Gases and the KineticMolecular Theory

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Chapter 5
Gases and the Kinetic-Molecular Theory
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Gases and the Kinetic Molecular Theory
5.1 An Overview of the Physical States of Matter
5.2 Gas Pressure and Its Measurement
5.3 The Gas Laws and Their Experimental
Foundations
5.4 Further Applications of the Ideal Gas Law
5.5 The Ideal Gas Law and Reaction Stoichiometry
5.6 The Kinetic-Molecular Theory A Model for
Gas Behavior
5.7 Real Gases Deviations from Ideal Behavior
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An Overview of the Physical States of Matter
The Distinction of Gases from Liquids and Solids
1. Gas volume changes greatly with pressure.
2. Gas volume changes greatly with temperature.
3. Gas have relatively low viscosity.

• Most gases have relatively low densites under
  normal conditions.
• Gas density varies greatly with T and P.

5. Gases are miscible.
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Figure 5.1
The three states of matter.
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A mercury barometer.
Figure 5.3
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Table 5.2 Common Units of Pressure
Atmospheric Pressure
Unit
Scientific Field
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Sample Problem 5.1
Converting Units of Pressure
SOLUTION
291.4mmHg
291.4torr
291.4torr
0.3834atm
0.3834atm
38.85kPa
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Boyle's Law

• Boyle found that doubling the pressure on a gas
  resulted in the gas having half of the original
  volume.
• Boyle's Law - at constant temperature the volume
  of a fixed amount of gas is inversely
  proportional to pressure
• V ? 1/P
• or V constant x 1/P
• or VP constant
• therefore P1V1 P2V2
• When T and n is constant

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Charles's Law

• Charles found that doubling the absolute temp.
  doubled the volume of the gas
• Charles's Law - at constant pressure the volume
  of a fixed amount of gas is directly proportional
  to it's temperature.
• V ? T
• or V constant x T
• or V/T constant
• Therefore V1 V2
• T1 T2
• When P and n is constant

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Avogadro's law

• Avogadro's law - at constant T and P, V of a gas
  is directly related to n
•  
• or V constant x n
• V ? n
• or V constant x n
• or V/n constant
• Therefore V1 V2
• n1 n2
• When T and P are constant

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Ideal Gas Law

• if V ? 1/P, and V ? T, and V ? n,
• then V ? nT/P
• or V constant x nT/P
• we symbolize the constant R, and call it the
  universal gas constant
• The value of R depends on the units of P,V, n,
  and T
• R .08206 L atm/mole K
• PVnRT
• Use this form for a static system to find 1 of
  the 4 variables.
• the law is named for an ideal gas because most
  real gases don't follow this relationship exactly.

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Ideal Gas Law

• Instead of memorizing individual laws, consider
• PVnRT so PV/nT R constant
• Therefore in any gas system under changing
  conditions
• P1V1 P2V2
• n1T1 n2T2
• Cross out any unchanging variables

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Ideal Gas Law Terminology

• Standard temperature and pressure or STP
• Standard T 0 C or 273.15K
• Standard P 1 atm or 760 torr
• At STP, the volume of 1 mol of any ideal gas is
  22.4L
• This is called standard molar volume.

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Standard Molar Volume
Figure 5.8
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Sample Problem 5.2
Applying the Volume-Pressure Relationship
PLAN
SOLUTION
P and T are constant
V1 in cm3
P1 1.12atm
P2 2.64atm
1cm31mL
V1 24.8cm3
V2 unknown
V1 in mL
24.8cm3
0.0248L
103mL1L
V1 in L
xP1/P2
P1V1 P2V2

