Lecture 1: The Gas Laws



A Molecular Model of Gases
observation
hypothesis
Gases are easy to expand
gas molecules don't strongly attract each other
Gases are easy to compress
gas molecules don't strongly repel each other
Gases have densities that are about 1/1000 of solid or liquid densities
molecules are much farther apart in gases than in liquids and solids
Gases completely fill their containers
gas molecules are in constant motion
Hot gases leak through holes faster than cold gases
the hotter the gas, the faster the molecules are moving
Pressure
  • definition: pressure = force/area
  • units
Unit
Symbol
Conversions
  • || pascal || Pa || 1 Pa = 1 N/m2 ||
  • || psi || lb/in2 || ||
  • || atmosphere || atm || 1 atm = 101325 Pa = 14.7 lb/in2 ||
  • || bar || bar || 1 bar = 100000 Pa ||
  • || torr || torr || 760 torr = 1 atm ||
  • || millimeters of mercury || mm Hg || 1 mm Hg = 1 torr ||

  • pressure exerted by a weight
    • How much pressure does an elephant with a mass of 2000 kg and a total footprint area of 5000 cm2 exert on the ground?
    • Estimate the total footprint area of a tyrannosaur weighing 16000 kg. Assume it exerts the same pressure on its feet that the elephant does.

  • measuring pressure
    • strategy: relate pressure to fluid column heights
    • observation: you can't draw water higher than 34 feet by suction- why?
    • hypothesis: atmospheric pressure supports the fluid column
pressure
=
force / area
  • || ||
  • || mass x g / area || (because force

  • weight of liquid = mass x g) ||
  • || ||
  • || density x volume x g / area || (because mass

  • density x volume) ||
  • || ||
  • || density x g x height || (because volume

  • area x height) ||

  • a barometer measures atmospheric pressure as a mercury column height
  • a manometermeasures gas pressure as a difference in mercury column heights
    • two types
      • closed manometer: difference in column heights gives absolute gas pressure
      • open manometer: difference in column heights gives difference between gas and atmospheric pressures

  • kinetic-molecular view of gas pressure
    • gas pressure arises from force of molecular collisions
    • anything that increases the number of collisions will increase pressure


The state of a gas
  • most behaviors of a pure gas sample can be related to just 4 physical properties:
property
symbol
convenient
  • units
property
  • type
  • || pressure || P || atm, torr, Pa || intensive ||
  • || volume || V || L, cm3 || extensive ||
  • || temperature || T || K || intensive ||
  • || moles || n || mol || extensive ||

  • any equation that relates P, V, T, and n for a material is called an equation of state
STP - Standard Temperature and Pressure
STP is commonly used to define standard conditions for temperature and pressure which is important for the measurements and documentation of chemical and physical processes:
  • STP - Standard Temperature and Pressure - is defined by IUPAC (International Union of Pure and Applied Chemistry) as air at 0oC (273.15 K, 32 oF) and 105 pascals
  • STP - commonly used in the Imperial and USA system of units - as air at 60 oF (520 oR) and 14.696 psia (15.6oC, 1 atm)
Note that the earlier IUAPC definition of STP to 273.15 K and 1 atm (1.01325 105 Pa) is discontinued.
1 Pa = 10-6 N/mm2 = 10-5 bar = 0.1020 kp/m2 = 1.02x10-4 m H2O = 9.869x10-6 atm = 1.45x10-4 psi (lbf/in2)

NTP - Normal Temperature and Pressure
NTP is commonly used as a standard condition for testing and documentation of fan capacities:
  • NTP - Normal Temperature and Pressure - is defined as air at 20oC (293.15 K, 68oF) and 1 atm (101.325 kN/m2, 101.325 kPa, 14.7 psia, 0 psig, 29.92 in Hg, 760 torr). Density 1.204 kg/m3 (0.075 pounds per cubic foot)
SATP - Standard Ambient Temperature and Pressure
SATP - Standard Ambient Temperature and Pressure is also used as a reference:
  • SATP - Standard Ambient Temperature and Pressure is a reference with temperature of 25 degC (298.15 K) and pressure of 101 kPa.
ISA - International Standard Atmosphere
ISA - International Standard Atmosphere is used as a reference to aircraft performance:
  • ISA - International Standard Atmosphere is defined to 101.325 kPa, 15 degC and 0% humidity.
ICAO Standard Atmosphere
Standard model of the atmosphere adopted by the International Civil Aviation Organization (ICAO):
  • Atmospheric pressure: 760 mmHg = 14.7 lbs-force/sq inch
  • Temperature: 15oC = 288.15 K = 59oF

Pressure-Volume Relationships - Boyle’s Law




An animated version of Boyle's law.Pressure times volume equals a constant.
An animated version of Boyle's law.Pressure times volume equals a constant.



