## 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/v^{2}**

### B. UNITS OF VOLUME, TEMPERATURE, AND PRESSURE

1. Volume – cc or L or m^{3}

2. Temperature – K = °C + 273.15

3. Pressure – Pa = 1 N/m^{3}

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_{b}N_{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 → V_{1}/T_{1} = V_{2}/T_{2}

b) Constant T → P_{1}V_{1} = P_{2}V_{2}

c) Constant V → P_{1}/T_{1} = P_{2}/T_{2}

d) Combined gas law:

(1) **P _{1}V_{1}/T_{1} = P_{2}V_{2}/T_{2}**

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 + an ^{2}/V^{2})(V – nb) = nRT**

a) **an ^{2}/V^{2}** → 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_{A}^{2})_{avg} = ½m_{B}(v_{B}^{2})_{avg} ⇒ (v_{A}^{2})_{avg}/(v_{B}^{2})_{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) = √

2. KE ∝ T → therefore, if kinetic energies of gases are equal, so are their temperatures