The Gas Laws AP Chemistry

e.g., gasoline vapors Basics on Gases composition of the atmosphere: ~78% N 2, ~21% O 2 properties of gases: expand to fill container compressible form homogeneous mixtures vapors: gases of substances that are normally liquids or solids -- due to gas particles being… 1) 2) far apart in constant, random motion

F  Equation for pressure: A  N m 2 P  N/m 2 = Pa 1 atm… = 760 mm Hg = 760 torr = kPa = bar (1 bar = 10 5 Pa) At a depth of 350 m (1150 ft), the hull pressure on a submarine is 3.4 x 10 6 Pa (36 tons/ft 2 ).

BAROMETER air pressure mercury (Hg) vacuum mercury barometer Close-up of an early mercury barometer, showing how the mercury column is supported by atmospheric pressure.

An aneroid barometer contains a small, pressure-sensitive metal box that has been evacuated of air. The box is prevented from collapsing by being connected to a spring that is also attached to the dial on the barometer. When the air pressure on the walls of the box changes, the box “flexes,” which moves the spring and the dial.

OPEN END MANOMETER CLOSED END MANOMETER air pressure Hg height difference sealed end confined gas SMALL + HEIGHT = BIGP gas = HEIGHT

The Gas Laws Boyle’s law: Robert Boyle (1627–1691) PV = constant (when T is constant) P V V 1/P

Charles’s law: Jacques Charles (1746–1823) V/T = constant (when P is constant) T V **T in K (0.00 K = – o C)

Gay-Lussac’s law: Joseph Louis Gay-Lussac (1778–1850) P/T = constant (when V is constant) T P

(1876–1956) i.e., Avogadro’s hypothesis: Equal volumes of gas at the same temperature and pressure have the same number of particles. Amedeo Avogadro (1776–1856) Avogadro’s law: Volume of a gas is proportional to the number of moles of gas. e.g., 22.4 L of ANY GAS at STP contains 6.02 x particles Amedeo Avocado

Combined Gas law: merges Boyle’s, Charles’s, and Gay-Lussac’s laws into one equation. **NOTE: For all gas law calculations, use the absolute temperature (in K).

Ideal Gas law: P V = n R T R = L-kPa/mol-K = L-atm/mol-K Conditions of standard temp. and pressure (STP): 0 o C ( K)1 atm Equations for gas density:  = molar mass of gas Other Equations and Constants same gas; two sets of conditions

To solve problems involving volumes of gases NOT at STP in chemical reactions: Combine PV = nRT and stoichiometry.

What volume of hydrogen will react w /carbon at 981 torr and 334 o C to yield 42.0 g of n-pentane? H 2 + CC 5 H g n-p () 1 mol n-p 72 g n-p P V = n R T = 135 = 3.5 mol H 2 L H 2 () 6 mol H 2 1 mol n-p

Dalton’s law of Partial Pressure P tot = P 1 + P partial pressures: total pressure of gaseous mixture pressure that each gas would exert by itself Other equations… Total moles of gas in a mixture: The mole fraction (X) of a gas in a mixture: n tot = n 1 + n 2 + …

Find the total pressure exerted by 38.0 g of carbon monoxide and 38.0 g of hydrogen in a 6.00-L container at 25 o C. n CO = 38 g () 1 mol 28.0 g 1. Find n tot. 2. Use PV = nRT to find P. = mol CO n H2 = 38 g () 1 mol 2.0 g = 19.0 mol H 2 n tot = mol = 8410kPa

With reference to the previous problem… What is the partial pressure exerted by each gas? P CO = () kPa= 560 kPa P H2 = () kPa= 7850 kPa The ratio of the partial pressures is the same as the mole ratio. mole fraction (38 g CO) (38 g H 2 )

Collecting Gases over Water filled w /water, initially before reaction gas being collected H 2 O levels even After rxn. is complete, raise or lower collecting vessel so H 2 O levels inside and out are the same. In this way... P atmos = P gas + P H2O during reaction reaction complete gas from reaction collected gas ( w /H 2 O vapor, too)

