*Gas Laws*
Physical Characteristics of Gases Physical Characteristics Typical Units Volume, V liters (L) Pressure, P atmosphere (1 atm = 1.015x105 N/m2) Temperature, T Kelvin (K) Number of atoms or molecules, n mole (1 mol = 6.022x1023 atoms or molecules)
“Father of Modern Chemistry” Chemist & Natural Philosopher Boyle’s Law Pressure and volume are inversely related at constant temperature. PV = K As one goes up, the other goes down. P1V1 = P2V2 “Father of Modern Chemistry” Robert Boyle Chemist & Natural Philosopher Listmore, Ireland January 25, 1627 – December 30, 1690
Boyle’s Law: P1V1 = P2V2
Boyle’s Law: P1V1 = P2V2
Jacques-Alexandre Charles Mathematician, Physicist, Inventor Charles’ Law Volume of a gas varies directly with the absolute temperature at constant pressure. V = KT V1 / T1 = V2 / T2 Jacques-Alexandre Charles Mathematician, Physicist, Inventor Beaugency, France November 12, 1746 – April 7, 1823
Charles’ Law: V1/T1 = V2/T2
Charles’ Law: V1/T1 = V2/T2
Avogadro’s Law At constant temperature and pressure, the volume of a gas is directly related to the number of moles. V = K n V1 / n1 = V2 / n2 Amedeo Avogadro Physicist Turin, Italy August 9, 1776 – July 9, 1856
Avogadro’s Law At the same P and T equal Volumes of gas contain the same # of molecules Na = 6.023X1023 molecules/ mole
Avogadro’s Law: V1/n1=V2/n2
Joseph-Louis Gay-Lussac Gay-Lussac Law (Pressure Law) At constant volume, pressure and absolute temperature are directly related. P = k T P1 / T1 = P2 / T2 Joseph-Louis Gay-Lussac Experimentalist Limoges, France December 6, 1778 – May 9, 1850
Eaglesfield, Cumberland, England Dalton’s Law The total pressure in a container is the sum of the pressure each gas would exert if it were alone in the container. The total pressure is the sum of the partial pressures. PTotal = P1 + P2 + P3 + P4 + P5 ... (For each gas P = nRT/V) John Dalton Chemist & Physicist Eaglesfield, Cumberland, England September 6, 1766 – July 27, 1844
Dalton’s Law
Only at very low P and high T Differences Between Ideal and Real Gases Ideal Gas Real Gas Obey PV=nRT Always Only at very low P and high T Molecular volume Zero Small but nonzero Molecular attractions Small Molecular repulsions
Johannes Diderik van der Waals Mathematician & Physicist Van der Waal’s equation Corrected Pressure Corrected Volume “a” and “b” are determined by experiment different for each gas bigger molecules have larger “b” “a” depends on both size and polarity Johannes Diderik van der Waals Mathematician & Physicist Leyden, The Netherlands November 23, 1837 – March 8, 1923
The Ideal Gas Ideal gas properties Volume of gas molecules is negligible compared with gas volume Forces of attraction or repulsion between molecules or walls of container are zero No loss of internal energy due to collisions
Derivation of the kinetic theory formula This proof was originally proposed by Maxwell in 1860. He considered a gas to be a collection of molecules and made the following assumptions about these molecules: molecules behave as if they were hard, smooth, elastic spheres molecules are in continuous random motion the average kinetic energy of the molecules is proportional to the absolute temperature of the gas
Assumptions contd. the molecules do not exert any appreciable attraction on each other the volume of the molecules is infinitesimal when compared with the volume of the gas the time spent in collisions is small compared with the time between collisions
Consider a volume of gas V enclosed by a cubical box of sides L Consider a volume of gas V enclosed by a cubical box of sides L. Let the box contain N molecules of gas each of mass m, and let the density of the gas be r. Let the velocities of the molecules be u1, u2, u3 . . . uN. (Figure 2)
Consider a molecule moving in the x-direction towards face A with velocity u1. On collision with face A the molecule will experience a change of momentum equal to 2mu1. It will then travel back across the box, collide with the opposite face and hit face A again after a time t, where t = 2L/u1.
The number of impacts per second on face A will therefore be 1/t = u1/2L. Therefore rate of change of momentum = [mu12]/L = force on face A due to one molecule. But the area of face A = L2, so pressure on face A = [mu12]/L3
But there are N molecules in the box and if they were all travelling along the x-direction then Total pressure on face A = [m/L3](u12 + u22 +...+ uN2)
But on average only one-third of the molecules will be travelling along the x-direction. Therefore: pressure = 1/3 [m/L3](u12 + u22 +...+ uN2)
If we rewrite Nc2 = [u12 + u22 + …+ uN2 ] where c is the mean square velocity of the molecules: pressure = 1/3 [m/L3]Nc2 But L3 is the volume of the gas and therefore: Pressure (P) = 1/3 [m/V]Nc2 and so PV = 1/3 [Nmc2] and this is the kinetic theory equation.
Pressure (P) = 1/3 [ρc2] The root mean square velocity or r.m.s. velocity is written as c r.m.s. and is given by the equation: r.m.s. velocity = c r.m.s. = [c2]1/2 = [u12 + u22 + …+ uN2 ]1/2/N We can use this equation to calculate the root mean square velocity of gas molecules at any given temperature and pressure.