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Gases Key Points

• States or Phases of Matter – Gas

• Compressible, variable volume and pressure• Expands into available space• Rapid mixing• No collective structure … except the container• Subject to condensation into liquid

– Liquid• Incompressible• Flows under pressure• Conforms to container• Slow mixing (relative to gases)• Cohesive collective structure• subject to evaporation into gas

Overview

• More States of Matter – Solid

• Retains shape, not container confined• Subject to cleavage, breakage, bending• More structure, higher degree of order

– Crystalline shapes are common– “Lattice Energy” holds things together

• No (or slow) mixing, diffusion limited– Plasma … the “4th state of matter”

• Not part of everyday experience• Ionized gases, charged particles, subatomic species

– Interior of the Sun, “Solar Wind”– Florescent or Neon gas electrical discharge– Arc Welding, “sputtering” of metals in vacuum– Plasma metal cutters, 30K degrees

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Gases in the minority of elements

• Less than 10% of 116 elements are gases– Six of these are non-reactive “noble gases”

• He, Ne, Ar, Kr, Xe, Rn

– Five remaining gases• H2, N2, O2, Cl2, F2

• All are diatomic, yielding octets of 8 electrons

• All are reactive, N2 and O2 in atmosphere

• All have some biologic role

The Gaseous Elements

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Atmospheric Gases– Atmosphere is homogeneous mixture of gases

• Nitrogen 76% - “inert” for most uses, but reactive– Numerous oxides “NOX“ components of “smog”– Anesthetics, nitrate fertilizer, nitrite preservatives

• Oxygen 21% - essential for animal life– Product of photosynthesis– Oxidizes food for chemical energy, kinetic energy, and heat

• Argon 0.93% - most abundant “noble” or inert gas– Used in light bulbs to prevent darkening by evaporating W– Used in gas discharge lamps (blue), lasers, protective package

• Carbon Dioxide 0.037% - breathing, fermentation, combustion– Required by plants to provide cellulose, sugars, and Oxygen– Basis of carbonated drinks … soda, beer, champagne– Most life on earth involves both CO2 and O2

• Methane 0.00017% - natural gas, animal waste– Very common natural product

» “swamp gas”, land fill decomposition, animal flatulation– Lighter than air, very little stays near the surface

Gas Behavior

• Gases– Intertwined relationships

• Compressibility relates volume & pressure• Amount of material controls volume of that material• Volume changes with Temperature

– Basis of “heat engines”• Subject to phase changes

– Water vapor into liquid water or ice– “Liquid Air”, sublimation of “dry ice”

– Models needed to explain & predict behavior• Started with simple “2 at a time” relationships

– P & V, moles & V, V & Temp, etc• Evolved to “Ideal Gas Law”, PV=nRT

– Includes all the variables in one equation– Reduces to named gas laws with omitted variables

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Gases & Gas Laws

• Gas laws with 2 variables– Boyle’s law, Charles’ law, Avagadro’s Law

• Combined gas law with 3 variables– PV/T=constant

• Ideal Gas Law with all 4 variables– PV=nRT

• Applications– Density and Lift– Air Bags, etc.

Gas Pressure

• Pressure Units of Measure– Air pressure is familiar concept

• High altitude, auto tires, skin diving, sailboats• Pressure defined as force per unit area

– Originally 1 Atmosphere = 760 mm Hg (sea level)

– 1 Atmosphere = 14.4 psi (Imperial system)

– 1 atmosphere ≡ 101,325 Pascal (mks)

– MKS units often inconvenient, leading to new units• 100,000 Pa = 1 Bar (not tied to atmosphere)• 1mm Hg = 1 Torr (1/760 = 0.13% of atmospheric pressure)

Units of Pressure

Atmosphere 1.00PSI 14.69inch Hg 29.92cm Hg 76

mm Hg 760 by definitiontorr 760 torr ≡ 1 mm of Hg

Pascals 101,300 1kg / square meter (0.1gm/cm2)

kPa 101.3 kilo Pascals

mPa 0.1013 mega Pascals

Bar 1.013 1 Bar ≡ 1E5 Pascal

Mercury Barometer

• Vacuum at top of glass– Zero pressure at glass top

• Mercury rises in tube– Air pressure pushes Hg up– Mercury height = pressure– 760mm Hg ≡ 1 atmosphere

