Gas Laws

1
Gas Laws

• M. L. Watson

2

Ideal Gas Law
An ideal gas is defined as one in which all
collisions between atoms or molecules are
perfectly elastic and in which there are no
intermolecular attractive forces. One can
visualize it as a collection of perfectly hard
spheres which collide but which otherwise do not
interact with each other. In such a gas, all the
internal energy is in the form of kinetic energy
and any change in internal energy is accompanied
by a change in temperature. An ideal gas can be
characterized by three state variables absolute
pressure (P), volume (V), and absolute
temperature (T). The relationship between them
may be deduced from kinetic theory and is called
the Where n number of moles R universal
gas constant 8.3145 J/mol K N number of
molecules k Boltzmann constant 1.38066 x
10-23 J/K 8.617385 x 10-5 eV/K k R/NA NA
Avogadro's number 6.0221 x 1023

3
Ideal Gas Law

• PVnRTZ
• n Mass
• R Universal gas constant
• T Temperature
• Z Supercompressability
• (P1V1/T1)Z1(P2V2/T2)Z2

4
Ideal Gas Law

• The ideal gas law can be viewed as arising from
  the kinetic pressure of gas molecules colliding
  with the walls of a container in accordance with
  Newton's laws. But there is also a statistical
  element in the determination of the average
  kinetic energy of those molecules. The
  temperature is taken to be proportional to this
  average kinetic energy this invokes the idea of
  kinetic temperature. One mole of an ideal gas at
  STP occupies 22.4 liters.

5
Ideal Gas Law

• An Ideal Gas (perfect gas)is one which obeys
  Boyle's Law and Charles' Law exactly.
• An Ideal Gas obeys the Ideal Gas Law (General gas
  equation)PV nRTwhere Ppressure, Vvolume,
  nmoles of gas, Ttemperature, R is the gas
  constant which is dependent on the units of
  pressure, temperature and volumeR 8.314 J K-1
  mol-1 if Pressure is in kilopascals(kPa), Volume
  is in litres(L), Temperature is in Kelvin(K)R
  0.0821 L atm K-1 mol-1 if Pressure is in
  atmospheres(atm), Volume is in litres(L),
  Temperature is in Kelvin(K)
• An Ideal Gas is modelled on the Kinetic Theory of
  Gases which has 4 basic postulates
• Gases consist of small particles (molecules)
  which are in continuous random motion
• The volume of the molecules present is negligible
  compared to the total volume occupied by the gas
• Intermolecular forces are negligible
• Pressure is due to the gas molecules colliding
  with the walls of the container
• Real Gases deviate from Ideal Gas Behaviour
  because
• at low temperatures the gas molecules have less
  kinetic energy (move around less) so they do
  attract each other
• at high pressures the gas molecules are forced
  closer together so that the volume of the gas
  molecules becomes significant compared to the
  volume the gas occupies
• Under ordinary conditions, deviations from Ideal
  Gas behaviour are so slight that they can be
  neglected
• A gas which deviates from Ideal Gas behaviour is
  called a non-ideal gas.

6
There are many applications to the Ideal Gas Law
Equation

• How can the ideal gas law be applied in dealing
  with how gases behave?
• PV nRT
• Used to derive the individual ideal gas laws
• For two sets of conditions initial and final set
  of conditions
• P1V1 n1RT1 and P2V2 n2RT2
• Solving for R in both equations gives
• R P1V1 / n1T1 and R P2V2 / n2T2
• Since they are equal to the same constant, R,
  they are equal to each other
• P1V1 / n1T1 P2V2 / n2T2
• For the Volume Pressure relationship (ie Boyle's
  Law)
• n1 n2 and T1 T2 therefore the n's and T's
  cancel in the above expression resulting in the
  following simplification
• P1V1 P2V2 (mathematical expression of Boyle's
  Law)
• For the Volume Temperature relationship (ie
  Charles's Law)
• n1 n2 and P1 P2 therefore the n's and the P's
  cancel in the original expression resulting in
  the following simplification
• V1 / T1 V2 / T2 (mathematical expression of
  Charles's Law)

7
Ideal Gas Law w/Constraints

• For the purpose of calculations, it is convenient
  to place the ideal gas law in the form
• where the subscripts i and f refer to the initial
  and final states of some process. If the
  temperature is constrained to be constant, this
  becomes
• which is referred to as Boyle's Law.
• If the pressure is constant, then the ideal gas
  law takes the form
• which has been historically called Charles' Law.
  It is appropriate for experiments performed in
  the presence of a constant atmospheric pressure.