V2 in L
P1V1
P2
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Sample Problem 5.3
Applying the Temperature-Pressure Relationship
PROBLEM
A 1-L steel tank is fitted with a safety valve
that opens if the internal pressure exceeds
1.00x103 torr. It is filled with helium at 230C
and 0.991atm and placed in boiling water at
exactly 1000C. Will the safety valve open?
PLAN
SOLUTION
P1(atm)
T1 and T2(0C)
P1 0.991atm
P2 unknown
1atm760torr
K0C273.15
T1 230C
T2 100oC
P1(torr)
T1 and T2(K)
x T2/T1
P2(torr)
753torr
949torr
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Sample Problem 5.4
Applying the Volume-Amount Relationship
PROBLEM
A scale model of a blimp rises when it is filled
with helium to a volume of 55dm3. When 1.10mol
of He is added to the blimp, the volume is
26.2dm3. How many more grams of He must be added
to make it rise? Assume constant T and P.
PLAN
We are given initial n1 and V1 as well as the
final V2. We have to find n2 and convert it from
moles to grams.
SOLUTION
n1(mol) of He
P and T are constant
x V2/V1
n1 1.10mol
n2 unknown
n2(mol) of He
V1 26.2dm3
V2 55.0dm3
subtract n1
mol to be added
x M
g to be added
4.84g He
2.31mol
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Sample Problem 5.5
Solving for an Unknown Gas Variable at Fixed
Conditions
PROBLEM
A steel tank has a volume of 438L and is filled
with 0.885kg of O2. Calculate the pressure of O2
at 210C.
PLAN
V, T and mass, which can be converted to moles
(n), are given. We use the ideal gas law to find
P.
SOLUTION
V 438L
T 210C (convert to K)
n 0.885kg (convert to mol)
P unknown
210C 273.15 294K
27.7mol O2
1.53atm
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Further Applications of Ideal Gas Law

• because n m/mw,
• we can substitute into the ideal gas law to find
  m, m.w., and gas densities(m/V).
• PV (m/m.w.)RT
• Gas Density m/V m.w.P/RT
• m m.w.PV/RT
• m.w. mRT/PV

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Sample Problem 5.6
Calculating Gas Density
PROBLEM
Calculate the density (in g/L) of carbon dioxide
and the number of molecules per liter (a) at STP
(00C and 1 atm) and (b) at ordinary room
conditions (20.0C and 1.00atm).
PLAN
Density is mass/unit volume substitute for
volume in the ideal gas equation. Since the
identity of the gas is known, we can find the
molar mass. Convert mass/L to molecules/L with
Avogardros number.
d mass/volume
PV nRT
V nRT/P
d

SOLUTION
1.96g/L
d
(a)
2.68x1022molecules CO2/L
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Sample Problem 5.6
Calculating Gas Density
continued
(b)
1.83g/L
2.50x1022molecules CO2/L
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Figure 5.11
Determining the molar mass of an unknown volatile
liquid
based on the method of J.B.A. Dumas (1800-1884)
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Sample Problem 5.7
Finding the Molar Mass of a Volatile Liquid
PROBLEM
An organic chemist isolates from a petroleum
sample a colorless liquid with the properties of
cyclohexane (C6H12). She uses the Dumas method
and obtains the following data to determine its
molar mass
Is the calculated molar mass consistent with the
liquid being cyclohexane?
PLAN
Use unit conversions, mass of gas and density-M
relationship.
SOLUTION
m (78.416 - 77.834)g
0.582g
x
M

M of C6H12 is 84.16g/mol and the calculated value
is within experimental error.
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Dalton's law of partial pressures

• the total pressure of a mixture of gases is equal
  to the sum of the pressures that each would exert
  alone
• Pt P1 P2 P3 P4 ...
• total pressure depends only on total moles, not
  the id of each gas
• Mole Fraction (c)- ratio of moles of a component
  to total moles in mixture c n1/nt
• because each gas exerts it's own part of the
  pressure we can relate pressure of each gas to
  it's percent or mole fraction
• P1 n1
• Pt nt or P1 c1 x Pt

