🥵Thermodynamics Unit 10 – Ideal Gases and Real Gases

Key Concepts and Definitions

• Ideal gas a hypothetical gas that perfectly follows the ideal gas law and assumptions
• Real gas a gas that deviates from ideal behavior due to intermolecular forces and finite molecular volume
• Equation of state a mathematical relationship between state variables (pressure, volume, temperature) that describes the behavior of a gas
• Compressibility factor ($Z$) a measure of how much a real gas deviates from ideal behavior, defined as $Z = \frac{PV}{nRT}$
  • For an ideal gas, $Z = 1$ at all conditions
  • For a real gas, $Z$ varies with pressure and temperature
• Critical point the highest temperature and pressure at which a substance can exist as a vapor and liquid in equilibrium
  • Critical temperature ($T_c$) the temperature above which a gas cannot be liquefied by pressure alone
  • Critical pressure ($P_c$) the pressure required to liquefy a gas at its critical temperature
• Intermolecular forces attractive and repulsive forces between molecules that cause deviations from ideal behavior (van der Waals forces)

Ideal Gas Laws and Equations

• Ideal gas law $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is number of moles, $R$ is the universal gas constant, and $T$ is absolute temperature
• Boyle's law $PV = \text{constant}$ at constant temperature, pressure and volume are inversely proportional
• Charles's law $\frac{V}{T} = \text{constant}$ at constant pressure, volume and temperature are directly proportional
• Gay-Lussac's law $\frac{P}{T} = \text{constant}$ at constant volume, pressure and temperature are directly proportional
• Avogadro's law $\frac{V}{n} = \text{constant}$ at constant pressure and temperature, volume and number of moles are directly proportional
• Dalton's law of partial pressures $P_{\text{total}} = P_1 + P_2 + \ldots + P_n$, where $P_{\text{total}}$ is the total pressure of a gas mixture and $P_1, P_2, \ldots, P_n$ are the partial pressures of each component gas
• Amagat's law of partial volumes $V_{\text{total}} = V_1 + V_2 + \ldots + V_n$, where $V_{\text{total}}$ is the total volume of a gas mixture and $V_1, V_2, \ldots, V_n$ are the partial volumes of each component gas

Properties of Ideal Gases

• Ideal gas particles are assumed to be point masses with no volume and no intermolecular forces
• Ideal gas particles have perfectly elastic collisions with each other and the container walls
• The average kinetic energy of ideal gas particles is directly proportional to the absolute temperature
• Ideal gases have a constant specific heat capacity that is independent of temperature
  • Specific heat capacity at constant volume ($C_v$) the amount of heat required to raise the temperature of a unit mass of gas by one degree at constant volume
  • Specific heat capacity at constant pressure ($C_p$) the amount of heat required to raise the temperature of a unit mass of gas by one degree at constant pressure
• The internal energy of an ideal gas depends only on its temperature and is independent of pressure and volume
• The entropy change of an ideal gas during a reversible process is given by $\Delta S = nR \ln \frac{V_2}{V_1}$ at constant temperature or $\Delta S = nC_v \ln \frac{T_2}{T_1}$ at constant volume

Limitations of the Ideal Gas Model

• The ideal gas model assumes that gas particles have no volume, but real gas molecules have a finite volume that becomes significant at high pressures
• The ideal gas model assumes no intermolecular forces, but real gas molecules experience attractive and repulsive forces that affect their behavior
• The ideal gas law becomes less accurate at high pressures and low temperatures, where intermolecular forces and molecular volume are more significant
• The ideal gas model does not account for the formation of liquid or solid phases, which can occur at high pressures and low temperatures
• The ideal gas model assumes that the specific heat capacities are constant, but they can vary with temperature for real gases
• The ideal gas model does not predict non-ideal behavior such as Joule-Thomson cooling or the inversion temperature
• The compressibility factor ($Z$) deviates from unity for real gases, indicating non-ideal behavior

Introduction to Real Gases

• Real gases are actual gases that exhibit deviations from ideal gas behavior due to intermolecular forces and finite molecular volume
• Intermolecular forces in real gases include dispersion forces (London forces), dipole-dipole interactions, and hydrogen bonding
  • Dispersion forces result from temporary fluctuations in the electron distribution of molecules and are present in all gases
  • Dipole-dipole interactions occur between molecules with permanent dipole moments (polar molecules)
  • Hydrogen bonding is a strong intermolecular force that occurs between molecules containing hydrogen bonded to highly electronegative atoms (N, O, F)
• The finite volume of real gas molecules becomes significant at high pressures, leading to a reduction in the available volume for the gas to occupy
• Real gases may undergo phase transitions (condensation or solidification) at high pressures and low temperatures
• The compressibility factor ($Z$) is used to quantify the deviation of real gases from ideal behavior
  • $Z > 1$ indicates repulsive intermolecular forces dominate (e.g., at high temperatures and low pressures)
  • $Z < 1$ indicates attractive intermolecular forces dominate (e.g., at low temperatures and high pressures)

