Gases and Their Properties

Effectsoftheatmosphericpressure

Standard Atmospheric Pressure

1.00 atm

760 mm Hg, 760 torr

101.325 kPa

1.01325 bar

1013.25 mbar

Measurements of Pressure: barometers

g • h • ρHg =

760 mm Hg = 760 torr =1atm =

980.665 cm • s-2 • 13.5951 • 10-3 kg • cm-3 • 76.0 cm =

1.033 kgp • cm-2 =

1.01325 × 105 N × m-2

P=FS

Manometers

Boyle’s Law

Robert Boyle (1627-1691)

Boyle’sLaw

�

PV = K

•  Boyle1662 PV = constant

P∝1V

Compressibility

γ =−1VΔVΔP

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

V K1=T

Charles’ Law

P K2 =T

Charles’ Law

Jacques Charles 1787

V = b T

Charles’s Law V∝T

Gay-Lussac 1802

V K3 =T

V K4 =n

�

PV = nRT

Charles’andAvogadro’sLaws

Dalton’s Law of Partial Pressure

Temperature 0 ºC ~ 273 K Pressure 1 atm

One mole of any gas, behaving ideally, occupies 22.4 L at normal conditions

Ntotal = NA + NB +… Ptotal = PA + PB +…

Ptotal = NtotalRT/V Ptotal = NART/V + NBRT/V +…

We see from this simple derivation that: it is the numbers of molecules, not the kind of molecules that are important.

Compression factor Z The variation of the compression factor Z = PV/RT with pressure for several gases at 0°C. A perfect gas has Z = 1 at all pressures. Notice that, although the curves approach 1 as P 0, they do so with different slopes.

The ideal gas law PV = nRT

PV = nRT

∂ pv( )∂p

⎡

⎣⎢⎢

⎤

⎦⎥⎥

T

= 0 p → 0

Theidealgaslaw

The virial expansion, also called the virial equation of state, is the most interesting and versatile of the equations of state for gases. The virial expansion is a power series in powers of the variable, n/V, and has the form,

PVnRT =Z=1+BT⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

nV +C T⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

nV⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2

+…

The coefficient, B(T), is a function of temperature and is called the "second virial coefficient”. C(T) is called the third virial coefficient, and so on. The expansion is, in principle, an infinite series, and as such should be valid for all isotropic substances. In practice, however, terms above the third virial coefficient are rarely used in chemical thermodynamics.

In thermodynamics, the Boyle temperature is defined as the temperature for which the second virial coefficient, B2(T) vanishes, i.e. B2(T) = 0. Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature. In any case, when the pressures are low, the second virial coefficient will be the only relevant one because the remaining concern terms of higher order on the pressure. We then have dZ/dp = 0 at p = 0, where Z is the compression factor.

The second virial coefficient, B(T), is an increasing function of temperature throughout most of the useful temperature range. (It does decrease slightly at very high temperatures.) B is negative at low temperatures, passes through zero at the so-called "Boyle temperature" and then becomes positive. The temperature at which B(T) = 0 is called the Boyle temperature because the gas obeys Boyle's law to high accuracy at this temperature. We can see this by noting that at the Boyle temperature the virial expansion looks like:

PVnRTB

=Z=1+0+C T⎛⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

nV⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2

+…

If the density is not too high the C term is very small so that the system obeys Boyle's law.

P

Vm

Gas ideale

liquido gas

T > Tc Tc T < Tc

Liquido + gas

Punto Critico

P

Vm

ValoriCriticiL’isotermacritica,perT=Tchaunruolospeciale:Per T > Tc non è possibile liquefare il sistema,indipendentementedallapressioneIvaloridipressioneevolume,corrispondentiallatemperaturacritica,vengonoindicaticonpceVcTcpceVcsichiamanoparametricritici

Vapore saturo Liquido saturo

Nature’s Forces

J. van der Waals, 1837-1923, Premio Nobel 1910.

EquazionediVanderWaals

È utile avere una equazione di stato approssimata che midescrivaigasrealiVarieequazioni,piùomenoempiriche,sonostatesviluppate.Lapiùnotaèstata ideatadaVanderWaals,nellasuatesidiLaurea

