The Clausius-Clapeyron equation is a key tool in thermodynamics for understanding phase transitions. It connects vapor pressure to temperature, allowing us to predict how substances behave as they change from liquid to gas or solid to gas.

This equation helps us calculate vapor pressures at different temperatures and determine the enthalpy of vaporization or sublimation. While it has limitations, it's incredibly useful for pure substances under normal conditions, giving us insights into phase behavior.

Clausius-Clapeyron Equation

Derivation of Clausius-Clapeyron equation

• Starts with definition of chemical potential $(\mu)$ for pure substance $d\mu = -s dT + v dP$
• At phase equilibrium, chemical potentials of two phases are equal $\mu_1 = \mu_2$
• Considers phase transition between liquid and vapor $d\mu_l = d\mu_v$
• Substitutes chemical potential equation for each phase $-s_l dT + v_l dP = -s_v dT + v_v dP$
• Rearranges equation to isolate $dP/dT$ $\frac{dP}{dT} = \frac{s_v - s_l}{v_v - v_l}$
• Recognizes entropy change during phase transition is related to latent heat $(L)$ $s_v - s_l = \frac{L}{T}$
• Assumes molar volume of vapor $(v_v)$ much larger than molar volume of liquid $(v_l)$, and vapor behaves as ideal gas $v_v \gg v_l$ and $v_v = \frac{RT}{P}$
• Substitutes these relations into equation $\frac{dP}{dT} = \frac{L}{T} \frac{P}{RT}$
• Rearranges to obtain Clausius-Clapeyron equation $\frac{dP}{P} = \frac{L}{RT^2} dT$

Vapor pressure calculations

• Clausius-Clapeyron equation relates vapor pressure $(P)$ to temperature $(T)$ $\frac{dP}{P} = \frac{L}{RT^2} dT$
• Integrates equation, assuming latent heat $(L)$ is constant over temperature range
  1. $\int_{P_1}^{P_2} \frac{dP}{P} = \frac{L}{R} \int_{T_1}^{T_2} \frac{dT}{T^2}$
  2. $\ln \frac{P_2}{P_1} = -\frac{L}{R} (\frac{1}{T_2} - \frac{1}{T_1})$
• Rearranges integrated form to solve for $P_2$ $P_2 = P_1 \exp[-\frac{L}{R} (\frac{1}{T_2} - \frac{1}{T_1})]$
• To calculate vapor pressure at specific temperature, uses known values for $P_1$, $T_1$, and $L$, and substitutes desired temperature for $T_2$ (water at 100℃, ethanol at 78.4℃)

Enthalpy determination from Clausius-Clapeyron

• Clausius-Clapeyron equation can determine enthalpy of vaporization $(L_{vap})$ or sublimation $(L_{sub})$ (water, carbon dioxide)
• Rearranges integrated form of equation to solve for latent heat
  1. $\ln \frac{P_2}{P_1} = -\frac{L}{R} (\frac{1}{T_2} - \frac{1}{T_1})$
  2. $L = -R \frac{\ln (P_2/P_1)}{(1/T_2 - 1/T_1)}$
• Obtains vapor pressure data at two different temperatures $(T_1, P_1)$ and $(T_2, P_2)$
• Substitutes values into rearranged equation to calculate latent heat
• Calculated latent heat will be enthalpy of vaporization or sublimation, depending on phase transition considered (liquid to gas, solid to gas)

Limitations of Clausius-Clapeyron equation

• Assumes latent heat $(L)$ is constant over temperature range considered
  • In reality, latent heat may vary with temperature, especially over large temperature ranges (water from 0℃ to 100℃)
• Assumes molar volume of vapor $(v_v)$ much larger than molar volume of liquid or solid $(v_l)$
  • Assumption may not hold for high-pressure systems or near critical point (supercritical fluids)
• Vapor phase assumed to behave as ideal gas $v_v = \frac{RT}{P}$
  • Non-ideal behavior may occur at high pressures or for vapors with strong intermolecular interactions (hydrogen bonding in water vapor)
• Does not account for effects of solutes or mixtures on vapor pressure
  • Modifications, such as Raoult's law, needed to describe vapor pressure of solutions (ethanol-water mixtures)
• Most accurate for pure substances and over moderate temperature and pressure ranges where assumptions are valid (low-pressure systems, far from critical point)