– 32 feet H2O ≈ 1 atmosphere

• Blowing versus Sucking– Which is stronger?– Establishes pump designs

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Two Variable Gas Laws

• Boyle’s Law– Pressure and Volume are inversely related– Pressure * Volume = constant

• assumes constant amount of material & temp• If pressure goes up, volume goes down .. & vice versa

• Charles’ Law– Adds temperature as a new variable– Volume proportion al to temperature – Volume = constant * Temperature (or V/T=constant)

• Assumes constant amount of material and pressure• If temperature goes up, so does volume

• Avagadro’s Law– Adds amount of material as a new variable – Volume of gas depends on moles

• Volume = constant * Moles (or V/M=constant)• If moles goes up, so does volume• Each mole of gas occupies approx 22.4 liters

Boyle’s LawPressure and Volume inversely related

see NASA animation related to this image

Gas Law Animations

NASA site for gas law animationshttp://www.grc.nasa.gov/WWW/K-12/airplane/Animation/gaslab/gastil.html– “stop” prior animation,

• lower left red box in the animation itself

– Select “New Case”, left-hand column– Freeze 2 of 4 variables– Select one of 2 cases to animate

Alternative Boyle’s Formulas

• P1V1 = P2V2 = constant

– Generally used relationship, most often quoted

• V2 = V1*(P1/P2) solving for Volume

– This rearrangement has Volumes directly related V = V*(ratio)– Note that Pressure dimensions cancel (psi, pascal, etc.)– Note that volumes must have same dimensions (liter, quart, mL)– Assumes NO CHANGE in temperature or amount of material

• P2 = P1*(V1*/V2) solving for Pressure

– Same idea as for volume, cancellation of units– Can use arbitrary units of measure, but must be consistent

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Boyle’s Law Calculation Example• Bag of potato chips San Jose Tahoe

– What is the volume at Tahoe pressure?• P1 = 1.0 atmosphere (14.7psi) in San Jose

• V1 = 1.0 liter air volume in San Jose• P2 = 0.75 atmosphere at Lake Tahoe (6225 feet)

• V2 = ?

• P1V1 = P2V2, or V2 = V1*(P1/P2)

– V2 = 1 Liter * (1 atmos / 0.75 atmos)

– V2 = 1/0.75 = 1.3 liters volume at Tahoe• Note that pressure units vanish, anything consistent is

OK (atm, psi, pascals, etc.). Same for volume

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Pressure vs Altitude6225 ft at Lake Tahoe, ≈ 0.75 atmosphere

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Temperature Change, Charles’ Law

• Charles’ Law: V1 / T1 = V2 / T2

– Adds temperature as a variable– Volume proportional to temperature– Volume / Temperature = constant

• Assumes constant amount of material & pressure• If temperature goes up, so does volume

– NASA graphic shows 4 liters at 300 degrees• Also shows 3 liters at 225 degrees• 4 / 300 = 0.013 3 / 225 = 0.013• Both divisions yield same value = a constant

Charles’ LawVolume of gas proportional to absolute temperature

See NASA animation of slide below

What Temperature Scale to use?

• Cannot use arbitrary scales– 2oF is NOT “twice as hot” as 1oF, equally cold– 0oC =melting ice, not absence of temperature– We use ratios in gas law calculations

• Temp. Ratio 1oC/0oC =∞… not very useful

• What’s needed – A scale with truly proportional temperatures

• Where 100o is actually “Twice as Hot” as 50o

– A scale which goes to true (absolute) zero• No negative temperatures

Traditional Temperature Scales

Proportional P vs T with scale through zeroProportional scale defined using “Kelvin” degrees

