Gas Laws

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Gas Laws

• L. Scheffler
• Lincoln High School

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Gases

• Variable volume and shape
• Expand to occupy volume available
• Volume, Pressure, Temperature, and the number of
  moles present are interrelated
• Can be easily compressed
• Exert pressure on whatever surrounds them
• Easily diffuse into one another

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Mercury Barometer

• Used to define and measure atmospheric pressure
• On the average at sea level the column of mercury
  rises to a height of about 760 mm.
• This quantity is equal to 1 atmosphere
• It is also known as standard atmospheric pressure
  

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Pressure Units

• The above represent some of the more common units
  for measuring pressure. The standard SI unit is
  the Pascal or kilopascal.
• The US Weather Bureaus commonly report
  atmospheric pressures in inches of mercury.
• Pounds per square inch or PSI is widely used in
  the United States.
• Most other countries use only the metric system.

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Boyles Law

• According to Boyles Law the pressure and volume
  of a gas are inversely proportional at constant
  pressure.
• PV constant.
• P1V1 P2V2

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Boyles Law

• A graph of pressure and volume gives an inverse
  function
• A graph of pressure and the reciprocal of volume
  gives a straight line

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Charles Law

• According to Charles Law the volume of a gas
  is proportional to the Kelvin temperature as long
  as the pressure is constant
• V kT

V1 T1
V2 T2
Note The temperature for gas laws must always
be expressed in Kelvin where Kelvin oC
273.15 (or 273 to 3 significant digits)
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Charles Law

• A graph of temperature and volume yields a
  straight line.
• Where that line crosses the x axis (x intercept)
  is defined as absolute zero

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Advogadros Law

• Equal volumes of a gas under the same temperature
  and pressure contain the same number of
  particles.
• If the temperature and pressure are constant the
  volume of a gas is proportional to the number of
  moles of gas present
• V constant n
• where n is the number of moles of gas
• V/n constant
• V1/n1 constant V2 /n2
• V1/n1 V2 /n2

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Universal Gas Equation

• Based on the previous laws there are four factors
  that define the quantity of gas Volume,
  Pressure, Kevin Temperature, and the number of
  moles of gas present (n).
• Putting these all together

PV nT
Constant R
The proportionality constant R is known as the
universal gas constant
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Universal Gas Equation

• The Universal gas equation is usually written as
• PV nRT
• Where P pressure
• V volume
• T Kelvin Temperature
• n number of moles

The numerical value of R depends on the pressure
unit (and perhaps the energy unit) Some common
values of R include R 62.36 dm3
torr mol-1 K-1 0.0821 dm3 atm mol-1
K-1 8.314 dm3kPa mol-1 K-1
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Standard Temperature and Pressure (STP)

• The volume of a gas varies with temperature
  and pressure. Therefore it is helpful to have a
  convenient reference point at which to compare
  gases.
• For this purpose standard temperature and
  pressure are defined as

Temperature 0oC 273 K Pressure
1 atmosphere 760 torr
101.3 kPa This point is often called STP
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Sample Problem

• Example What volume will 25.0 g O2 occupy
• at 20oC and a pressure of 0.880 atmospheres?

(25.0 g) n ----------------- 0.781
mol (32.0 g mol-1)
Data Formula Calculation Answer
V ? P 0.880 atm T (20 273)K
293K R 0.08205 dm-3 atm mol-1 K-1
PV nRT so V nRT/P
V (0.781 mol)(0.08205 dm-3 atm mol-1
K-1)(293K) 0.880 atm V 21.3 dm3
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Universal Gas Equation Alternate Forms
Density (d) Calculations
m is the mass of the gas in g
d
M is the molar mass of the gas
Molar Mass (M ) of a Gaseous Substance
d is the density of the gas in g/L
M
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Sample Problem
A 2.10 dm3 vessel contains 4.65 g of a gas at
1.00 atmospheres and 27.0oC. What is the molar
mass of the gas?
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Sample Problem
A 2.10 dm3 vessel contains 4.65 g of a gas at
1.00 atmospheres and 27.0oC. What is the molar
mass of the gas?
M
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Daltons Law of Partial Pressures

• The total pressure of a mixture of gases is
  equal to the sum of the pressures of the
  individual gases (partial pressures).
• PT P1 P2 P3 P4 . . . .
• where PT total pressure
• P1 partial pressure of gas 1
• P2 partial pressure of gas 2
• P3 partial pressure of gas 3
• P4 partial pressure of gas 4

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Daltons Law of Partial Pressures

• Applies to a mixture of gases
• Very useful correction when collecting gases over
  water since they inevitably contain some water
  vapor.

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Sample Problem A

• Henrietta Minkelspurg generates Hydrogen gas
  and collected it over water.
• If the volume of the gas is 250 cm3 and the
  barometric pressure is 765.0 torr at 25oC, what
  is the pressure of the dry hydrogen gas at STP?
  
• (PH2O 23.8 torr at 25oC)

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Sample Problem A -- Solution

• Henrietta Minkelspurg generates Hydrogen gas
  and collected it over water.
• If the volume of the gas is 250 cm3 and the
  barometric pressure is 765.0 torr at 25oC, what
  is the pressure of the dry hydrogen gas at STP?
  
• (PH2O 23.8 torr at 25oC)

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Sample Problem B

• Henrietta Minkelspurg generated Hydrogen gas and
  collects it over water. If the volume of the gas
  is 250 cm3 and the barometric pressure is 765
  torr at 25oC, what is the volume of the dry
  oxygen gas at STP?