8
Boyles Law

• At constant temperature, the volume of a given
  quantity of gas is inversely proportional to its
  pressure V 1/PSo at constant temperature, if
  the volume of a gas is doubled, its pressure is
  halved.ORAt constant temperature for a given
  quantity of gas, the product of its volume and
  its pressure is a constant PV constant, PV
  k
• At constant temperature for a given quantity of
  gas PiVi PfVfwhere Pi is the initial
  (original) pressure, Vi is its initial (original)
  volume, Pf is its final pressure, Vf is its final
  volumePi and Pf must be in the same units of
  measurement (eg, both in atmospheres), Vi and Vf
  must be in the same units of measurement (eg,
  both in litres).
• All gases approximate Boyle's Law at high
  temperatures and low pressures. A hypothetical
  gas which obeys Boyle's Law at all temperatures
  and pressures is called an Ideal Gas. A Real Gas
  is one which approaches Boyle's Law behaviour as
  the temperature is raised or the pressure
  lowered.

9
Boyles Law

• P1V1P2V2

10
Charles Law

• At constant pressure, the volume of a given
  quantity of gas is directly proportional to the
  absolute temperature V T (in Kelvin)So at
  constant pressure, if the temperature (K) is
  doubled, the volume of gas is also doubled.ORAt
  constant pressure for a given quantity of gas,
  the ratio of its volume and the absolute
  temperature is a constant V/T constant, V/T
  k
• At constant pressure for a given quantity of gas
  Vi/Ti Vf/Tfwhere Vi is the initial
  (original) volume, Ti is its initial (original)
  temperature (in Kelvin), Vf is its final volme,
  Tf is its final tempeature (in Kelvin)Vi and Vf
  must be in the same units of measurement (eg,
  both in litres), Ti and Tf must be in Kelvin NOT
  celsius.temperature in kelvin temperature in
  celsius 273 (approximately)
• All gases approximate Charles' Law at high
  temperatures and low pressures. A hypothetical
  gas which obeys Charles' Law at all temperatures
  and pressures is called an Ideal Gas. A Real Gas
  is one which approaches Charles' Law as the
  temperature is raised or the pressure lowered.As
  a Real Gas is cooled at constant pressure from a
  point well above its condensation point, its
  volume begins to increase linearly. As the
  temperature approaches the gases condensation
  point, the line begins to curve (usually
  downward) so there is a marked deviation from
  Ideal Gas behaviour close to the condensation
  point. Once the gas condenses to a liquid it is
  no longer a gas and so does not obey Charles' Law
  at all.Absolute zero (0K, -273oC approximately)
  is the temperature at which the volume of a gas
  would become zero if it did not condense and if
  it behaved ideally down to that temperature.

11
Charles Law

• V1/V2T1/T2
• P1V1/T1P2V2/T2

12
Supercompressability

• Why? Molecular Squish
• - under 1 for pressures under 1500 PSI
• - different for each gas mixture
• - similar to alcohol water or sand gravel

Theoretical
Actual
13
Isentropic Expansion

• Thermodynamics!
• P1V1kP2V2k
• k(Cp/Cv)
• kRatio of specific heat at a constant pressure
• to the specific heat at a constant volume
• K1.3 for natural gas
• Specific Heat kJ/kg

14
All the possible states of an ideal gas can be
represented by a PvT surface as illustrated. The
behavior when any one of the three state
variables is held constant is also shown.
15
Molecular Constants

• In the kinetic theory of gases, there are certain
  constants which constrain the ceaseless molecular
  activity. A given volume V of any ideal gas will
  have the same number of molecules. The mass of
  the gas will then be proportional to the
  molecular mass. A convenient standard quantity is
  the mole, the mass of gas in grams equal to the
  molecular mass in amu. Avogadro's number is the
  number of molecules in a mole of any molecular
  substance.

The average translational kinetic energy of any
kind of molecule in an ideal gas is given by
16
State Variables

• A state variable is a precisely measurable
  physical property which characterizes the state
  of a system, independently of how the system was
  brought to that state. It must be inherently
  single-valued to characterize a state. For
  example in the heat-work example, the final state
  is characterized by a specific temperature (a
  state variable) regardless of whether it was
  brought to that state by heating, or by having
  work done on it, or both.
• Common examples of state variables are the
  pressure P, volume V, and temperature T. In the
  ideal gas law, the state of n moles of gas is
  precisely determined by these three state
  variables. If a property, e.g., enthalpy H, is
  defined as a combination of other state
  variables, then it too is a state variable.
  Enthalpy is one of the four "thermodynamic
  potentials", and the other three, internal energy
  U, Helmholtz free energy F and Gibbs free energy
  G are also state variables. The entropy S is also
  a state variable.
• Some texts just use the term "thermodynamic
  variable" instead of the description "state
  variable".