Kelvin Scale is simple idea• Absolute zero is absence of all motion

– Cannot go any lower than 0oK– Close to zero is boiling point of helium at 4oK

• Kelvin degree “size” same as Centigrade– Zero Kelvin becomes -273oC

– Conversion is oK = oC+273– Going the other way oK-273 = oC

• Some die-hards like Fahrenheit degrees– Conversion is “Rankine” scale oR = oF+492– Could be handy if you have lots of Fahrenheit data

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Charles’ Law Calculation Example

• Balloon from ski area into heated lodge

• V/T = constant

• V1/T1 = V2/T2 = constant

– V1 = 1 liter

– T1 = -10oC at ski lift (-10+273=263oK)

– T2 = 25oC in lodge (25+273 = 298oK)

– V2 = V1*(T2/T1) = 1 Liter* (298/263)

– V2 = 1.13 Liters (constant pressure)

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Gay-Lussac Law

• Same as Charles’ law, substitute Pressure

• P = k*T P & T proportional, k=constant

• P1/T1 = P2/T2 assumes constant volume

• P2/P1 = T2/T1 P’s and T’s together

– Useful because easy to see that units cancel

• P1*T2 = P2*T1 avoids division in formula

An example

• What is pressure inside a tennis ball going from warm room to winter tennis court

• P1/T1 = P2/T2, P2=P1*(T2/T1)

– P1 = 2 atmospheres inside ball (assumed)

– no change to tennis ball volume

– T1 = 298oK (25oC) , T2 = 263oK (-10oC)

• P2=P1*(T2/T1)

– P2 = 2atm*(263/298) = 2*0.88 atm

– P2 = 1.76 atmospheres (less bounce)29

Combination 3-Variable Gas Law• Can combine Charles’ and Boyle’s Laws

– Boyle’s Law• P1V1 = P2V2 (constant temperature & mass)

– Charle’s Law• V / T = constant

– Algebraic substitution & yields• P*V = constant, also V/T = constant

– Both are related to same variables, so

• P1V1 / T1 = P2V2 / T2

• Can calculate any quantity if other 5 are known

Formula Variationskey point that start = end metric, other ratios cancel

P1V1 / T1 = P2V2 / T2

• P2 = P1 (V1 /V2)*(T2 / T1)– Pressure change given by ∆T and inverse of ∆V

• V2 = V1 (P1 /P2)*(T2 / T1)– Volume change given by ∆T and inverse of ∆P

• T2 = T1 (P2 /P1)*(V2 /V1)– Temperature change given by ∆P and ∆V

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Combined Law Calculation

• Automobile driven to Death Valley– Temperature changed 70oF 120oF

• 70oF 21oC 294oK• 120oF 49oC 322oK

– Tire volume changed 20 21 liters

– Tire pressure 30psi in S.Jose ? In Death V.• P2 = P1 (V1 /V2)*(T2 / T1)

• P2 = 30 ( 20/21)*(322 / 294)

• P2 = 31.3 psi

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What about mass change?

• Avogadro’s Law– Adds amount of material as a new variable

• Intuitive that volume related to material amount

– Volume of gas depends on moles• Volume / Moles = constant• Or …. Volume = constant * moles• If moles goes up, so does volume• Each mole of gas occupies approx 22.4 liters

Avagadro’s Gas LawChange of mass is added variable, see animation

more material more pressure and/or more volume

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Gas Law Summary

– Four variables involved for Gases• Pressure, can be “atmospheres” or Pascals

– For ratios involving 2 pressures, dimensions cancel

• Volume, can be liters or cubic meters– For ratios involving 2 volumes, dimensions cancel

• Temperature, always in Kelvin, oC+273=oK– Kelvin is linear, “0” is really zero (not so for oC, oF)

• Mass, usually in moles– For ratios involving 2 masses, dimensions cancel

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Gas Law Summary

• Why have 4 laws?– Bad News

• 4 different names and relationships to remember

• PV=c, V/T=c, V/n=constant, combo P1V1 / T1 = P2V2 / T2

• Three People’s names, which one goes where?