Equations of State for Real Gases

• Van der Waals equation $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$, where $a$ and $b$ are gas-specific constants accounting for intermolecular forces and molecular volume
  • $a$ is a measure of the strength of attractive intermolecular forces
  • $b$ is the volume occupied by the molecules themselves (excluded volume)
• Redlich-Kwong equation $P = \frac{RT}{V-b} - \frac{a}{\sqrt{T}V(V+b)}$, an improved equation of state that better represents the behavior of real gases at high pressures and low temperatures
• Soave-Redlich-Kwong (SRK) equation $P = \frac{RT}{V-b} - \frac{a\alpha}{V(V+b)}$, where $\alpha$ is a temperature-dependent function that improves the accuracy of the Redlich-Kwong equation
• Peng-Robinson equation $P = \frac{RT}{V-b} - \frac{a\alpha}{V(V+b)+b(V-b)}$, another widely used equation of state that accurately predicts the behavior of real gases and liquids
• Virial equation $\frac{PV}{nRT} = 1 + \frac{B}{V} + \frac{C}{V^2} + \ldots$, an equation of state that expresses the compressibility factor as a power series in inverse volume, where $B$, $C$, etc., are virial coefficients dependent on temperature and the gas species

Comparing Ideal and Real Gas Behavior

• At low pressures and high temperatures, real gases behave more like ideal gases because intermolecular forces are less significant and molecular volume is negligible compared to the total volume
• As pressure increases and temperature decreases, real gases deviate more from ideal behavior due to increased intermolecular forces and the significance of molecular volume
• The compressibility factor ($Z$) is used to compare the behavior of real gases to ideal gases
  • For an ideal gas, $Z = 1$ at all conditions
  • For a real gas, $Z$ varies with pressure and temperature, indicating deviations from ideal behavior
• Real gases may exhibit non-ideal behavior such as Joule-Thomson cooling, where the temperature of a gas changes upon expansion through a porous plug or valve
  • The Joule-Thomson coefficient $\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H$ determines whether a gas will cool or heat up during expansion
  • The inversion temperature is the temperature at which the Joule-Thomson coefficient changes sign (cooling to heating or vice versa)
• Real gases have a critical point, beyond which the distinction between liquid and gas phases disappears
  • The critical point is characterized by the critical temperature ($T_c$), critical pressure ($P_c$), and critical volume ($V_c$)
  • At the critical point, the compressibility factor is approximately 0.3 for most gases
• The behavior of real gases can be visualized using compressibility factor charts (Z-charts) or reduced pressure-volume-temperature (PVT) diagrams, which show the deviation from ideal behavior as a function of pressure and temperature

Applications and Problem-Solving

• Ideal gas law calculations involve using the equation $PV = nRT$ to solve for unknown properties of a gas, such as pressure, volume, temperature, or number of moles
  • Example: Calculate the volume occupied by 2 moles of an ideal gas at 300 K and 1 atm pressure
• Real gas calculations often require the use of equations of state (e.g., van der Waals, Redlich-Kwong, SRK, Peng-Robinson) to account for deviations from ideal behavior
  • Example: Determine the pressure of 1 mole of carbon dioxide at 350 K and a volume of 0.5 L using the van der Waals equation with $a = 3.592 \text{ L}^2\text{atm}/\text{mol}^2$ and $b = 0.04267 \text{ L}/\text{mol}$
• Compressibility factor calculations involve determining the deviation of a real gas from ideal behavior using the equation $Z = \frac{PV}{nRT}$
  • Example: Calculate the compressibility factor for nitrogen at 200 atm and 300 K, given that the molar volume is 0.035 L/mol
• Joule-Thomson effect calculations involve determining the temperature change of a gas upon expansion using the Joule-Thomson coefficient $\mu_{JT}$ and the pressure change $\Delta P$
  • Example: A gas with $\mu_{JT} = 0.2 \text{ K}/\text{atm}$ expands from 100 atm to 1 atm. Calculate the temperature change assuming the process is Joule-Thomson expansion
• Critical point calculations involve determining the critical properties of a gas using experimental data or equations of state
  • Example: Estimate the critical temperature and pressure of water using the Redlich-Kwong equation and the following data: $T_c = 647.1 \text{ K}$, $P_c = 220.6 \text{ atm}$, and $V_c = 0.056 \text{ L}/\text{mol}$
• Gas mixture calculations involve applying Dalton's law of partial pressures or Amagat's law of partial volumes to determine the properties of a mixture of gases
  • Example: A gas mixture contains 2 moles of nitrogen and 3 moles of oxygen at a total pressure of 2 atm. Calculate the partial pressure of each gas in the mixture