AttrazioneMolecolare

EquazionediVanderWaals

Partiamodall’equazionedeigasidealipV=nRTLemolecoleacortadistanzasirespingono,quindiilvolumeaccessibilealgasnonèVma(V-nb)Lapressionedipendedallafrequenzadicollisioneconilrecipiente,edallaforzaconcuiavvengonogliurti.Ambeduequesti fattori vengonoattenuatidallapresenzadi forzeattrattive, proporzionalmente al numero di molecole, per unfattorepariaa(n/V)2

IsotermediVanderWaals

∂p∂V

⎛⎝⎜

⎞⎠⎟T

= 0

∂2p∂V 2

⎛

⎝⎜⎞

⎠⎟T

= 0

⎧

⎨⎪⎪

⎩⎪⎪

∂p∂V

⎛⎝⎜

⎞⎠⎟T

= − RT

V - b( )2 + 2aV 3 = 0

∂2p∂V 2

⎛

⎝⎜⎞

⎠⎟T

= 2RT

V - b( )3 − 6aV 4 = 0

⎧

⎨

⎪⎪

⎩

⎪⎪

ParametriCriticidiVanderWaals

Calcoliamoilvaloredeiparametricritici.Pendenza e curvatura dell’isoterma, per pressione e volumecritici,sononulle

ParametriCriticiVanderWaals

Vc = 3b; pc =a

27b2 ; Tc =8a

27Rb;

Zc =pcVcRTc

= a27b2 3b( ) 27Rb

R8a= 3

8= 0.375

PrincipiodegliStatiCorrispondenti

Unatecnicageneralenellascienzaperconfrontareoggetti“simili”èquelladitrovareunaproprietà“intrinseca”fondamentale,eriferirelevarieproprietàaquella,usatacome“unitàdimisura”Ognigaspossiededeiparametricriticiunici.Possiamo sperare che, usando questi parametri come unità dimisura,ilcomportamentodeivarigasrealipossaessereunificatoVarderWaalsprovòperprimoquestoapproccio

Van der Waals osservò che, sperimentalmente, molte sostanze obbediscono alla stessa equazione di stato, se scritta in termini di variabili ridotte.

Vr =

VVc

pr =ppc

Tr =TTc

VariabiliRidotte

Introduciamolecosiddettevariabiliridotte

PrincipiodegliStatiCorrispondenti

I parametri a e b sono scomparsi!!

pr = f (Tr ,Vr ) pr =

8Tr

3Vr −1− 3

Vr2

PrincipiodegliStatiCorrispondenti

Gas reali, allo stesso volume ridotto e alla stessa temperatura ridotta, esercitano la stessa pressione ridotta

PVm

nRT =Z =1+B T⎛⎝⎜⎞⎠⎟

1Vm

+C T⎛⎝⎜⎞⎠⎟

1Vm

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+…

P = RTVm − b

− aVm2

P = RTVm

1

1− bVm

− aRTVm

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

P = RTVm

1

1− bVm

− aRTVm

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

1− x{ }−1 =1+ x + x2 + ...

P = RTVm

1+ bVm

− aRTVm

⎧⎨⎩

⎫⎬⎭= RTVm

1+b − a

RTVm

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

P = RTVm

1+ bVm

− aRTVm

⎧⎨⎩

⎫⎬⎭= RTVm

1+b − a

RTVm

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

B T( ) = b − a

RT= 0

TB =

abR

= 278TC

Vc = 3b; pc =a

27b2

Tc =8a

27Rb; Zc =

pcVcRTc

= 38= 0.375

While it has been widely applied to describe interactions between atoms and molecules, a major shortcoming of the Lennard-Jones Potential is that it is spherically symmetric. Important non-bonded interactions, such as hydrogen bonding, are often highly directional in character, and require more elaborate potentials with angle dependent terms.