– Good News• Simple formulas, only a few variables involved• Dimensions tend to cancel• Not forced into MKS or self-consistent unit system

– Can mix any units where dimensions cancel

– PSI1/PSI2 = Pascal1/Pascals2 = Atmosph1/Atmosph2

• Best to choose SIMPLEST formula to solve a problem

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Ideal Gas LawBringing it all together

Combination of P, V, n, and T • Pressure = Pascals (MKS definition, or atmos)• Volume = Liters, or cubic meters• n = moles of material• Temperature = degrees Kelvin (Centigrade + 273)• Constant = R (depends on dimensions used)

– R = 8.314 Joules/(mole-degree K)– R = 0.082 Liter-Atmospheres/(mole-degree K)

PV = nRT

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Ideal Gas Law

• PV=nRT … 5th & final Gas Law– Good News

• Combines all 4 variables in one relationship• One formula can solve any gas problem• Linear relationship, no exponents or logs

– Bad News• Dimensions of “R” must be consistent with inputs

– R= can be 0.082 (liter-atmos)/(mole-oK) commonly used– R= can be 8.315 (Joule-oK)/mole MKS units

• Cannot mix systems of units, no ratios to cancel• Need to know 3 variables to get the 4th.

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Ideal Gas Law

• PV=nRT extremely useful– Handles all 4 variables (including a constant)– Can determine 4th variable if other 3 are known

• Moles of methane in tank of known P, V, T• Pressure on piston if V, n, and T are known• Temperature of a system if P, V, and n are known• Volume of gas if P, n, and T are known

– Lots of examples in text and homework– These are linear relationships, no exponents

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Ideal Gas• Ideal Gas Law PV=nRT

– Simplifies to Boyle’s Law when n and T are constant• PV = nRT = constant “k”

– Simplifies to Charles’ Law when n and P are constant• V/T = nR/P = constant “k”

– Simplifies to Avagadro’s Law when T and P are constant

• V/n = RT/P = constant “k”

– PV=nRT Very useful• handles many gas calculations

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Gas Density• Ideal Gas Law P*V=n*R*T

– Define “MM” = MolarMass (formula wt.)= grams/mole– n = moles = grams / MolarMass– Plug into ideal gas law … P*V= (grams/MM) R*T– MM = grams*R*T/ P*V (or MM=mRT/PV)

• Density definition = m/V (typically gm/ml)– From ideal law, P*V=(gram/MM)R*T,

• Rearrange to …. M = grams = P*V*MM/(R*T)

– Density = grams/V = P*MM/(R*T)• Can be a handy way to find density of a gas

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Ideal Gas Law

• Standardized Conditions– Any measurement unit can be a variable– Need a “standard” set of conditions to compare– Use “STP” (not the engine additive or Hippie drug)

• Abbreviation for “Standard Temperature & Pressure”• Reference temperature is 0oC (usually) or 273oKelvin• Reference pressure is 1 atmosphere (101,325 Pascals)

– Pressure often cancels in calculations

– Results from using pressure ratios, so dimensions cancel

– Can use any consistent dimensions in those cases

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Automobile Air Bag

Reaction = 2NaN3 + ignition 2Na + 3 N2

• Gas volume from 145 grams NaN3

– Conditions are 1.15 atmos. Press at 30oC

• Reaction ratios are always in moles – NaN3 = 65 grams/mole, 145 gram=2.23 moles

– Mole ratio for N2 = (3/2)*2.23 = 3.35 moles

– How to get to volume from moles ?

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Automobile Air Bag Reaction = 2NaN3 + ignition 2Na + 3 N2

• Gas volume from 145 grams NaN3

– Conditions are 1.15 atmos. Press, 30oC?• Reaction rations are always in moles

– NaN3 = 65 grams/mole, 145 gram=2.23 moles– Mole ratio for N2 = (3/2)*2.23 = 3.35 moles

• PV=nRT … or V=nRT/P– N = 3.35 mole– R = 0.0821(Liter-atmosphere)/(oK-mole)– T = 30oC+273 = 303oK– P = 1.13 atmospheres

• V=nRT/P = 3.35*0.0821*303 / 1.13 = 72.4 liters

Now to our experiment

• We will evaluate Charles Law today– Measure gas volume at different temperatures– Compare our data to theoretical– How close do we come to Charles law?

