Gas Pressure

Gas Pressure

Air Pressure

Pressure Units

• Units of pressure: atmosphere (atm)

Pa (N/m2, 101,325 Pa = 1 atm)Torr (760 Torr = 1 atm)

bar (1.01325 bar = 1 atm)

mm Hg (760 mm Hg = 1 atm)

lb/in2 (14.696 lb/in2 = 1 atm)

in Hg (29.921 in Hg = 1 atm)

Universal Gas Behavior

• Unlike solids and liquids, gas behavior is generally independent of chemical identity.

• Depends on four things only:– Absolute temperature– Pressure– Volume– Amount (moles)

Kinetic Molecular Theory

• This theory presents physical properties of gases in terms of the motion of individual molecules.

• Kinetic Theory (in this class) will be based upon six assumptions:

• Average Kinetic Energy Kelvin Temperature

• Gas molecules are points separated by a great distance

• Particle volume is negligible compared to gas volume

• Gas molecules are in rapid random motion

• Gas collisions are perfectly elastic

• Gas molecules experience no attraction or repulsion

Gas Behavior:Gases in a Box

• Insert 1 mole of gas into a fixed volume container. Then:

1. Gas expands to fill the container. Why?

2. The pressure becomes whatever value the gas laws dictate for that volume, mole, and temperature combination.

Gas Behavior:Gases in a Piston

• Insert 1 mole of gas into a piston. Then:

1.Gas fills the piston. Why?

2.The piston changes volume until the pressure inside is equal to the pressure outside. Why?

Understanding the Gas Laws

• Two keys to understanding the gas laws:– Understand which parameters are changing– Understand which are NOT changing

Boyle’s Law• Pressure–Volume Law (Boyle’s Law):

Boyle’s Law• Pressure–Volume Law (Boyle’s Law):

• The volume of a fixed amount of gas maintained at constant temperature is inversely proportional to the gas pressure.

Pressure1

Volume

XPV 11

Charles’ Law

• Temperature–Volume Law (Charles’ Law):

Charles’ Law• Temperature–Volume Law (Charles’ Law):

• The volume of a fixed amount of gas at constant pressure is directly proportional to the Kelvin temperature of the gas.

V T

XT

V

1

1

Avogadro’s Law

• The Volume–Amount Law (Avogadro’s Law):

Avogadro’s Law• The Volume–Amount Law (Avogadro’s Law):

• At constant pressure and temperature, the volume of a gas is directly

proportional to the number of moles of the gas present.

nV

Xn

V

1

1

Collecting the Gas Laws

• Mathematically one can combine all of the statements we’ve made about gases.

• Two equivalent equations come from this:– Combined gas law– Ideal gas law

Combined Gas Law• Combining the law gives:

• But if it equals a constant, then after any change it will still be equal to the constant:

• We write it this way:

• Nothing needs to be held constant now• Remember that anything that does stay constant can be

cancelled.

XTn

VP

1

1

11

2

22

1

11 Tn

VP

Tn

VP

21

4

44

3

33

2

22

1

11 X Tn

VP

Tn

VP

Tn

VP

Tn

VP

4321

Ideal Gas Law

• This constant “X” is just a number.

• Units of (pressure * volume) / (moles * temp)

• That is, L·atm·K–1·mol–1

• Numerically, this constant has a value of R = 0.08206 L·atm·K–1·mol–1

Ideal Gas Law

• The equation then becomes

We usually write it this way instead:

PV = nRT

RTn

VP

STP

• Standard temperature: 273.15 K

• Standard pressure: 1 atm

Ideal gas law vs. combined gas law

• Ideal gas law– Under unchanging conditions

• Combined gas law– Under changing conditions

What is the volume of one mole of helium gas at STP?

22.4 L

What is the volume of one mole of argon gas at STP?

22.4 L

What is the volume of one mole of radon gas at STP?

22.4 L

What is the density of one mole of helium gas at STP?

4.003 g / 22.4 L = 0.179 g/L

What is the volume of one mole of argon gas at STP?

39.948 g / 22.4 L = 1.78 g/L

What is the volume of one mole of radon gas at STP?

222 g / 22.4 L = 9.91 g/L

What information would you need to calculate the molar mass of a gas?

• Mass / moles (m / n)• Enough information to get mass• P,V,T to use ideal gas law to get n

• What is the molar mass of a gas with a density of

1.342 g/L–1 at STP?

mole

g

mole

STPatL

L

g06.30

1

4.22

1

342.1

Funky questions

• At what temperature do you have 0.1 moles/atm of helium in a 1 L pure helium sample?

• In one mole of chlorine gas at STP, how many Kelvins are there per liter?

K

KmolatmL

moles

Latm

nR

PVT 9.121

08206.01.0

11

L

K

KmolatmL

mol

atm

nR

P

V

T2.12

08206.01

1

Gas-phase stoichiometry

• We have a new route to moles PV=nRT

• But we need to know first how two different gases behave when in the same space

Gas Mixtures

• Two gases in the same container have the same volume—whatever the volume of the container is.

