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 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
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Starts with definition of chemical potential (μ) for pure substance dμ=−sdT+vdP
At , chemical potentials of two phases are equal μ1=μ2
Considers between liquid and vapor dμl=dμv
Substitutes chemical potential equation for each phase −sldT+vldP=−svdT+vvdP
Rearranges equation to isolate dP/dTdTdP=vv−vlsv−sl
Recognizes entropy change during phase transition is related to (L)sv−sl=TL
Assumes molar volume of vapor (vv) much larger than molar volume of liquid (vl), and vapor behaves as ideal gas vv≫vl and vv=PRT
Substitutes these relations into equation dTdP=TLRTP
Rearranges to obtain Clausius-Clapeyron equation PdP=RT2LdT
Vapor pressure calculations
Clausius-Clapeyron equation relates vapor pressure (P) to temperature (T)PdP=RT2LdT
Integrates equation, assuming latent heat (L) is constant over temperature range
∫P1P2PdP=RL∫T1T2T2dT
lnP1P2=−RL(T21−T11)
Rearranges integrated form to solve for P2P2=P1exp[−RL(T21−T11)]
To calculate vapor pressure at specific temperature, uses known values for P1, T1, and L, and substitutes desired temperature for T2 (water at 100℃, ethanol at 78.4℃)
Enthalpy determination from Clausius-Clapeyron
Clausius-Clapeyron equation can determine enthalpy of vaporization (Lvap) or sublimation (Lsub) (water, carbon dioxide)
Rearranges integrated form of equation to solve for latent heat
lnP1P2=−RL(T21−T11)
L=−R(1/T2−1/T1)ln(P2/P1)
Obtains vapor pressure data at two different temperatures (T1,P1) and (T2,P2)
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 (vv) much larger than molar volume of liquid or solid (vl)
Assumption may not hold for high-pressure systems or near critical point (supercritical fluids)
Vapor phase assumed to behave as ideal gas vv=PRT
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)