Experiment Procedure

• (1) Measure air temperature in dry flask– Access path is through cork in flask– Convert hot temp. to Kelvin (add 273o)

• (2) Measure temperature of ice water– Thumb on cork until flask under water– Release thumb, shrinking air sucks in water– Convert temperature of cold bath to Kelvin– (3) Measure amount of water sucked in– (4) Measure volume of flask up to cork

Sample Calculation• Temperature

– Hot temperature = 70oC+273=343oK– Cold Temperature = 3oC+273= 276oK

• Volumes– Water sucked into flask = 30 mL– Total Flask = 145 mL– Cold air volume (total – water) = 145 – 30 = 115 mL

• Apply Charles Law:– Vcold = Vhot * (T2 / T1) – Vcold = 145mL * (276/343) – Calculated volume = 116.7 mL– Error = Calc – Exper = 116.7 - 115 = 1.7mL– Error % = [(calc-expt) / calc]*100 – Error % = [(116.7 – 115) / 116.6] *100 = 1.4%

Experiment summary

• Measure volume of water sucked into flask– Cold air volume = (empty flask) – (water sucked in)– Know ratios of hot/cold temperatures & volumes– Compare theoretical versus actual

• Repeat experiment for second trial data– Be sure to record temperatures & volumes

• Will share data on white board– Make sure your data makes sense before leaving

• Answer post-lab questions– 4 items; Charles, Boyle, and combination gas laws

Charles law Experiment

• End of 32A lab lecture

Sample Calculation

A T1 = Temperature of HOT Water 70 oC 343 oK

B T2 = Temperature of COLD Water 3 oC 276 oK

C V1 = Total volume of flask 145 mL

D Volume of water drawn into flask 30 mL

E=C-D V2 = volume of cold air in flask 115 mL

TRIAL #1

Show Calculations, Charles Law

F=C*(B/A) V2 =V1*(T2/T1)

= 116.7

Show Calculations, Error %

Err = (Charles - Expt) / Charles

G=(E-F)/F = 1.4%

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Dalton's Law of Partial Pressures

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Partial Pressures

• What about mixtures of gases?– Total pressure = sum of component pressures

• Each component gas has a “partial pressure”• Partial pressure is the contribution from each gas• Sum is1 atmosphere (air at sea level)

– Nitrogen in air = 0.780 partial atmospheres– Oxygen in air = 0.209 partial atmospheres– Argon in air = 0.009 partial atmospheres– Carbon Dioxide = 0.00037 partial atmospheres– TOTAL = 1.00 atmospheres (sum of partials)

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Partial Pressures

– Partial pressures depend on mole fractions• PV = nRT, so pressure also depends on “n”• Calculations will involve moles of gas• Oxygen and Nitrogen = 99% of air, similar mass

– O2 at 32 gm/mole, N2 at 28 gm/mole

– Atmospheric pressure not linear with altitude• Most of the air near the planet surface

– Gravity keeps the air from escaping into space– Pressure vs altitude non-linear due to compression– Lake Tahoe about 0.75 Atmosphere, OK for people– Aircraft at 30,000 feet at 0.30 Atmosphere, can’t breathe

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Pressure vs AltitudeLake Tahoe 6225 ft, ≈ 0.75 atmosphere

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Partial Pressures

• Homework Example (textbook 8.84)– Partial Pressure of oxygen given as 160mm Hg at 1.0

atmosphere. So what is PP on top of Mt.Whitney where atm. Pressure is 440mm Hg? Assume same % oxygen.

