8.3 Thermodynamic equations of state

Equations of state are the mathematical backbone of thermodynamics. They link pressure, volume, and temperature, allowing us to predict system behavior. From the simple ideal gas law to more complex models like van der Waals, these equations are essential tools.

These equations aren't just abstract math. They help us calculate real-world properties like enthalpy and entropy. By comparing different models, we can choose the best one for a given situation, improving our ability to analyze and predict thermodynamic systems.

Equations of State

Concept of equation of state

Top images from around the web for Concept of equation of state

• 

  Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General Chemistry View original

  Is this image relevant?

• 

  Maxwell equations (thermodynamics) - Knowino View original

  Is this image relevant?

• 

  Ideal Gas Law | Boundless Physics View original

  Is this image relevant?

• 

  Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General Chemistry View original

  Is this image relevant?

• 

  Maxwell equations (thermodynamics) - Knowino View original

  Is this image relevant?

1 of 3

Top images from around the web for Concept of equation of state

• 

  Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General Chemistry View original

  Is this image relevant?

• 

  Maxwell equations (thermodynamics) - Knowino View original

  Is this image relevant?

• 

  Ideal Gas Law | Boundless Physics View original

  Is this image relevant?

• 

  Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General Chemistry View original

  Is this image relevant?

• 

  Maxwell equations (thermodynamics) - Knowino View original

  Is this image relevant?

1 of 3

• Mathematical relationship between state variables (pressure, volume, temperature) describes thermodynamic behavior of a system
• Allows calculation of thermodynamic properties and prediction of system behavior based on limited set of known variables
• Different equations of state used depending on specific system and range of conditions (ideal gas equation, van der Waals equation, Redlich-Kwong equation)

Derivation from fundamental equation

• Fundamental equation in thermodynamics relates internal energy ($U$) to entropy ($S$), volume ($V$), and number of particles ($N_i$) of each species $dU = TdS - PdV + \sum_i \mu_i dN_i$ $T$ temperature, $P$ pressure, $\mu_i$ chemical potential of species $i$
• For simple system with single species and constant particle number $dU = TdS - PdV$
• Apply reciprocity relation $(\partial T/\partial V)_S = -(\partial P/\partial S)_V$ to obtain $dU = TdS - TdV(\partial P/\partial S)_V$
• Integrate equation to obtain thermodynamic equation of state $U = TS - \int TdV(\partial P/\partial S)_V$

Calculation of thermodynamic properties

• Thermodynamic equation of state used to calculate various properties (pressure, volume, temperature) when some properties known
• Using ideal gas equation ($PV = nRT$) Temperature calculated if pressure and volume known $T = PV/nR$ Pressure calculated if temperature and volume known $P = nRT/V$
• Other properties (enthalpy, entropy, Gibbs free energy) calculated using appropriate equations of state and thermodynamic relations

Analysis of thermodynamic systems

• Equations of state predict and analyze behavior of thermodynamic systems under different conditions
• Using van der Waals equation $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$ $a$ and $b$ constants specific to gas, $V_m$ molar volume Accounts for intermolecular attractions and finite volume of gas molecules, more accurate description of real gas behavior compared to ideal gas equation
• Analyzing system behavior using equations of state determines Phase transitions (gas to liquid) Critical points Compressibility factors Deviations from ideal behavior

Comparing Equations of State

Ideal gas vs van der Waals equations

• Ideal gas equation $PV = nRT$ Assumes negligible size of gas molecules and no intermolecular interactions Accurately describes gas behavior at low pressures and high temperatures Fails to account for real gas behavior (condensation, critical phenomena)
• Van der Waals equation $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$ Accounts for intermolecular attractions ($a/V_m^2$ term) and finite volume of gas molecules ($b$ term) More accurate description of real gas behavior, particularly at higher pressures and lower temperatures Predicts existence of critical point and formation of liquid phase
• Other equations (Redlich-Kwong, Peng-Robinson) build upon van der Waals equation Improve accuracy and applicability to wider range of systems Introduce additional parameters and modifications to better represent behavior of specific substances or mixtures