• The molecules are very close to

one another, thus their volume is important.

• The molecular interactions also become important.

Real gases behave ideally at ordinary temperatures and pressures. At low temperatures and high pressures real gases do not behave ideally.

Real Gases Van Der Waals Equation

Real gases do not follow PV = nRT perfectly. The van der Waals equation corrects for the nonideal nature of real gases.

a corrects for interaction between atoms.

b corrects for volume occupied by atoms.

Real Gases Van Der Waals Equation

• A non-zero volume of molecules = “nb” (b is a constant depending on the type of gas, the 'excluded volume‘, it represents the volume occupied by “n” moles of molecules).

• The molecules have less free space to move around in, so replace V in the ideal gas equation by V - nb

• Very roughly, b 4/3 r3 where r is the molecular radius.

Real Gases :Van Der Waals Equation

The attractive forces between real molecules, which reduce the pressure:

p wall collision frequency and p change in momentum at each collision.

Both factors are proportional to concentration, n/V, and p is reduced by an amount a(n/V)2, where a depends on the type of gas.

[Note: a/V2 is called the internal pressure of the gas]. n2a/V2 represents the effect on pressure to intermolecular attractions or repulsions.

If sulfur dioxide were an “ideal” gas, the pressure at 0°C exerted by 1.000 mol occupying 22.41 L would be 1.000 atm. Use the van der Waals equation to estimate the “real” pressure.

Example:

Solving the van der Waals equation for pressure.

R= 0.0821 L. atm/mol. KT = 273.2 KV = 22.41 L

a = 6.865 L2.atm/mol2b = 0.05679 L/mol

The “real” pressure exerted by 1.00 mol of SO2 at STP is slightly less than the “ideal” pressure.

Work and Heating (“Heat”)We are often interested in U , not U. U is due to:

Q - energy flow between a system and its environment due to T across a boundary and a finite thermal conductivity of the boundary

– heating (Q > 0) ---------cooling (Q < 0)W - any other kind of energy transfer across boundary ( work )

Heating/cooling processes:conduction: the energy transfer by molecular contact – fast-moving molecules transfer energy to slow-moving molecules by collisions;convection: by macroscopic motion of gas or liquidradiation: by emission/absorption of electromagnetic radiation.

HEATING

WORKWork and Heating are both defined to describe energy transfer across a system boundary.