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)     KE_av ∝ 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 = k_bN_a → k_b is Boltzman’s constant and N_a 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:  P_real < P_ideal

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:  V_real < V_ideal

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.     P_tot = p_a + p_b + p_c

 

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.     ½m_A(v_A²)_avg = ½m_B(v_B²)_avg ⇒ (v_A²)_avg/(v_B²)_avg = m_B/m_A ⇒ (rms v_A)/(rms v_B) = √(m_B/m_A)

3.     simplify → v_A/v_B = √(m_A/m_B) → 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) = √