I. 8.1: GASES AND THE KINETIC-MOLECULAR THEORY
A. KINETIC-MOLECULAR THEORY
A model for describing the behavior of gases (ideal gases), is based on the following assumptions:
1. The molecules of gas are so small compared to the average spacing between them that the molecules themselves take up essentially no volume
2. The molecules of a gas are in constant motion, moving in straight lines at constant speeds and in random directions between collisions; the collisions of the molecules within the walls of the container define the pressure of a gas, and all the collisions are elastic (total KE is conserved)
3. Since the molecules move at constant speed between collisions and the collisions are elastic, the molecules of a gas experience no intermolecular forces
4. The molecules of a gas span a distribution of speeds, and the average KE of the molecules is directly proportional to the absolute temperature (K) of the sample:
a) KEav ∝ T → T ∝ 1/v2
B. UNITS OF VOLUME, TEMPERATURE, AND PRESSURE
1. Volume – cc or L or m3
2. Temperature – K = °C + 273.15
3. Pressure – Pa = 1 N/m3
a) 101,300 Pa = 101.3 kPa = 1 atm = 760 torr = 760 mmHg
4. STP – 0° C (273 K) and 1 atm
II. 8.2: THE IDEAL GAS LAW
A. PV = nRT
1. P = pressure of the gas in atm
2. V = volume of the container in L
3. n = number of moles of the gas
4. R = universal gas constant (0.0821 L-atm/K-mol or J/(mol*K)
a) R = kbNa → kb is Boltzman’s constant and Na is Avogadro’s number
5. T = absolute temperature (K)
B. PVT GAS LAWS IN SYSTEMS WHERE N IS CONSTANT
1. Charles’s Law – if the pressure remains constant, then a gas will expand when heated and contract when cooled
a) If pressure is constant, V/T = k (k is a constant); ∴ V ∝ T
2. Boyle’s Law – if volume decreases, then pressure will increase, and vice versa
a) If the temp is constant, PV = k (k is a constant); ∴ P ∝ 1/V
3. Equations:
a) Constant P → V1/T1 = V2/T2
b) Constant T → P1V1 = P2V2
c) Constant V → P1/T1 = P2/T2
d) Combined gas law:
(1) P1V1/T1 = P2V2/T2
4. Avogadro’s law – the same number of particles exist in containers of the same size with the same temperature and pressure, regardless of the identity of the particles
a) V/n = k (k is constant)
b) Standard molar volume of any ideal gas = 22.4 L
III. 8.3: DEVIATIONS FROM THE IDEAL GAS BEHAVIOR
A. Review these two assumptions from kinetic-molecular theory:
1. No intermolecular forces
a) Some gases have intermolecular forces (like water vapor). The resulting pressure would therefore be smaller than the ideal pressure: Preal < Pideal
2. Volumeless particles
a) Remember, the volume of a gas is defined as the free space the particles have to move around; therefore, when particles at very high pressures actually take up volume, the free space decreases: Vreal < Videal
B. Summary of Deviations:
1. Attractive forces between particles cause a decrease in pressure
2. Particle volume (large particles) causes a decrease in free space (system volume)
C. Van der Waals equation
Accounts for differences in the observed behavior of real gases and calculated properties of ideal gases
1. (P + an2/V2)(V – nb) = nRT
a) an2/V2 → serves as a correction for the intermolecular forces that generally result in lower pressures for real gases
(1) This term approaches 0 at higher temperatures because gases behave more like ideal gases at high temperatures
b) nb → corrects for the physical volume that individual particles occupy in a real gas
c) van der Waals constants – a and b
(1) a is greater when molecules experience greater intermolecular forces
(2) b is greater when molecules are larger and therefore take up larger volumes
2. Effects of high pressures and low temps
a) ↑ P – ↑ intermolecular forces, causing ↓ P (less than predicted)
b) ↓ T – ↑ intermolecular forces, causing ↓ V (less than predicted)
3. Most ideal gases have smallest weights/volumes and weakest intermolecular forces
IV. 8.4: DALTON’S LAW OF PARTIAL PRESSURES
A. Partial pressues
Pressure of one gas in a container of multiple gases
B. Dalton’s law of partial pressures
The total pressure is the sum of the partial pressures
1. Ptot = pa + pb + pc
V. 8.5: GRAHAM’S LAW OF EFFUSION
A. Effusion
The escape of a as molecule through a very tiny hole into an evacuated region
B. Consider 2 gases, gas A and gas B, in the same container:
1. Gas A is lighter than gas B, but they have the same KE:
2. ½mA(vA2)avg = ½mB(vB2)avg ⇒ (vA2)avg/(vB2)avg = mB/mA ⇒ (rms vA)/(rms vB) = √(mB/mA)
3. simplify → vA/vB = √(mA/mB) → note the opposite placement of the A/B on each side
C. Graham’s law of effusion:
1. (rate of effusion gas A)/(rate of effusion gas B) = √
2. KE ∝ T → therefore, if kinetic energies of gases are equal, so are their temperatures