2.1 Molecular Model of an Ideal Gas

Gases are everywhere, and understanding their behavior is crucial. The ideal gas law connects pressure, volume, temperature, and the number of gas particles, helping us predict how gases will act under different conditions.

But gases aren't always ideal. Real gases can behave differently, especially at high pressures or low temperatures. The molecular model of gases helps explain these behaviors, treating gas particles as tiny, fast-moving objects that collide with each other and container walls.

Ideal Gas Law and Molecular Model

Gas behavior and ideal gas law

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• Ideal gas law relates pressure ($P$), volume ($V$), temperature ($T$), and number of moles ($n$) of a gas: $PV = nRT$
  • $R$ is the ideal gas constant, which has a value of 8.314 J/(mol·K)
• At constant temperature and number of moles, pressure is inversely proportional to volume (###Boyle's_Law_0###)
  • Doubling the volume of a gas at constant temperature and moles will halve its pressure
• At constant pressure and number of moles, volume is directly proportional to temperature (###Charles's_Law_0###)
  • Increasing the temperature of a gas from 300 K to 600 K at constant pressure will double its volume
• At constant volume and temperature, pressure is directly proportional to the number of moles (###Avogadro's_Law_0###)
  • Adding more gas to a container at constant volume and temperature increases the pressure
• Combined gas law relates pressure, volume, and temperature changes between two states: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
  • Used to calculate changes in gas properties when multiple variables are changing simultaneously (pressure, volume, and temperature)

Mole and molecule conversions

• One mole of a substance contains Avogadro's number ($N_A$) of particles (atoms or molecules)
  • $N_A = 6.022 \times 10^{23}$ particles per mole
• Number of moles ($n$) is related to the number of particles ($N$) by: $n = \frac{N}{N_A}$
  • 1 mole of carbon dioxide ($CO_2$) contains $6.022 \times 10^{23}$ $CO_2$ molecules
• Molar mass ($M$) is the mass of one mole of a substance, expressed in grams per mole (g/mol)
  • Molar mass of water ($H_2O$) is 18.02 g/mol
• Mass ($m$) of a substance is related to the number of moles ($n$) by: $m = nM$
  • 2 moles of sodium chloride (NaCl, molar mass 58.44 g/mol) has a mass of 116.88 g

Limitations of ideal gas law

• Ideal gas law assumptions break down at high pressures, low temperatures, and near the point of condensation
  • High pressure: gas particles are close together, and intermolecular forces become significant
  • Low temperature: gas particles move slowly, and intermolecular forces become more important relative to kinetic energy
• Real gases deviate from ideal behavior due to the finite volume of gas particles and intermolecular forces between them
  • Finite volume: gas particles occupy space, reducing the available volume for motion
  • Intermolecular forces: attractive forces between gas particles cause them to stick together, reducing pressure
• Van der Waals equation corrects for non-ideal behavior: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$
  • $a$ accounts for intermolecular attraction, and $b$ accounts for the finite volume of gas particles
  • Used for more accurate calculations of gas behavior under non-ideal conditions (high pressure or low temperature)

Molecular model of ideal gases

• Gas particles are treated as point masses with negligible volume, moving randomly and independently
  • Particle size is much smaller than the average distance between particles
• Collisions between particles and the container walls are elastic (no energy loss)
  • Kinetic energy is conserved during collisions, and no energy is transferred to or from the container walls
• No intermolecular forces (attraction or repulsion) between gas particles
  • Gas particles do not interact with each other, only with the container walls during collisions
• Average kinetic energy of gas particles is proportional to the absolute temperature: $\frac{1}{2}mv_{rms}^2 = \frac{3}{2}kT$
  • $m$ is the mass of a gas particle, $v_{rms}$ is the root-mean-square velocity, and $k$ is the Boltzmann constant ($k = \frac{R}{N_A}$)
  • Higher temperature means higher average kinetic energy and faster particle motion
• Mean free path is the average distance a gas particle travels between collisions
  • Longer mean free path indicates less frequent collisions and lower gas density

Statistical mechanics and gas behavior

• Maxwell-Boltzmann distribution describes the probability distribution of particle speeds in an ideal gas
  • It shows that not all particles have the same speed, but rather a range of speeds centered around an average value
• Equipartition theorem states that energy is equally distributed among all degrees of freedom in a system
  • Each degree of freedom (e.g., translational, rotational, vibrational) contributes $\frac{1}{2}kT$ to the average energy per particle
• Degrees of freedom represent the independent ways a molecule can store energy
  • Monatomic gases have 3 translational degrees of freedom, while diatomic gases have additional rotational degrees of freedom
• Pressure in a gas results from the collective impact of particles colliding with the container walls
  • Higher particle velocity and more frequent collisions lead to higher pressure