17
The Mole

• A mole (abbreviated mol) of a pure substance is a
  mass of the material in grams that is numerically
  equal to the molecular mass in atomic mass units
  (amu). A mole of any material will contain
  Avogadro's number of molecules. For example,
  carbon has an atomic mass of exactly 12.0 atomic
  mass units -- a mole of carbon is therefore 12
  grams. For an isotope of a pure element, the mass
  number A is approximately equal to the mass in
  amu. The accurate masses of pure elements with
  their normal isotopic concentrations can be
  obtained from the periodic table.
• One mole of an ideal gas will occupy a volume of
  22.4 liters at STP (Standard Temperature and
  Pressure, 0C and one atmosphere pressure).
• Avogadro's number

18
Standard Temperature and Pressure

• STP is used widely as a standard reference point
  for expression of the properties and processes of
  ideal gases. The standard temperature is the
  freezing point of water and the standard pressure
  is one standard atmosphere. These can be
  quantified as follows
• Pressure ( P ) is the ratio of the force applied
  to a surface (F) to the surface area ( A ).
• P F / A
• Standard temperature 0C 273.15 K 32 F
• Standard pressure 1 atmosphere 760 mmHg
  101.3 kPa 14.696 psi
• Standard volume of 1 mole of an ideal gas at STP
  22.4 liters

19
Gauge Pressure

• Does the flat tire on your automobile have zero
  air pressure? If it is completely flat, it still
  has the atmospheric pressure air in it. To be
  sure, it has zero useful pressure in it, and your
  tire gauge would read zero pounds per square
  inch. Most gauges read the excess of pressure
  over atmospheric pressure and this excess is
  called "gauge pressure". While a useful
  measurement for many practical purposes, it must
  be converted to absolute pressure for
  applications like the ideal gas law.
• Since a partial vacuum will be below atmospheric
  pressure, the phrase "negative pressure" is often
  used. Certainly there is no such thing as a
  negative absolute pressure, but small decreases
  in pressure are commonly used to entrain fluids
  in sprayers, in carburetors for automobiles, and
  many other applications. In the case of
  respiration, we say that the lungs produce a
  negative pressure of about -4 mmHg to take in
  air, which of course means a 4 mmHg decrease from
  the surrounding atmospheric pressure.

20
Example

• Here is a sample problem involving a single set
  of conditions
• A sample of chlorine gas weighs 1.31 g at STP.
  Calculate the volume this sample of chlorine
  would occupy under the following new conditions.
• 3.20atm and 0.00 C
• Here are the steps that will lead to an answer
• Calculate the moles of Cl2 from 1.31 grams Cl2
  using the formula weight of 2 x at weight of Cl
• 1.31 grams Cl2 X 1 mole Cl2 / 71 grams Cl2
  0.0184 moles Cl2
• Check the temperature and convert to Kelvin if
  necessary
• K C 273 0.00 273 273 K
• Check the pressure given and convert to
  atmospheres unit
• Pressure is already in atmospheres, 3.20 atm
• Use the value of R 0.0821 liter-atm/mole-K
• Using the PV nRT plug in the moles,
  temperature, pressure, and R and solve for the
  Volume in liters
• V nRT / P (0.0184 moles) ( 0.0821 liter-atm /
  mol-K) (273 K) / 3.20 atm 0.129 liters

21
Example

• Let's let you try one
• A sample of chlorine gas weighs 1.31 g at STP.
  Calculate the volume this sample of chlorine
  would occupy under the following new conditions.
• 760 torr and -23.0C
• Use the Ideal Gas Law Equation to solve Gas
  stoichiometry problems involving conversion of
  moles of gas to volume or the conversion of a
  given volume of a gas sample under a stated
  temperature and pressure to moles or mass of
  another component in the process under a stated
  temperature and pressure.
• Use of Ideal Gas Law Equation to determine the
  density of a gas.
• PV nRT if n mass of gas / molecular mass
• then PV (mass of gas/molecular mass) R T
  Solving for molecular mass
• molecular mass (mass of gas) R T / P V
• Since Density mass / V Then
• molecular mass D R T / P
• and solving for Density D (molecular mass) (P)
  / R T
• Using the equation to solve for Partial Pressure
  of a known amount of gas in a gas mixture
• P1 n1 R T / V and P2 n2 R T / V

22
Example

• Ideal Gas Law Calculations
• Calculating Volume of Ideal Gas V (nRT) P
• What volume is needed to store 0.050 moles of
  helium gas at 202.6kPa and 400K? PV nRT
• P 202.6 kPan 0.050 molT 400KV ? LR
  8.314 J K-1 mol-1 202.6V0.050x8.314x400202.6 V
  166.28V 166.28 202.6V 0.821 L (821mL)

23
Notes
24
Notes
25
Notes