Thermodynamics

🥵Thermodynamics Unit 10 – Ideal Gases and Real Gases

Ideal gases and real gases are fundamental concepts in thermodynamics. Ideal gases follow simple laws and equations, providing a useful model for understanding gas behavior. Real gases, however, deviate from ideal behavior due to intermolecular forces and finite molecular volume. This unit covers key concepts, equations, and properties of ideal and real gases. It explores the limitations of the ideal gas model, introduces real gas behavior, and examines equations of state for real gases. The unit also compares ideal and real gas behavior and discusses practical applications in problem-solving.

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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 (ZZ) a measure of how much a real gas deviates from ideal behavior, defined as Z=PVnRTZ = \frac{PV}{nRT}
    • For an ideal gas, Z=1Z = 1 at all conditions
    • For a real gas, ZZ 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 (TcT_c) the temperature above which a gas cannot be liquefied by pressure alone
    • Critical pressure (PcP_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=nRTPV = nRT, where PP is pressure, VV is volume, nn is number of moles, RR is the universal gas constant, and TT is absolute temperature
  • Boyle's law PV=constantPV = \text{constant} at constant temperature, pressure and volume are inversely proportional
  • Charles's law VT=constant\frac{V}{T} = \text{constant} at constant pressure, volume and temperature are directly proportional
  • Gay-Lussac's law PT=constant\frac{P}{T} = \text{constant} at constant volume, pressure and temperature are directly proportional
  • Avogadro's law Vn=constant\frac{V}{n} = \text{constant} at constant pressure and temperature, volume and number of moles are directly proportional
  • Dalton's law of partial pressures Ptotal=P1+P2++PnP_{\text{total}} = P_1 + P_2 + \ldots + P_n, where PtotalP_{\text{total}} is the total pressure of a gas mixture and P1,P2,,PnP_1, P_2, \ldots, P_n are the partial pressures of each component gas
  • Amagat's law of partial volumes Vtotal=V1+V2++VnV_{\text{total}} = V_1 + V_2 + \ldots + V_n, where VtotalV_{\text{total}} is the total volume of a gas mixture and V1,V2,,VnV_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 (CvC_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 (CpC_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 ΔS=nRlnV2V1\Delta S = nR \ln \frac{V_2}{V_1} at constant temperature or ΔS=nCvlnT2T1\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 (ZZ) 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 (ZZ) is used to quantify the deviation of real gases from ideal behavior
    • Z>1Z > 1 indicates repulsive intermolecular forces dominate (e.g., at high temperatures and low pressures)
    • Z<1Z < 1 indicates attractive intermolecular forces dominate (e.g., at low temperatures and high pressures)

Equations of State for Real Gases

  • Van der Waals equation (P+an2V2)(Vnb)=nRT\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT, where aa and bb are gas-specific constants accounting for intermolecular forces and molecular volume
    • aa is a measure of the strength of attractive intermolecular forces
    • bb is the volume occupied by the molecules themselves (excluded volume)
  • Redlich-Kwong equation P=RTVbaTV(V+b)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=RTVbaαV(V+b)P = \frac{RT}{V-b} - \frac{a\alpha}{V(V+b)}, where α\alpha is a temperature-dependent function that improves the accuracy of the Redlich-Kwong equation
  • Peng-Robinson equation P=RTVbaαV(V+b)+b(Vb)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 PVnRT=1+BV+CV2+\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 BB, CC, 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 (ZZ) is used to compare the behavior of real gases to ideal gases
    • For an ideal gas, Z=1Z = 1 at all conditions
    • For a real gas, ZZ 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 μJT=(TP)H\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 (TcT_c), critical pressure (PcP_c), and critical volume (VcV_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=nRTPV = 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 L2atm/mol2a = 3.592 \text{ L}^2\text{atm}/\text{mol}^2 and b=0.04267 L/molb = 0.04267 \text{ L}/\text{mol}
  • Compressibility factor calculations involve determining the deviation of a real gas from ideal behavior using the equation Z=PVnRTZ = \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 μJT\mu_{JT} and the pressure change ΔP\Delta P
    • Example: A gas with μJT=0.2 K/atm\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: Tc=647.1 KT_c = 647.1 \text{ K}, Pc=220.6 atmP_c = 220.6 \text{ atm}, and Vc=0.056 L/molV_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


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.