Study Notes on Gases and the Kinetic-Molecular Theory

Gas:
- Particles are far apart.
- Particles move freely.
- Fill the available space.
Liquid:
- Particles are close together but move around one another.
Solid:
- Particles are close together in a regular array.
- Do not move around one another.

Distinguishing gases from liquids and solids:
- Gas volume changes significantly with pressure.
- Solid and liquid volumes are not greatly affected by pressure.
- Gas volume changes significantly with temperature.
- Gases expand when heated and shrink when cooled.
- The volume change is 50 to 100 times greater for gases than for liquids and solids.
- Gases flow very freely.
- Gases have relatively low densities.
- Gases form a solution in any proportions.
- Gases are freely miscible with each other.

Definition of Pressure:
- Pressure is defined as $P = \frac{\text{force}}{\text{area}}$ .
- Atmospheric Pressure:
- Arises from the force exerted by atmospheric gases on the Earth's surface.
- Decreases with altitude.
Units of Measurement:
- $1 ext{ Newton/m}^2 = 1 ext{ pascal (Pa)}$
- $1 ext{ standard atmosphere} = 101,325 ext{ Pa}$
- $1 ext{ atmosphere} \approx 10^5 ext{ Pa}$

Unit	Value
Pascal (Pa)
Kilopascal (kPa)	$1 ext{ kPa} = 1000 ext{ Pa}$
Atmosphere (atm)	$1 atm = 101.325 ext{ kPa}$
Millimeters of mercury (mmHg)	$760 ext{ mmHg} = 1 atm$
torr	$1 ext{ torr} = 1 ext{ mmHg}$
Pounds per square inch (lb/in² or psi)	$14.7 ext{ lb/in}^2 = 1 ext{ atm}$
Bar	$1 ext{ bar} = 1.01325 imes 10^5 ext{ Pa}$

Variables in Gas Laws:
- Pressure (P)
- Temperature (T)
- Volume (V)
- Amount (number of moles, n)
Ideal Gas:
- An ideal gas exhibits linear relationships among these variables.
- No ideal gas actually exists; however, most simple gases behave nearly ideally at ordinary temperatures and pressures.

Definition:
- At constant temperature, the volume occupied by a fixed amount of gas is inversely proportional to the external pressure.
- Mathematically expressed as: $V \propto \frac{1}{P}$ or $PV = ext{constant}$
- At fixed T and n:
- As volume (V) increases, pressure (P) decreases.
- As pressure (P) increases, volume (V) decreases.

Definition:
- At constant pressure, the volume occupied by a fixed amount of gas is directly proportional to its absolute (Kelvin) temperature.
- Mathematically expressed as: $V \propto T$ or $\frac{V}{T} = ext{constant}$
- At fixed P and n:
- As volume (V) increases, temperature (T) decreases.
- As temperature (T) increases, volume (V) decreases.

Definition:
- At fixed temperature and pressure, the volume occupied by a gas is directly proportional to the amount of gas.
- Mathematically expressed as: At fixed temperature and pressure, equal volumes of any ideal gas contain equal numbers of particles (or moles).

Respiration:
- Illustrates the principle of gas laws: the diaphragm creates changes in volume, leading to changes in pressure, enabling air to flow in and out of lungs.

STP (Standard Temperature and Pressure):
- Specified as a pressure of $1 ext{ atm} (760 ext{ torr})$ and a temperature of $0°C (273.15 K)$ .
- Standard Molar Volume:
- The volume of 1 mol of an ideal gas at STP is $22.414 ext{ L}$ or approximately $22.4 ext{ L}$ .

Expression of the Ideal Gas Law:
- Mathematically defined as: $PV = nRT$
- Where
  - R = the universal gas constant = $0.0821 \frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}}$
The ideal gas law can also be expressed by the combined equation: $\frac{P1V1}{T1} = \frac{P2V2}{T2}$

Boyle's Law Problem:
- If 22.5 L of nitrogen at 748 mm Hg are compressed to 725 mm Hg at constant temperature, what is the new volume?
- Use Boyle's Law formula to find the new volume, which leads to $V_2$ calculation.
Charles's Law Problem:
- Calculate the decrease in temperature when 6.00 L at 20.0 °C is compressed to 4.00 L.
- The new temperature must be calculated using Charles's Law adjustments.
Avogadro's Law Problem:
- A gas occupies a volume of 12.4 L at 23°C and 0.956 atm; find the volume at 40°C and 0.956 atm using Avogadro's Law.

Postulate 1: Gas particles are tiny with large spaces between them. The volume of each particle is negligible compared to the total volume of the gas.
Postulate 2: Gas particles are in constant, random, straight-line motion except when they collide with each other or with the walls of the container.
Postulate 3: Collisions among particles are elastic, meaning colliding particles exchange energy but do not lose energy due to friction; their total kinetic energy remains constant.

Definition: Effusion is the process by which gas escapes through a small hole into an evacuated space. Graham’s Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass: $\text{Rate of effusion} \propto \frac{1}{\sqrt{\mathcal{M}}}$ where $\mathcal{M}$ represents the molar mass.

Real gases deviate from ideal gas behavior due to:
- Real volume of gas particles.
- Attractive and repulsive forces between particles.
- Deviations are more significant at low temperature and high pressure.

Van der Waals Adjustments: The ideal gas law can be adjusted through the van der Waals equation to account for the volume of gas particles and the intermolecular forces:
$\left(P + \frac{a n^2}{V^2}\right)(V-nb) = nRT$
where constants a and b account for attractions and volume, respectively.