Standard T & P (STP)

22.4 L

The standard temperature and pressure for gases is:T = 273 K (0oC)P = 1 atm = 101.325 kPa = 1.01325 bar

At STP, 1 mol of any ideal gas occupies 22.4 L

Using STP in problemsA 5.00 L sample of Ar gas at STP is heated until its final pressure and temperature are 2.50 atm and 75.50°C. Calculate the new volume of the gas.

= ; V2 = ?P1 = 1 atm, V1 = 5.00 L, T1 = 273 K;

P2 = 2.50 atm, T2 = 273 + 75.50 = 348.5 KV2 = = = 2.55 L

Using PV = nRTHow many moles of NO2 gas occupy a volume of 5.00 L at 50.00°C and 725 mmHg?

PV = nRT; n = P = 725 mmHg = 0.954 atm; V = 5.00 L, T = 273 + 50 = 323 K; n = n = 0.18 moles

What mass of CO gas occupies a volume of 75.0 L at 35.00°C and 2.50 atm?

Using PV = nRT

PV = nRT; n = P = 2.50 atm; V = 75.0 L, T = 273 + 35 = 308 K; n = n = 7.42 molesmw of CO = 28.01 g/molmass = 7.42 moles * 28.01 g/mol

= 207.8 g

Gas Density• We can use PV = nRT to determine

the density of gases.• What are the units of density?

mass/volume

• What does this suggest about gas density?

It will depend strongly on T, P, and the mass of the gas molecules.

• Contrast with liquids and solids, whose densities depend somewhat on T, but far less on P.

He balloons

Hot-air balloon

What is the density of O2 gas in g/L at 25oC and 0.850 atm?Calculate # moles in 1 L, use MW of O2 to get g, divide

by V.P = 0.850 atmV = 1 L n = ?R = 0.08206 L·atm/mol·KT = 25°C + 273 = 298 K

0.850 atm 1 L

n = L atm0.08206 298 Kmol K

2 = 0.0347 mol O

2 2? g/L O = 0.0347 mol O 2

2

32.00 g O1 mol O

11 L

2

o

= 1.11 g/L O

@ 25 C, 0.850 atm

Note: A value for gas density is meaningful only if accompanied by the T and P at which it was measured.

Gas Density Example

PV = nRT n =

Gas Density and Molar MassWe can develop an expression relating density and molecular weight using PV = nRT.

Substituting:

Therefore, the density of an ideal gas can be related to the T, P, V and molecular weightAND we can use the density of a gas to determine its molar mass.

Density!!

P*MW = dRT

n = PV = nRT

PV = RT P*MW = RT

P*MW = dRT ExampleA 0.76 g sample of an unknown solid was vaporized in a 345-mL vessel. If the vapor has a pressure of 985 mmHg at 148oC, what is the molecular weight of the solid?

Heat applie

d

V = 345 mL

mass of solid = 0.76 g

P = 985 mmHg

T = 148oC

mass of gas = 0.76 g

= 1.296 atm

= 421 K

0.76 g0.345 L

d =

= 2.20 g/LMW = ?

R = 0.08206 L·atm/mol·K

= 58.65 g/mol

MW  = dRTP

 =(2.20g

L )( 0.08206  L   atmmol   K )(421 K )

1.296  atm

Chemical Equations and Calculations

Reactants ProductsMoles

MassMolarMass g/mol

Atoms (Molecules)Avogadro’sNumber 6.022 x 1023 mol-1

Solutions

Molaritymoles / L

Gases

PV = nRT

Gas Stoichiometry ExampleWhat volume of N2(g) is produced when 70.0 g NaN3 is decomposed at a pressure of 735 mmHg and at 26oC?

2 NaN3(s) 2 Na(l) + 3 N2(g)70.0 g ? L

64.99 g/mol

28.02 g/mol

735 mmHg26oC

1. The Stoichiometry Part: mass NaN3 mol NaN3 mol N2

2. The Gas Law Part: mol N2, P, T V of N2(g)

3 mol N22 mol NaN3

Example: The Stoichiometry Part

? mol N2

= 1.616 mol N2

1 mol NaN364.99 g

NaN3

= 70.0 g NaN3

What volume of N2(g) is produced when 70.0 g NaN3 is decomposed at a pressure of 735 mmHg and at 26oC?

