Properties of Gases

Unit 1: Properties of Gases

Part B: Physical Chemistry IIIA

PCA216X

Dr. Maphoru MV

Ideal Gas Law Formula

The Ideal Gas Law is expressed by the equation:
$PV = nRT$
where:

• $P$ = Pressure

• $V$ = Volume

• $n$ = Number of moles of gas

• $R$ = Universal gas constant (8.314 J/(mol K))

• $T$ = Temperature (in Kelvin)

As per the law, when temperature is kept constant and volume decreases from $V1$ to $V2$, the pressure increases from $P1$ to $P2$.

Learning Outcomes

The learning outcomes for this unit include:

1. Defining Effusion and Diffusion: Understanding and distinguishing between the two concepts.

2. Deriving Graham's Law: Ability to derive Graham's Law of Effusion and use it in calculations.

3. Understanding Kinetic Molecular Theory: Discussing the Kinetic Molecular Theory and its application in explaining the behavior of an ideal gas.

4. Temperature Effects: Discussing how temperature affects the distribution of molecular speeds of gases.

5. Defining Root-Mean-Square Speed: Defining and understanding the root-mean-square speed of a gas.

Graham's Law of Effusion

Definitions

• Effusion: The process by which gas molecules escape through a tiny hole into an evacuated space. An example is a tire puncture.

• Diffusion: The process through which one substance spreads throughout another substance or space, such as perfume diffusing throughout a room.

Graham's Law of Effusion

Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its density (or molar mass) when compared at identical pressures and temperatures. Mathematically, it is often represented as:
$r<em>1 imes rac{1}{ ext{sqrt}(d</em>1)} = r<em>2 imes rac{1}{ ext{sqrt}(d</em>2)}$
or equivalently,
$rac{r<em>1}{r</em>2} = rac{ ext{sqrt}(M<em>2)}{ ext{sqrt}(M</em>1)}$
where:

• $r1$, $r2$ = rates of effusion of gas 1 and gas 2 respectively

• $M1$, $M2$ = molar masses of gas 1 and gas 2 respectively.

Tutorials

Tutorial 1: Given gases O2, NH3, He, determine which gas will effuse at a higher rate at the same temperature and pressure and justify the answer.

Tutorial 2: Ammonia effuses at a rate that is 2.93 times larger than that of an unknown gas. Find the molar mass of the unknown gas using Graham's Law.

Tutorial 3: Assess the densities of CO2 (1.96 g/L) and N2 (1.25 g/L). Determine which gas will effuse faster and calculate the ratio of their rates of effusion.

Kinetic Molecular Theory (KMT)

The Ideal Gas Laws explain the behavior of gases; however, they do not delve into the reasons behind these behaviors. The Kinetic Molecular Theory was developed to provide a model that helps understand the physical properties of gases. It is summarized by the following hypotheses:

1. Gases are composed of individual particles, typically atoms or molecules, whose sizes are negligible compared to the distances between them.

2. The particles are in constant and random motion, thus possessing kinetic energy.

3. There are no attractive or repulsive forces between the particles.

4. The volume of gas molecules is negligible relative to the overall volume of the gas.

5. Energy can be transferred between molecules during collisions, but the average kinetic energy remains constant as long as the temperature is kept constant (the collisions are perfectly elastic).

6. The average kinetic energy of gas molecules is directly proportional to the absolute temperature.

KMT and Kinetic Energy

The average kinetic energy (KE) can be expressed as
$ext{KE} = rac{3}{2}kT$
where:

• $k$ = Boltzmann constant ($1.38064852 imes 10^{-23} ext{m}^2 ext{kg}^{-1} ext{K}^{-1}$)

• $T$ = Absolute temperature in Kelvin.

Molecular Velocity

• Root Mean Square Velocity ($u{rms}$):
  The $u{rms}$ speed can be expressed mathematically as:
  $u_{rms} = ext{sqrt} rac{3RT}{M}$

• The relationship between mass and temperature indicates that lighter gases have higher $u{rms}$ than heavier gases at the same temperature, and higher temperatures yield higher $u{rms}$.

Kinetic Theory and Boyle's Law

According to Boyle's Law, which states that the volume of a gas is inversely proportional to its pressure (provided temperature is constant):
$PV = C$
The interactions explained by KMT support that decreasing the volume of a gas increases the frequency of collisions between gas molecules and the container walls, thereby increasing pressure.

Kinetic Theory and Gay-Lussac's Law

Gay-Lussac's Law states that the pressure of a fixed quantity of gas at constant volume is directly proportional to its Kelvin temperature:
$P = C imes T$
This is consistent with the increase in average velocity of gas particles at higher temperatures.

Kinetic Theory and Charles's Law

Charles's Law indicates that the volume of a gas is directly proportional to its absolute temperature when pressure is held constant:
$V = C' imes T$
This law is justified through KMT, as raising the gas temperature raises the average kinetic energy of particles, thus increasing volume if pressure remains constant.

Kinetic Theory and Dalton's Law

Dalton's Law states that the total pressure of a mixture of gases equals the sum of the partial pressures of each gas in the mixture:
$P<em>{total} = P</em>1 + P<em>2 + P</em>3 + …$
This derives from the principle that gas particles act independently due to negligible intermolecular forces reflected in KMT.

Kinetic Theory and Graham's Law

Graham's Law mirrors the earlier statements of effusion, reiterating that the rate of effusion of a gas is inversely proportional to the square root of its molar mass (or density). When comparing two gases at the same temperature, their kinetic energies are equal:
$(KE)<em>{1} = (KE)</em>{2} ext{, such that } rac{1}{2} M<em>{1} u</em>{1}^2 = rac{1}{2} M<em>{2} u</em>{2}^2$

• This connection highlights that lighter gases will effuse faster than heavier gases.

Kinetic Theory and Absolute Zero

Absolute zero represents the lowest possible temperature, where thermal energy is effectively zero. The energy of gas molecules ceases, thus halting all motion. In terms of kinetic theory, average kinetic energy is proportional to absolute temperature, encapsulated as:
$ext{Average KE} ext{ is proportional to } T$
At absolute zero, the gas molecules cease to possess average kinetic energy, indicating that the gas reaches an electrically static state—an impossibility to achieve experimentally per current understanding but theoretically proposed based on KMT principles.

Tutorials

Tutorial 1: Calculate the average kinetic energy of nitrogen gas in a $22.8 ext{ g}$ container at $500 ext{ K}$.
Tutorial 2: Determine the temperature of HCl gas with a root mean square velocity of $647.5 ext{ m/s}$.
Tutorial 3: Rank the following samples of gas in order of increasing average speed of gas molecules:

• a. 2.0 moles of neon gas at 1.1 atm and -32 degrees Celsius

• b. 3.0 moles of neon gas at 1.6 atm and -54 degrees Celsius

• c. 2.3 moles of argon gas at 1.4 atm and -54 degrees Celsius

• d. 1.8 moles of argon gas at 2.3 atm and -76 degrees Celsius

Tutorial 4: For the urms speed of $65 ext{ g}$ of oxygen gas at $254 ext{ m/s}$, compute the average kinetic energy.
Tutorial 5: Evaluate the accuracy of statements regarding the rate of diffusion concerning gas temperature and RMS speed.
Tutorial 6: Determine the ratio of $u_{rms}$ values for helium and xenon at $30^ ext{C}$ and substantiate the result.