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Sample Problem B -- Solution

• Henrietta Minkelspurg generated Hydrogen gas and
  collects it over water. If the volume of the gas
  is 250 cm3 and the barometric pressure is 765
  torr at 25oC, what is the volume of the dry
  oxygen gas at STP?
• From the previous calculation the adjusted
  pressure is 742.2 torr

P1 PH2 742.2 torr P2 Std Pressure 760
torr V1 250 cm3 T1 298K T2 273K V2
? (V1P1/T1) (V2P2/T2) therefore V2
(V1P1T2)/(T1P2)
V2 (250 cm3)(742.2 torr)(273K)
(298K)(760.torr) V2 223.7 cm3
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Kinetic Molecular Theory

• Matter consists of particles (atoms or molecules)
  in continuous, random motion.
• Particles in continuous, random, rapid motion
• Collisions between particles are elastic
• Volume occupied by the particles has a negligibly
  small effect on their behavior
• Attractive forces between particles have a
  negligible effect on their behavior
• gases have no fixed volume or shape, take the
  volume and shape of the container
• The average kinetic energy of the particles is
  proportional to their kelvin temperature

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Maxwell-Boltzman Distribution

• Molecules are in constant motion
• Not all particles have the same energy
• The average kinetic energy is related to the
  temperature
• An increase in temperature spreads out the
  distribution and the mean speed is shifted upward

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Velocity of a Gas
The distribution of speeds for nitrogen gas
molecules at three different temperatures
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Diffusion
Gas diffusion is the gradual mixing of molecules
of one gas with molecules of another by virtue of
their kinetic properties.
HCl 36.5 g/mol
NH3 17.0 g/mol
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DIFFUSION AND EFFUSION

• Diffusion is the gradual mixing of molecules of
  different gases.

• Effusion is the movement of molecules through a
  small hole into an empty container.

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Grahams Law

• Grahams law governs effusion and diffusion
  of gas molecules.
• KE1/2 mv2

The rate of effusion is inversely proportional to
its molar mass.
Thomas Graham, 1805-1869. Professor in Glasgow
and London.
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Grahams Law Problem 1

• 1 mole of oxygen gas and 2 moles of ammonia
  are placed in a container and allowed to react at
  850oC according to the equation
• 4 NH3(g) 5 O2(g) ? 4 NO(g) 6 H2O(g)
• Using Graham's Law, what is the ratio of the
  effusion rates of NH3(g) to O2(g)?

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Grahams Law Problem 1

• 1 mole of oxygen gas and 2 moles of ammonia
  are placed in a container and allowed to react at
  850oC according to the equation
• 4 NH3(g) 5 O2(g) ? 4 NO(g) 6 H2O(g)
• Using Graham's Law, what is the ratio of the
  effusion rates of NH3(g) to O2(g)?

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Grahams Law Problem 2

• What is the rate of effusion for H2 if 15.00
  cm3 of CO2 takes 4.55 sec to effuse out of a
  container?

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Grahams Law Problem 2

• What is the rate of effusion for H2 if 15.00
  cm3 of CO2 takes 4.55 sec to effuse out of a
  container?

Rate for CO2 15.00 cm3/4.55 s 3.30 cm3/s
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Grahams Law Problem 3

• What is the molar mass of gas X if it effuses
  0.876 times as rapidly as N2(g)?

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Grahams Law Problem 3

• What is the molar mass of gas X if it effuses
  0.876 times as rapidly as N2(g)?

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Ideal Gases v Real Gases

• Ideal gases are gases that obey the Kinetic
  Molecular Theory perfectly.
• The gas laws apply to ideal gases.
• In reality there is no perfectly ideal gas.
• Under normal conditions of temperature and
  pressure many real gases approximate ideal gases.
• Under more extreme conditions more polar gases
  show deviations.

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In an Ideal Gas ---

• The particles (atoms or molecules) in continuous,
  random, rapid motion.
• The particles collide with no loss of momentum
• The volume occupied by the particles is
  essentially zero when compared to the volume of
  the container
• The particles are neither attracted to each other
  nor repelled
• The average kinetic energy of the particles is
  proportional to their Kelvin temperature
• At normal temperatures and pressures gases
  closely approximate idea behavior

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Real Gases

• For ideal gases the product of pressure and
  volume is constant. Real gases deviate somewhat
  as shown by the graph pressure vs. the ratio of
  observed volume to ideal volume below.

• These deviations occur because
• Real gases do not actually have zero volume
• Polar gas particles do attract if compressed

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van der Waals Equation
The van der Waals equation shown below
includes corrections added to the universal gas
law to account for these deviations from ideal
behavior

• (P n2a/V2)(V - nb) nRT

where a gt attractive forces between
molecules b gt residual volume or
molecules The van der Waals constants for some
elements are shown below
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Gas Stoichiometry
What is the volume of CO2 produced at 370 C and
1.00 atm when 5.60 g of glucose are used up in
the reaction C6H12O6 (s) 6O2 (g)
6CO2 (g) 6H2O (l)
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Gas Stoichiometry
What is the volume of CO2 produced at 370 C and
1.00 atm when 5.60 g of glucose are used up in
the reaction C6H12O6 (s) 6O2 (g)
6CO2 (g) 6H2O (l)
5.60 g C6H12O6
0.187 mol CO2
V
4.76 dm3
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