2 NaN3(s) 2 Na(l) + 3 N2(g)

What volume of N2(g) is produced when 70.0 g NaN3 is decomposed? (P = 735 mmHg, T = 26oC)

P = 735 mmHg = 0.9671 atmV = ? L n = 1.616 mol N2R = 0.08206 L.atm/mol.KT = 26oC + 273 = 299 K

nRTV = P

PV = nRT

Example: The Gas Law Part

1 atm760

mmHg

V = (1.616  mol  N2)( 0.08206  L   atm

mol  K )(299 K )

0.9671 atm

V = 40.999 L = 41.0 L

ExampleThe active agent in many hair bleaches is hydrogen peroxide, H2O2. The amount of H2O2present can be determined by titration with a standard permanganate solution: 2 MnO4

– (aq) + 5 H2O2 (aq) + 6 H+

(aq) 5 O2 (g) + 2 Mn+2 (aq) +

8 H2O (l) Calculate the molarity of hydrogen peroxide if 28.75 mL of hydrogen peroxide produced 695 mL of oxygen gas at 0.950 atm and 315K?P = 0.950 atm; V = 0.695 L; T = 315 K

n = PV/RT = (0.950 atm)(0.695 L) (0.08206 Latm/molK)(315 K)

n= 0.0255 moles O2 = moles H2O2 (1:1 ratio)Molarity = 0.0255 moles H2O2= 0.888 M

0.02875 L

An unknown gas having a mass of 6.150 g occupies a volume of 5.00 L at 874 torr and 23.50°C. Calculate the molar mass of the unknown gas.

Example

P = 874 torr = 1.15 atm; V = 5.00 L; T = 23.50+273.15 =296.65 Kn = PV/RT = (1.15 atm)(5.00 L)

(0.08206 Latm/molK)(296.65 K)n= 0.236 moles gas MW = 6.150 g gas = 26.04 g/mol C2H2 (acetylene)? 0.236 moles gas

Dalton’s Law of Partial Pressures

For a mixture of ideal gases in a container:

total pressure = the sum of the individual gas pressures.

Recall that according to the ideal gas law, gas molecules are non-interacting point particles.Increasing the number of point particles increases the pressure by an amount that is proportional to the number of particles.

PHe = 200 torr PAr = 500 torr Ptotal = 700 torr

total totalP V = n RT

total 1 2 3n = n + n + n

total 1 2 3P V = n + n + n RT

total 1 2 3P V = n RT + n RT + n RT

31 2total

n RTn RT n RTP = + + V V V

total 1 2 3P = P + P + P

Say we have a container with some amount of three different gases inside, at a certain T and P.

total 1 2 3P = P + P + P

Dalton’s Law says that the total pressure exerted by the three gases is the sum of the individual pressures.

Dalton’s Law of Partial Pressures

Partial Pressures ExampleMixtures of He and O2 are used in scuba tanks to help prevent “the bends.” For a particular dive, 12 L of O2 at 25oC and 1.0 atm was pumped along with 46 L of He at 25oC and 1.0 atm into a 5.0-L tank. What is the partial pressure of each gas? What is the total pressure?

1. Find the number of moles of each gas that were delivered to the tank.

2. Find the partial pressure of each gas in the tank.

3. Add them up!

Mixtures of He and O2 are used in scuba tanks to help prevent “the bends.” For a particular dive, 12 L of O2 at 25oC and 1.0 atm was pumped along with 46 L of He at 25oC and 1.0 atm into a 5.0-L tank. What is the partial pressure of each gas? What is the total pressure? O2 Data:

P = 1.0 atmV = 12 L n = ? molR = 0.08206 L·atm/mol·KT = 25oC = 298 K

He Data:P = 1.0 atmV = 46 L n = ? molR = 0.08206 L·atm/mol·KT = 25oC = 298 K

2O

1.0 atm 12 Ln =

L atm0.08206 298 Kmol K2 = 0.49 mol O

He

1.0 atm 46 Ln =

L atm0.08206 298 Kmol K = 1.9 mol He

PVn = RT

Using the moles of each gas, the temperature, and volume of the tank we can now calculate the partial pressure of each gas, then add them to get the total pressure.

2

2

O

L atm0.49 mol O 0.08206 298 Kmol KP = 5.0 L

= 2.4 atm

= 9.3 atm

nRTP = V

He

L atm1.9 mol He 0.08206 298 Kmol KP = 5.0 L

Partial pressures of O2 and He in the 5.0 L scuba tank

totalP = 11.7 atm

Mole Fraction and Partial Pressure

• In the last example, we determined the total pressure by adding the partial pressures.

• We could have also added the moles of each gas, and determined a total pressure.

• These two approaches suggest that a relationship exists between the moles of each gas and the total pressure.