From bal. eq., n CaC2 = n C2H2, so g CaC2 = mol For the reaction CaC 2 (s) + H 2 O(l) C 2 H 2 (g) + CaO(s)... If L of acetylene are collected over water at 20.0 o C, find the moles of acetylene collected and the grams of calcium carbide used. The barometric pressure is torr torr = P C2H2 + P H2O o C (from p. 1111) So P C2H2 = torr = atm = mol C 2 H 2 () 1 mol 64.1 g = 2.13 g CaC 2

Find the total pres. in container Z, assuming constant T. PXPX VXVX VZVZ P X,Z A B C AB ZC 1.3 L 2.6 L 3.8 L 2.3 L 3.2 atm1.4 atm 2.7 atm X atm 3.2 atm1.3 L 2.3 L 1.81 atm 1.4 atm2.6 L1.58 atm 2.7 atm3.8 L4.46 atm 7.9 atm =

The Kinetic-Molecular Theory (the theory of moving particles) 1. Gas particles are in constant, random motion. 2. The volume of the particles is negligible compared to the container volume. 3. The attractive-repulsive forces between particles are negligible. 4. Collisions are elastic. 5. KE avg of particles is proportional to absolute temperature. (i.e., no energy is lost).

At a given temp., the gas particles of Sample A have the same avg. KE as the gas particles of Sample B. pressure = “ ” and “ ” gas particles collide with the sides of the container how hardhow often container wall A container wall B

1 mol 25 o C in a 5.0-L container… 1 mol 350 o C in a 5.0-L container… 1 mol 25 o C in a 5.0-L container… 5 mol 25 o C in a 5.0-L container…

Particle-Velocity Distribution (___________, same T and P) # of particles Speed of particles (m/s) (SLOW) most massive gas (e.g., CO 2 ) least massive gas (e.g., He) various gases (FAST) rms (root-mean-square) speed a gas of intermediate mass (e.g., N 2 )

Particle-Velocity Distribution (same gas, same P, ________) # of particles Speed of particles (m/s) (SLOW) coldest gas warmest gas various T (FAST)

T = 80.0 o C Equation for the rms speed of a gas: R = J/mol-K  = molar mass, in kg KE avg for a particle = ½ m u 2 in kg Find the rms speed of chlorine gas at 80.0 o C. = 353 K  = 71.0 g = kg = 352 m/s

diffusion: effusion: the net movement of a substance from high to low conc. the escape of gas particles through a tiny hole into an evacuated space NET MOVEMENT more massive = slow; less massive = fast [ HIGH ][ LOW ]

For gases, rates of diffusion & effusion obey Graham’s law: ** To use Graham’s Law, both gases must be at the… same temp. r = rates  = molar masses The rate of diffusion of gases is slower than the molecular speeds because of... collisions. -- ~10 x 10 9 collisions/sec per particle The mean free path is the average distance traveled by a particle between collisions. -- it is shorter when the pressure is… high t = times

He Ne Ar Kr Xe Rn 86 (222) mm = 16.0 g/mol = 39.9 g/mol Ar CH 4 moves 1.58 times faster than which noble gas? “Ar?” “Aahhrrrr! Buckets o’ blood! Swab de decks, ye scurvy dogs!” Ne 2 or Ar?

Real Gases: Deviations from Ideal Behavior All real gases deviate, to some degree, from PV = nRT. The deviations are most pronounced at... high P and/or low T. DO occupy space Real gas particles...1) 2) DO experience attractive forces

real gas (van der Waals equation): ideal gas (ideal gas equation): correction for particle volume correction for particle attraction Johannes Diderik van der Waals (1837 – 1923) (particles w /SOME vol. exert a bigger P than those w /zero vol.) (attraction makes P smaller than ideal gas law predicts)

ideal gas particles real gas particles mass m, speed v, ~NO volume mass m, speed v, YES volume **GREATER pressure than PV = nRT predicts shorter mean free path = more collisions/sec. Effect of Particle Volume

The constants a and b are unique for each gas. -- a is large when interparticle attractions are large -- b is large for large gas particles a and b both tend to increase with mass e.g., both are larger for Xe than for Ne