– By definition 1 atmosphere is 760 mm Hg, so air pressure on Mt Whitney is 440/760 = 0.579 atmosphere

– At 1 atmosphere, oxygen is 160mm Hg– Oxygen percent does not change with altitude, so same

ratio applies to air on the mountain tip• 160mmHg*0.579 = 92.64 mmHg on Mountain

– Alternatively a ratio can be used• (160mmHg O2)/(760mmHg air) = (X mmHg O2)/(440mmHgair)• X = 92.64 mmHg O2

2-variable Gas Laws

• Boyle’s Law– Pressure and Volume are inversely related– Pressure * Volume = constant

• assumes constant amount of material & temp• If pressure goes up, volume goes down .. & vice versa

• Charles’ Law– Adds temperature as a new variable– Volume proportional to temperature– Volume / Temperature = constant

• Assumes constant amount of material and pressure• If temperature goes up, so does volume

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Gas Laws, Boyle

• Boyle’s Law– Pressure and Volume inversely related

• P1V1 = P2V2

• Pressure * Volume = constant • Pressure = constant * 1/Volume• Pressure goes up, volume goes down• NASA graphic shows 4 liters at 1 atmosphere• Converts to 3 liters at 1.33 atmospheres

– 4*1 = 4, 3*1.33 = 4 – both P*V products yield same value = a constant

Gas Laws

• Boyle’s Law– Pressure and Volume inversely related– Pressure * Volume = constant – Pressure goes up, volume goes down

• NASA graphic shows 4 liters at 1 atmosphere• Converts to 3 liters at 1.33 atmospheres

– 4*1 = 4, 3*1.33 = 4 – both P*V products yield same value = a constant

– Assumes SAME amount of material present– Assumes SAME temperature both cases

Effect of Temperature Change

• Charles’ Law– Adds temperature as a variable– Gas volume proportional to temperature– Volume / Temperature = constant

• Assumes constant amount of material & pressure• If temperature goes up, so does volume

– NASA graphic shows 4 liters at 300 degrees• Also shows 3 liters at 225 degrees• 4 / 300 = 0.013 3 / 225 = 0.013• Both divisions yield same value = a constant

Examples

– Boyle’s Law: PV = constant• P1V1 = P2V2 (constant temperature & mass)

• Can solve for 4th condition when other 3 are known– P1 = P2 V2 / V1

– V1 = P2V2 / P1

– P2 = P1V1 / V2

– V2 = P1V1 / P2

• Alternatively, can have ratios of same quantity– P1/P2 = V2/V1

– Important to note DIMENSIONS are UNIMPORTANT» But must be consistent to cancel

All 4 temperature scales

Gases & Gas Law Summary

• Avagadro’s Gas Law– Addition of mass as a variable

• Ideal Gas Law with all 4 variables– PV=nRT

• Gas Density & Volume

• Stoichiometry, STP

• Applications– Auto air Bags, dirigibles, hot-air balloons

Gas Law Summary

– Boyle’s Law: PV = constant• P1V1=constant=P2V2 • no changes in temperature or mass• Only 2 variables to consider

– Charles’ Law: V/T = constant• Volume inversely related to absolute temperature• V = constant * T (no change in pressure or mass)

– says rising temperature increases volume– Assumes constant amount of material & pressure

• Only 2 variables involved

Gas Law Summary

– Avagadro’s law: V/n = constant• Volume inversely related to amount of material• V = constant * moles

– no changes in pressure or Temperature– More moles provides larger volume– Assumes constant Temperature and Pressure

• Only 2 variables involved

– Combined Gas Law P1V1/T1 = P2V2/T2

• Handles 3 variables• Mass not included

Why a Gas?• Gas molecules have kinetic energy

– Energy proportional to Kelvin temperature

• Gas molecules have full octets

• Gas molecules have little mutual attraction

• Energy of collisions keeps molecules apart

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Total of 4 Gas Law Variables

– Variables involved for Gases• Pressure, can be “atmospheres” or Pascals

– For ratios involving 2 pressures, dimensions cancel

• Volume, can be liters or cubic meters– For ratios involving 2 volumes, dimensions cancel

• Temperature, always in Kelvin, oC+273=oK– Kelvin is linear, “0” is really zero (not so for oC, oF)

• Mass, usually in moles– For ratios involving 2 masses, dimensions cancel