Mole Fraction and Partial Pressure

1 11

total 1 2 3

n nn n n n

total

ii

nn

i

total

VPRTVPRT

i

total

PP

i i totalP P

The fraction of moles of a certain gas in a mixture is equal to the ratio of its partial pressure to the total pressure of the mixture.

Mole Fraction (): ratio of the number of moles of a component in a mixture to the total number of moles in the mixture.

Dalton’s Law

Mixtures of He and O2 are used in scuba tanks to help prevent “the bends.” For a particular dive, 12 L of O2 at 25oC and 1.0 atm was pumped along with 46 L of He at 25oC and 1.0 atm into a 5.0-L tank. What is the partial pressure of each gas? What is the total pressure?

totaln = 2.39 mol

totalP = 11.7 atm

2

2O

0.49 mol O = 2.39 mol gas

Partial pressures of O2 and He in the 5.0 L scuba tank

2On = 0.49 mol

Hen = 1.9 mol

He1.9 mol He = 2.39 mol gas

= 0.205

= 0.795

HeP = 0.795 11.7 atm

2OP = 0.205 11.7 atm = 2.39 atm

= 9.28 atm

2.4 atm

9.3 atmPrevious results.

ExampleA mixture of gases contains 4.465 mol of neon, 0.741 mol of argon, and 2.154 mol of xenon. Calculate the partial pressures of all the gases if the total pressure is 2.00 atm at a given temperature.nNe = 4.465 mol nAr = 0.741 mol nXe = 2.154 mol

nTotal = 7.36 molχNe = 4.465 mol/7.36 mol = 0.607χAr = 0.741 mol/7.36 mol = 0.101χXe = 2.154 mol/7.36 mol = 0.293

PNe = 0.607 (2.00atm) = 1.213 atmPAr = 0.101 (2.00 atm) = 0.201 atmPXe = 0.293 (2.00 atm) = 0.586 atm

Kinetic Molecular Theory (KMT)

• The gas laws of Boyle, Charles, and Avogadro are empirical, meaning they are based on observation of a macroscopic property.

• These laws offer a general description of behavior based on many experiments.

• The empirical gas laws can tell you what happens to an ideal gas under certain conditions, but not why it happens.

• KMT is a theoretical, molecular-level model of ideal gases, which can be used to predict the macroscopic behavior of a gaseous system.

• KMT Simulation: http://www.falstad.com/gas/

Postulates of KMTGas particles are so small that their volume is

negligible.

Gas particles are in constant, random motion. Gas molecules constantly collide with each other and with the container walls. The collisions of the particles with the container walls are the cause of the pressure exerted by the gas. Collisions are elastic.

The particles are assumed to exert no forces on each other; they neither attract or repel their neighbors.

This motion is associated with an average kinetic energy that is directly proportional to the Kelvin temperature of the gas.

KMT: Central Points• The main ideas you should take from KMT are

that we can describe temperature and pressure from a molecular perspective.

• Pressure: arises from molecules banging into the container walls.

• Temperature: is directly related to the kinetic energy of the gas molecules. The more KE they have, the greater their temperature.

KMTLet’s consider the average KE per molecule and see how it determines molecular speed.

212muAverage KE per molecule:

Note: T is measured in Kelvin; R = 8.3145 J/mol·K; (J = kg·m2/s2)

Where u is an average of molecular velocity, and m is the mass of one molecule.We are apportioning the total

KE in the mole of gas among all the molecules in an average fashion.

The root-mean-square speed:

urms is the speed of a molecule that has the average KE. urms gives us a formal connection between average gas speed, T, and M.

Distribution of Molecular Speeds

“Maxwell-Boltzmann” curve (a statistical distribution)

um – most probable speeduavg – average speedurms – the speed of a molecule with the average molecular kinetic energy

This plot represents the fraction of gas molecules in a sample that are traveling at a given velocity.

(m/s)

MRTurms

3

Increased T increased average KE increased urmsIncreased M decreased urms

The higher the molar mass of a particle, the slower the particle moves.

Increased T increased average KE increased urms– Maximum of curve shifts to higher u, and distribution spreads

out.– Distribution of speeds will be “shorter” and “fatter” at higher

temperatures.

Increased M decreased urms– Heavier molecules have lower average speed than lighter

molecules at a given temperature.– Distribution of speeds for heavier gases will be “taller” and

“skinnier” than for lighter molecules.

NOTE: There are always some molecules with low velocity in a Boltzmann distribution!!