• Two gases in the same container have the same temperature—whatever the temperature is inside the container.

Gas Mixtures

• Two gases in the same container do NOT have the same pressure.

• They have whatever pressure they would have if they were in the container alone.

• That is, solve PV=nRT for each gas in the mixture separately.

Gas Mixtures

• The total pressure inside the container is the sum of the pressures of the individual gases.

• Dalton’s Law of Partial Pressures

i

itotal PP

New Density Unit: Mole Fraction

• For a two-component system, the moles of components A and B can be represented by the mole fractions (XA and XB).

1 BABA

BB

BA

AA

XX

nn

nX

nn

nX

Gas Stoichiometry

• In gas stoichiometry, for a constant temperature and pressure, volume is proportional to moles.

• Assuming no change in temperature and pressure, calculate the volume of O2 (in liters) required for the complete

combustion of 14.9 L of butane (C4H10):

2 C4H10(g) + 13 O2(g) 8 CO2(g) + 10 H2O(l)

Molecular Speed

• It can be shown that:

• So then for neon:

M

RTvrms

3

Molar mass

hr

milesm

molg

KmolKJ

M

RTvrms 3000sec136000

00.4

298314.833

Mean Molecular Speeds

Collisions

• It can be shown that:

• A room temp gas collides billions of times per second

• The mean free path is less than 100 nm.

kT

Pvz mean

P

kT

2

Collision frequency Mean free path

Maxwell speed distribution curves.

Same Behavior vs. Different Behavior

• Most gas behaviors are based upon comparisons of their relative energies (temperatures)– Same temperature = same behavior

• Some gas behaviors are based upon comparisons of their relative speeds– Same speed = same behavior

• Diffusion is the mixing of different gases by random molecular motion and collision.

Graham’s Law

Graham’s Law

• Effusion is when gas molecules escape without collision, through a tiny hole into a vacuum.

Graham’s Law

• Graham’s Law: Rate of effusion is proportional to its rms speed, vrms.

• For two gases at same temperature and pressure:

M

RTRate rms

3 v

Rate1

Rate2

M2

M1

M2

M1

Behavior of Real Gases

• Test of ideal gasbehavior.

• Z = PV/RT

Compressibility factor

This plot assumes room temperature.

Real Gases

• All the assumptions of kinetic molecular theory break down when explored in sufficient detail.

• Two assumptions break down first:– The volume of gas molecules is negligible– There are no attractive or repulsive forces

between molecules

Non-negligible volumes

• The volume of molecules affects pressure-volume behavior more than temperature-pressure behavior.

• For a given small volume, the pressure will be higher than the ideal gas suggests..

Behavior of Real Gases

• Test of ideal gasbehavior.

Volume non-idealities seen here!

Non-negligible interactions

• The long-range interactions of particles are attractions, not repulsions.

• Thus a real gas sample takes up less space than the ideal gas law suggests, when the molecules are not crowded together.

• This effect fades as molecules move faster.

Behavior of Real Gases

• Test of ideal gasbehavior.

Attractive force non-idealities seen here!

Behavior of Real Gases• Corrections for non-ideality require a non-ideal gas

law. The van der Waals equation is one of them:

nRTbnVV

naP –

2

2

IntermolecularAttractions

ExcludedVolume

Van der Waals Constants

Gas a

(L2 atm / mole2)

b

(L / mole)

Helium (He) 0.03412 0.02370

Ammonia (NH3)

4.170 0.03707

Hydrogen (H2) 0.2444 0.02661

n-octane 37.32 0.2368

Water 5.464 0.03049

Carbon dioxide 3.592 0.04267

Other gas laws

• van der Waals:

• Peng-Robinson:

• Redlich-Kwong:

B

nV

nV

T

A

BnVRT

P

nV

nV

nV

nVRT

P

nV

RT

a

bnVRT

P

Unifying the Gas Laws

• Under normal temperatures you can liquefy a gas simply by raising the pressure

• Above a certain critical temperature (Tc) you cannot liquefy a gas under any pressure. The pressure and volume of that “last” liquid are Pc and Vc

Critical Constants

Species Tc (K) Pc (atm) Vc (L)

Helium 5.195 2.2452 0.0578

Ammonia 405.3 109.84 0.0725

Water 647.126 217.66 0.05595

“Critical” adjustments

• Now we stop using temperature (and pressure and volume) in the gas laws.

• Instead we write the reduced temperature (TR) as a fraction of the critical temperature (Tc).

• That is TR = T / Tc

Compressibility factor plots redone

Atmosphere

Smog (Inversions)

32

2

OOO

ONOhNO

Brownish haze

Acid Rain

4223

322

22

22

SOHOHSO

SOOSO

SOOS

Global Warming