Lennard-Jones Potential

Kinetic Theory of Gases

Assumptions

1.  An ideal gas comprises discrete particles (molecules or atoms);

2.  The particles are far apart and are assumed to occupy zero volume;

3.  The particles are in constant motion described by well-defined laws;

4.  There are no attractive forces between gas molecules or between molecules and the sides of the container with which they collide;

5.  Molecules collide with one another and the sides of the container;

6.  Energy is conserved in these collisions, although one molecule may gain energy at the expense of the other.

�

P = forcearea

= forcecollision

× number of collisions area

�

P = forcearea

= momentum transfercollision

⎡

⎣ ⎢ ⎤

⎦ ⎥ ×number of collisions

time × area⎡

⎣ ⎢ ⎤

⎦ ⎥

Newton’s second law states:force = time rate of change of momentum

The rate of collision per unit area must be proportional to: u  t h e n u m b e r o f m o l e c u l e s ( N f o r

convenience) per unit volume (V); u  the molecular speed (ν). The faster they

travel, the more can hit the wall.

�

P = forcearea

= momentum transfercollision

⎡

⎣ ⎢ ⎤

⎦ ⎥ ×number of collisions

time × area⎡

⎣ ⎢ ⎤

⎦ ⎥

P = 2(m × )[(N/V) × ] = 2(N × m × 2)/V P × V = 2 N m 2

A more detailed analysis would show that, when taking into account the 3 directions of this random motion, and the six walls of cubical box, we are better served by:

We really need to be concerned with the average speed (or more precisely the average of the squares of the speeds).

PV = 2N × m × <ν2>

PV = (1/3)N × m × <2>

PV = nRT

PV = 13 ×2

2N × m × <ν2> =

23 N × (

12m <ν2>) =

23 N < εk>

Ek = 32 PV =

32 nRT

Thus we find that the average KE of a gas depends only on T.

Distribution of Molecular Speeds

We see that the molecular speed is faster at higher T and slower with large molecules.

�

νrms = 3RTM

<ν2 >= 3 RT/M (R = 8.31 j/(mol K); M = mass of a mole in kg)

ν

Besides the root-mean-square speed it would be useful a more complete picture of entire distribution of molecular speed.

ΔN f(n) Δn=N

νm is the modal speed

νav is the simple average

νrms is the root mean square

�

νrms = 3RTM

�

νav = 8RTπM

�

νm = 2RTM

�

f ν( ) = Kν 2e−mν 2

2k BT ; K = 4π m2πkBT⎛

⎝ ⎜

⎞

⎠ ⎟

32

�

ν2

0

∞∫ e−αν 2

dν = 14

πα3

�

f ν( )0

∞∫ dν = 1

�

Ik = ν k0

∞

∫ e−αν2dν

I0 = e−αν2

0

∞

∫ dν = 12

πα

I1 = ν0

∞

∫ e−αν2dν = 1

2α

�

I2 = −dI0dα

I3 = −dI1dα

I4 = +d2I0dα2

I5 = +d2I1dα2

�

νrms = 3kBTm

= 3RTM

�

ν = 8kBTπm

= 8RTπM

νmp : ν : νrms=1.000:1.128:1.225

�

νmp = 2kBTm

= 2RTM

Go to Maxwell

Distribution of Molecular Speeds

M3RTurms =

Determining Molecular Speed

•  Diffusion –  Net rate is proportional to molecular

speed. •  Effusion

–  A related phenomenon.

Gas Properties Relating to the Kinetic-Molecular Theory

•  Only for gases at low pressure (natural escape, not a jet). •  Tiny orifice (no collisions) •  Does not apply to diffusion.

A

BA

Brms

Arms

MM

3RT/MB3RT/M

)(u)(u ===

BofeffusionofrateAofeffusionofrate

Graham’s Law

•  Ratio used can be: –  Rate of effusion (as above) –  Molecular speeds –  Effusion times

–  Distances traveled by molecules –  Amounts of gas effused.