MRTurms

3

Molecular Speed (cont.)

• Let’s determine urms for N2 at 298 K.

• For a sense of scale, this is on the order of the speed of sound (~320 m s-1)….which isn’t a coincidence.

12-1 -1

-1-1

3 8.314 J mol K 298 K3 515 m s0.0280 kg molrms

RTuM

Smaller mass = greater speed

2

2

kg m1 Js

Comparison of urms for He and N2

At 25oC, which gas will travel faster, He or N2?

M(He) = 4.0 g/mol urms(He) = 1400 m/sM(N2) = 28 g/mol urms(N2) = 520 m/s

A car travelling at 60 mph,

ucar = 26.8 m/s

If gases travel so fast, why does it take so long for you to smell perfume from across the room?

ExampleMatch each of the following gases at the same temperature with its distribution curve in the figure at the left: N2O, Kr, and H2.

H2 – lowest MW, highest rms speed

Kr – highest MW, lowest rms speed

N2O – mid-range MW, mid-range rms speed

Diffusion• Gas molecules travel in a

straight line only until they collide with a container wall or another gas molecule.

• Gas molecules do not have an uninterrupted path in front of them.

• They are constantly colliding with other gas molecules.

• Rate of diffusion is proportional to urms. So lighter particles will have a higher rate of diffusion, and vice versa.

Diffusion (cont.)

Diffusion is the process of mixing gases.In a closed container, diffusion will eventually lead to a homogeneous mixture.http://www.youtube.com/watch?v=H7QsDs8ZRMI

Diffusion Examples• Circle the pair of gases in each set below that

diffuse faster.

a. Ne & F2 or Ar & Cl2

b. b. Kr & Ar or O2 & Cl2

Diffusion ExampleNitrous oxide, N2O, also known as laughing gas, is a colorless gas and has been used as a weak anesthetic. Hydrogen cyanide, HCN, is a poisonous, colorless gas that can cause a quick death. If both of these gases were accidentally released at the same time in the front of a theater full of moviegoers, would the people die laughing?

MW of N2O = 44.02 g/molMW of HCN = 27.029 g/mol

HCN is lighter, so faster. No.

Effusion

Effusion is a special case of diffusion, which exploits the difference in velocities of lighter gas molecules.

This process was used during the Manhattan Project to separate 235U and 238U isotopes.

Effusion (cont.)

• Effusion is dependent on molecular speed. The molecular speed is in turn inversely dependent on the atomic or molar mass. Recall:

• Graham’s Law of Effusion: Rates of effusion are inversely dependent on the square root of the mass of each gas:

3rms

RTuM

21

12

Rate of effusionRate of effusion

MM

Diffusion Reactions• By similar arguments, the distance a molecule

travels is inversely proportional to mass:

• Example: reaction of NH3 with HCl

21

12

distancedistance

MM

Exp. 1.3

(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒)𝑁𝐻3

(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒)𝐻𝐶𝑙=

√𝑀𝐻𝐶𝑙

√𝑀 𝑁𝐻3

=√ 36.517

=1.5

Real Gases• Generally speaking, there is

no such thing as an “Ideal Gas.”

• There are conditions under which a gas will behave ideally…– low P – moderate to high T

• van der Waals developed some corrections to the Ideal Gas law, based on a molecular picture, to explain these observed deviations.

N2

Real Gases (cont.)• At high P, the volume of

the individual gas molecules becomes non-negligible.

• Macroscopic gas is compressible, individual gas molecules are not.

• Under high P conditions, the space available for a gas molecule to move is decreased by its neighbors, so the volume of the system is reduced relative to the ideal case.

eff idealV V nb

Number of moles of gas

Empirical constant… different for each gas; increases with size of molecule.

Real Gases (cont.)• In the Ideal Gas theory, we

assume that gas molecules do not interact.

• But under high P, gas molecules get very close to each other and interact.

• Further, at low T the molecular speed drops also increasing the importance of intermolecular interactions.

Under high P and/or low T conditions, the molecules don’t collide with the container as frequently, so the pressure of the system is

reduced relative to the ideal case.

Concentration of the gas

Empirical constant… different for each gas; increases with increasing intermolecular attraction2

VnaPP idealobs

Real Gases (cont.)

• b generally increases with the size of the molecule• a generally increases with the strength of

intermolecular forces.

vdW equation corrects two major flaws in ideal gas theory:

• Gas molecules have finite volume which becomes important at high P.

• Gas molecules have non-trivial attractions that become important at low T and high P.

2

obsnP a V nb nRTV