Gas Laws

  1. Gases and Pressure
    1. Gases have no constant volume or pressure.  Gases have basically no IMF, which is why they are gases - there is little attraction between atoms and molecules so they don't stay together. Gases are very compressible, unlike solids and liquids. 
    2. Air is a mixture of many gases:  78% nitrogen, 21% oxygen, 1% argon, the rest is mainly the other noble gases, carbon dioxide and methane
    3. Carbon dioxide is a greenhouse gas.  It's % in air has gone up drastically in the past century and is now linked scientifically with an increase in Earth temperature.  NASA reported that 2005 was the hottest year on record, a fact the Bush administration tried to censure and a bitter battle ensued over the right of the public to NASA information. Why the concern over carbon dioxide?  CO2 results from burning fossil fuels (oil, gas, coal) in electric plants and it comes from car exhaust. Also cutting down all the tropical forests does not help as plants take CO2 out of the air and return O2. 
    4. All gases mix together (they are all soluble in each other) 
    5. Pressure is how hard gas molecules hit their container walls. Atmospheric pressure is the force the air has pushing down on us.  Pressure decreases with elevation.  If you climb Mount Everest, the pressure is about 1/2 that of normal sea level. 
      1. Pressure units are atmosphere (atm), torr and mm Hg which are the same, Pascals, bars and psi.  Pressure is measured by barometer.
      2. 1.000 atm = 760.0 torr = 760.0 mm Hg (we will use this as an exact conversion) Know how to convert between these units. 
      3. Standard pressure is 1.00 atm which is about 1 bar also.
  2. Gaw Laws
    1. Variables used:  P = pressure in atm, V = volume in L, n = moles, T = temperature in K
    2. Boyle's Law:  V goes up as P goes down (n and T constant), they are inversely proportional thus V1P1=V2P2  This makes sense because if we smush a gas the pressure will go up.
    3. Charles's Law:  T goes up as V goes up (n and P constant), they are directly proportional thus V1/ T1=V2/ T2  This makes sense because if we heat a gas up there in more KE so the atoms move faster and hit harder pushing their container out thus increasing the volume
    4. Avogadro's Law:  V goes up as n increases (P and T constant), they are directly proportional thus V1/ n1=V2/ n2  This makes sense as more molecules need more space. 
    5. Gay Lussac's Law:  T goes up as P goes up (V and n constant), they are directly proportional thus P1/ T1=P2/ T2  This makes sense if the container is a fixed size (glass flask or steel tank) if the temp increases the molecules will hit harder.
    6. (Note for the above know how the variables depend on each other, but don't memorize the math equations)
    7. STP - standard temp and pressure = 0oC and 1.00 atm.  One mole (6.02 x 1023 molecules) of any gas at STP takes up 22.4 L of volume.  Why?  Well gases are 99.9% empty space anyway so the size of the molecules is basically negligible so the volume is about the same for ANY gas.
    8. When a gas changes from time 1 to time 2 (ie. pressure goes up, temperature goes down, gas leaks out, etc...) use this equation derived by putting all the Laws above together:  P1V1 / n1T1 = P2V2 / n2T2   noting that not all four variable usually change at the same time. Memorize this equation. If one of the variables does not change, cancel it out. 
  3. Ideal Gas Law
    1. Ideal Gas Law:  PV = nRT where R is the gas constant 0.08206 (L atm / mol K)  Memorize this equation. 
    2. Note "ideal" means zero IMF which is not quite true.  There are tiny attractions between some gases.  But the approximation of the ideal gas is good enough for most purposes, plus we don't have to use calculus.
    3. Problems using the ideal and combined gas laws:
      1. How many moles of CO2 gas are in a 785 mL sample at 23.0 oC and 742 torr?  Answer:  n = PV / RT = (0.976 atm)(0.785 L) / (0.0821 L atm / mol K)(296 K) = 3.15 x 10-2 mol
      2. What is the pressure inside a 4.38 L balloon filled with 21.7 g He gas at 25.2oC?  Answer:  P = nRT / V = (5.43 mol)(0.0821 L atm / mol K)(298.2 K) / (4.38 L) = 30.4 atm 
      3. A gas sample contains 72.3 g oxygen gas in a steel tank at 24.0oC and 950.0 torr.  What happens to the pressure if the temperature is raised by 10.0oC?  Answer:  note that volume and moles did not change.  P2 = P1T2 / T1 = (1.25 atm)(307 K) / (297 K) = 1.29 atm
      4. 5.00 moles of Ar gas are in a 2.50 L balloon at 25.0oC and 1.00 atm.  What is the new volume if the temperature raises to 26.0oC and the pressure to 780.0 torr?  Answer:  Note that moles did not change.  V2 = P1V1T2 / T1P2 = (1.00 atm)(2.50 L)(299 K) / (298 K)(1.03 atm) = 2.44 L
    4. More problems to try below.
  4. Gas Stoichiometry
    1. In reactions if we know moles we can find the V of a gas involved in the reaction by using PV=nRT and vice versa. 
    2. Example: What volume of ammonia gas at STP is produced if 45.0 L of hydrogen gas reacts with enough nitrogen gas?  Well we need the balanced reaction:  3 H2(g) + N2(g)  g  2 NH3(g)   Now remember we must ALWAYS get moles to use the balanced reaction ratios.  So we need moles of hydrogen gas:  n = PV / RT = (1.00 atm)(45.0L) / (0.08206 Latm/molK)(273K) = 2.009 moles H2.   Now mole ratio 2.009 H2 (2 NH3 / 3 H2) = 1.339 moles NH3.  Now back to volume V = nRT / P = (1.339 mol)(0.08206 Latm/molK)(273K) / 1.00 atm = 30.0 L NH3
    3. We can also calculate density and molecular mass.
    4. Propylene is used in plastics production.  If 0.1654 grams of propylene gas with a volume of 98.41mL is collected at 740.3 mmHg and 24.0oC what is the molecular mass?  First we solve for moles n = PV / RT = (0.9741atm)(0.09841L) / (0.08206 Latm/molK)(297K) = 0.003933 moles.  Now to find MM we divide grams by moles = 0.1654 g / 0.003933 moles  = 42.05 g/mol
  5. Dalton's Law of Partial Pressures
    1. The total pressure is the sum of each gas's pressure.  Ptot = P1 + P2 + P3 and so forth for each gas
    2. The total moles is the sum of each gas's moles.  ntot = n1 + n2 + n3 and so forth for each gas
    3. The mole fraction (X) is the moles of one gas / total moles:  X1 = n1 / ntot  Thus P1 = X1Ptot  This makes sense - the partial pressure due to gas one must equal it's part or mole fraction of the total pressure. 
    4. The total pressure of a sample is 1.48 atm.  If the pressure due to N2 is 0.94 atm and H2 is 0.17 atm, what is the pressure of the other gas present CO2?  P due to CO2 = 1.48 - 0.94 - 0.17 = 0.37 atm.
    5. What are the partial pressures of hydrogen and helium gas in a mixture of 1.0 g hydrogen gas and 5.00 grams helium gas at 20oC and 8.4 atm?  First convert each to moles:  0.495 moles H2 and 1.25 moles He.  The total moles is 1.745 moles.  PH2 = (0.495 / 1.745)(8.4 atm) = 2.4 atm.  And PHe = (1.25 / 1.745)(8.4 atm) = 6.0 atm. Check - the partial pressure add up to the total?  Yes they do! We rock!
  6. Kinetic Molecular Theory of Gases
    1. The theory states that:
      1. Gases have basically zero IMF
      2. Gas atoms and molecules move randomly in straight lines until they collide with their container's atoms or other gas atoms
      3. They bounce off each other and walls of the container when they collide similar to pool balls
      4. KE is proportional to temperature
      5. Volume of the atoms is really negligible since gases are 99.9% space anyway
    2. Note that real gases do have tiny IMF and there are some volume differences.  But the ideal theory works for most purposes. 
    3. As temperature increases, the speed of the gas molecules increases also
    4. As molecular mass increases speed decreases, the smaller molecules are faster
    5. Diffusion is when gases spread out rapidly in their container.  This is why we smell things like perfume or other unmentionable smells. 
    6. Speed is proportional to 1 / MM1/2 (the inverse of the square root of the molar mass)