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Figure 5.12
Collecting a water-insoluble gaseous reaction
product and determining its pressure.
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Sample Problem 5.9
Calculating the Amount of Gas Collected Over Water
PLAN
The difference in pressures will give us the P
for the C2H2. The ideal gas law will allow us to
find n. Converting n to grams requires the molar
mass, M.
SOLUTION
(738-21)torr 717torr
Ptotal
0.943atm
717torr
H2O
x M
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Sample Problem 5.9
Calculating the Amount of Gas Collected Over Water
continued
0.943atm
0.523L
x
0.203mol
x
296K
0.529 g C2H2
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Sample Problem 5.8
Applying Daltons Law of Partial Pressures
PLAN
SOLUTION
mol 18O2
0.040
divide by 100
c 18O2
0.030atm
multiply by Ptotal
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Figure 15.13
Summary of the stoichiometric relationships among
the amount (mol,n) of gaseous reactant or product
and the gas variables pressure (P), volume (V),
and temperature (T).
amount (mol) of gas B
amount (mol) of gas A
P,V,T of gas A
P,V,T of gas B
ideal gas law
ideal gas law
molar ratio from balanced equation
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Sample Problem 5.10
Using Gas Variables to Find Amount of Reactants
and Products
SOLUTION
mass (g) of Cu
divide by M
35.5g Cu
0.559mol H2
mol of Cu
molar ratio
0.559mol H2
22.6L
mol of H2
use known P and T to find V
L of H2
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Sample Problem 5.11
Using the Ideal Gas Law in a Limiting-Reactant
Problem
SOLUTION
x
5.25L
n
Cl2
0.414mol KCl formed
0.435mol K
0.435mol KCl formed
Cl2 is the limiting reactant.
0.414mol KCl
30.9 g KCl
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Kinetic-Molecular Theory

• 1. Gases - large of molecules in continuous
  random motion
• 2. The volume of gas particles are negligible
  compared to the volume they occupy.
• 3. The attractive and repulsive forces between
  gases are negligible.
• 4. Energy is transferred by collisions, but
  average K.E. of a gas does not change with const
  T.
• 5. Average K.E. of molecules of any gas is
  proportional to Absolute T
• Pressure caused by collisions of gas with
  container
• Absolute temp is a measure of avg KE of a gas
• Avg K.E. of a gas molecule 1/2mu2 where u
  root-mean-square speed
• the speeds in a particular sample of gas vary

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Distribution of molecular speeds at three
temperatures.
Figure 5.14
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A molecular description of Boyles Law
Figure 5.15
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Figure 5.17
A molecular description of Charless Law
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Figure 5.18
A molecular description of Avogadros Law
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Figure 5.16
A molecular description of Daltons law of
partial pressures.
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Effusion and Diffusion

• The Average KE of any gas is the same at a given
  T.
• A lighter gas, one with a low m.w., must be
  moving faster at a given temp than a heavier gas.
• u (3RT/mw)1/2

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Figure 5.19
Relationship between molar mass and molecular
speed.
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Effusion and Diffusion

• effusion- the escape of molecules through a tiny
  hole into an evacuated space
• lighter gases move quicker at a given temp and
  therefore are more likely to find a hole in the
  container so their rate of effusion is higher.
• Graham's law - rate of effusion is inversely
  proportional to mw
• r1/r2 (mw2/mw1)1/2 u1/u2
•  
• diffusion - spread of a substance through a space
  or another substance
• rate also higher for light molecules, gas speeds
  N2 1150 mi/hr at RT
• gases collide often so they can't travel very far
  in a straight line
• mean free path- avg distance between collisions.
  sea level 6x10-6 cm

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Sample Problem 5.12
Applying Grahams Law of Effusion
SOLUTION
M of CH4 16.04g/mol
M of He 4.003g/mol
2.002
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Real Gases

• Real gases deviate from ideal behavior
• the deviation is greater at higher P and at lower
  T
•  
• Gases behave approximately ideally at Plt10 atm
  and T gtgtTb
• Why? The ideal gas law assumes no volume for
  molecules and no attractions between them
•  
• van der Waals equation for non-ideal gases
•  
• (P n2a/V2)(V-nb) nRT
•  
• constants for a and b are tabulated for gases

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