Robert Boyle was an English physicist and chemist, who, in 1662, published the results of his gas experiments and began an era of gas experimentation that would stretch on for over a century. Boyle’s experimentation with gases led to observations about the relationship between the volume of hissamples and the pressure of his samples, if he held the temperature and quantity of gas constant. His apparatus was essentially a cylinder with a moveable top (a crude piston). He realized that there was an indirect relationship between the two – if the pressure of the gas was increased, the volume of the gas would decrease (the piston would f all); conver sely, a decrease in pressure resulted in an increase in the volume (the piston would rise). Mathematically, Boyle represented his relationship as the equality between the products of the pressure and volume for each of two conditions:

external image Boyles%20Law.gif






Temperature-Volume Relationships – Charles’ Law



An animated version of Charles law.Volume equals a constant times the temperature.
An animated version of Charles law.Volume equals a constant times the temperature.


The French physicist Jacques Charles invested a great deal of time observing the effects of temperature change on the volume of a sample of gas with a fixed mass. By 1787, he had made enough consistent observations and collected enough experimental data to state that there is a direct relationship between the two. Using an apparatus similar in design to the one used by Boyle, he observed that as the temperature of an enclosed gas (this is, one with a constant pressure and mass) increased, the volume of the gas would increase (the piston would rise). Conversely, he realized that as the temperature of the sample was reduced, the volume of the sample would decrease (the piston would fall).

external image 9682f75ffd644c1e723156ad5919c8a6.png

Pressure-Temperature Relationships – Gay-Lussac’s Law



Jean Louis Gay-Lussac was a French chemist and physicist known most prominently f or his work in quantifying the properties of water-alcohol mixtures. However, in 1802, his work with gases led him to formulate a law describing the relationship between the pressure and temperature of gases. If a sample of gas of constant mass in a container of fixed volume was heated, Gay-Lussac noted that the pressure inside the container would increase as the temperature increased. Conversely, he noted that as he reduced the temperature of the sample, the pressure would decrease.


external image 6c6a35ce4507d81696a7a01125715174.png

Combined Gas Law


A derivation of the combined gas law using only elementary algebra can contain surprises. For example, starting from the three empirical laws
 P = k_v, T ,!
P = k_v, T ,!
............(1) Gay-Lussac's Law, volume assumed constant
 V = k_p T ,!
V = k_p T ,!
............(2) Charles's Law, pressure assumed constant
 P V = k_t ,!
P V = k_t ,!
............(3) Boyle's Law, temperature assumed constant
where kv, kp, and kt are the constants, one can multiply the three together to obtain
 PVPV = k_v T k_p T k_t  ,!
PVPV = k_v T k_p T k_t ,!

Taking the square root of both sides and dividing by T appears to produce of the desired result
 frac {PV}{T} = sqrt{k_p k_v k_t}  ,!
frac {PV}{T} = sqrt{k_p k_v k_t} ,!
Thus, PV=RT

Ideal Gas Equation



The ideal gas law can be derived from combining two empirical gas laws: the combined gas law and Avogadro's law. The combined gas law states that
frac{PV}{T}= C
frac{PV}{T}= C

where C is a constant which is directly proportional to the amount of gas, n (Avogadro's law). The proportionality factor is the universal gas constant, R, i.e. C = nR.
Hence the ideal gas law
 PV = nRT ,
PV = nRT ,


pvt.png
external image pvtgas.gif


PVTsurface for an ideal gas


Each point on the curved surface represents a possible combination of (P,V,T) for an arbitrary quantity of an ideal gas. The three sets of lines inscribed on the surface correspond to states in which one of these three variables is held constant.The red curved lines, being lines of constant temperature, orisotherms, are plots of Boyle's law. These isotherms are also seen projected onto the P-V plane at the top right.The yellow lines are isochors and represent changes of the pressure with temperature at constant volume.The green lines, known as isobars, and projected onto the V-Tplane at the bottom, show how the volumes contract to zero as the absolute temperature approaches zero, in accordance with the law of Charles and Gay-Lussac.

References:

http://www.grc.nasa.gov

[The link takes you to Boyle's Law Page. The animation on this page are from this website. A very easy to understand language as well.]

http://en.wikipedia.org/wiki/Combined_gas_law
[The link contains a couple of different approaches to combined gas law.]


Interesting Tools:
1. http://www.goldenkstar.com/chemistry-software-info/boyles-law-animation-software-chemistry.htm
This link contains an interesting software that lets you understand the effect of varying various elements of combined gas law equation.
external image gas-laws-chemistry-software.gif

Lecture 2


Gas constant, Internal energy, Relation between Cp and Cv,
Enthalpy,

Lecture 3


Non flow process, Constant volume process, Constant pressure process, Isothermal process,
Poly-tropic process, Adiabatic process.

Lecture 4




Lecture 5