 

Practice Problems

  1. In which state of matter is kinetic energy much higher than intermolecular forces?
  2. How many atmospheres is 552 torr? How many L is 345 mL?  Convert 24.5oC to Kelvin.
  3. As temperature increases, what happens to kinetic energy?
  4. As pressure increases what happens to volume if temperature is constant?
  5. As volume decreases, what happens to temperature if pressure is constant?
  6. What is the temperature of 3.24 grams of oxygen gas in a 0.767 L tank at 603 torr?
  7. What is the mass in grams of a sample of CO gas that occupies 4.24 L at 24.5oC and 1.05 atm?
  8. What is the pressure in a 789 mL steel tank at 34.5oC containing 65.0 grams of nitrogen gas?
  9. What is the volume of 0.333 moles of methane gas at -14.0oC and 0.989 atm?
  10. A gas in a 2.45 L balloon exists at 30.0oC and 777 torr.  If the temperature remains constant but the pressure changes to 905 torr, what happens to the volume?
  11. A gas sample is in a 14.4 L steel tank at 0.789 atm and 25.0oC.  What happens to the pressure if the temperature raises to 44.0oC?
  12. 29.0 g of helium gas is in a 345 mL balloon at 33.0oC and 766 torr.  Later the temperature is 35.5oC and the balloon is 357 mL.  What is the new pressure?
  13. A gas in a steel tank originally is at 299 K and 1.23 atm.  Later the pressure is 1.29 atm.  What is the new temperature?

Answers: 

  1. gas
  2. 522 torr (1 atm / 760 torr) = 0.697 atm, 345 mL (1 L / 1000 mL) = 0.345 L, 24.5oC + 273 = 298 K
  3. KE increases
  4. V decreases
  5. T decreases
  6. T = PV / nR = (0.793 atm)(0.767 L) / (0.101 mol)(0.0821 Latm/molK) = 73.3 K
  7. n = PV / RT = (1.05 atm)(4.24 L) / (0.0821 Latm/molK)(297.5 K) = 0.182 mol then convert to grams 0.182 mol(28.0 g/mol) = 5.10 g
  8. P = nRT / V = (2.32 mol)(0.0821 Latm/molK)(307.5 K) / (0.789 L) = 74.2 atm
  9. V = nRT / P = (0.333 mol)(0.0821 Latm/molK)(259 K) / (0.989 atm) = 7.16 L
  10. T and n don't change so cancel them out leaving P1V1 = P2V2, solving for V2 = (1.02 atm)(2.45 L) / (1.19 atm) = 2.10 L
  11. a steel tank can't change it's volume so V and n are constant leaving P1 / T1 = P2 / T2 solving for P2 = (0.789 atm)(317 K) / (298 K) = 0.839 atm
  12. n is the only constant leaving P1V1/T1 = P2V2/T2 solving for P2 = (1.01 atm)(0.345 L)(308.5 K) / (306 K)(0.357 L) = 0.984 atm
  13. n and V are constant leaving P1 / T1 = P2 / T2 solving for T2 = (1.29 atm)(299 K) / (1.23 atm) = 314 K