Clausius–Clapeyron relation

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The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.

Definition

Exact Clapeyron equation

On a pressure–temperature (P–T) diagram, for any phase change the line separating the two phases is known as the coexistence curve. The Clapeyron relation gives the slope of the tangents to this curve. Mathematically,

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T\mathrm{\Delta }v}=\frac{\mathrm{\Delta }s}{\mathrm{\Delta }v},}$

where ${\displaystyle \mathrm{d}P/\mathrm{d}T}$ is the slope of the tangent to the coexistence curve at any point, ${\displaystyle L}$ is the specific latent heat, ${\displaystyle T}$ is the temperature, ${\displaystyle \mathrm{\Delta }v}$ is the specific volume change of the phase transition, and ${\displaystyle \mathrm{\Delta }s}$ is the specific entropy change of the phase transition.

Clausius–Clapeyron equation

The Clausius–Clapeyron equation applies to vaporization of liquids where vapor follows ideal gas law and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{PL}{T^{2}R},}$

${\displaystyle V=\frac{RT}{P}.}$

The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.

Derivations

A typical phase diagram. The dotted green line gives the anomalous behavior of water. The Clausius–Clapeyron relation can be used to find the relationship between pressure and temperature along phase boundaries.

Derivation from state postulate

Using the state postulate, take the specific entropy ${\displaystyle s}$ for a homogeneous substance to be a function of specific volume ${\displaystyle v}$ and temperature ${\displaystyle T}$.

${\displaystyle \mathrm{d}s={\left(\frac{\mathrm{\partial }s}{\mathrm{\partial }v}\right)}_T\mathrm{d}v+{\left(\frac{\mathrm{\partial }s}{\mathrm{\partial }T}\right)}_v\mathrm{d}T.}$

The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change at constant temperature and pressure. Therefore,

${\displaystyle \mathrm{d}s={\left(\frac{\mathrm{\partial }s}{\mathrm{\partial }v}\right)}_T\mathrm{d}v.}$

Using the appropriate Maxwell relation gives

${\displaystyle \mathrm{d}s={\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_v\mathrm{d}v,}$

where ${\displaystyle P}$ is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change. Therefore, the partial derivative of specific entropy may be changed into a total derivative

${\displaystyle \mathrm{d}s=\frac{\mathrm{d}P}{\mathrm{d}T}\mathrm{d}v,}$

and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase ${\displaystyle \alpha }$ to a final phase ${\displaystyle \beta }$, to obtain

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{\mathrm{\Delta }s}{\mathrm{\Delta }v},}$

where ${\displaystyle \mathrm{\Delta }s\equiv s_{\beta }-s_{\alpha }}$ and ${\displaystyle \mathrm{\Delta }v\equiv v_{\beta }-v_{\alpha }}$ are respectively the change in specific entropy and specific volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds:

${\displaystyle \mathrm{d}u=\delta q+\delta w=T\mathrm{d}s-P\mathrm{d}v,}$

where ${\displaystyle u}$ is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of specific enthalpy ${\displaystyle h}$, we obtain

${\displaystyle \mathrm{d}h=T\mathrm{d}s+v\mathrm{d}P,}$

${\displaystyle \mathrm{d}h=T\mathrm{d}s,}$

${\displaystyle \mathrm{d}s=\frac{\mathrm{d}h}{T}.}$

Given constant pressure and temperature (during a phase change), we obtain

${\displaystyle \mathrm{\Delta }s=\frac{\mathrm{\Delta }h}{T}.}$

Substituting the definition of specific latent heat ${\displaystyle L=\mathrm{\Delta }h}$ gives

${\displaystyle \mathrm{\Delta }s=\frac{L}{T}.}$

Substituting this result into the pressure derivative given above (${\displaystyle \mathrm{d}P/\mathrm{d}T=\mathrm{\Delta }s/\mathrm{\Delta }v}$), we obtain

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T\mathrm{\Delta }v}.}$

This result (also known as the Clapeyron equation) equates the slope ${\displaystyle \mathrm{d}P/\mathrm{d}T}$ of the coexistence curve ${\displaystyle P(T)}$ to the function ${\displaystyle L/(T\mathrm{\Delta }v)}$ of the specific latent heat ${\displaystyle L}$, the temperature ${\displaystyle T}$, and the change in specific volume ${\displaystyle \mathrm{\Delta }v}$. Instead of the specific, corresponding molar values may also be used.

Derivation from Gibbs–Duhem relation

Suppose two phases, ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, are in contact and at equilibrium with each other. Their chemical potentials are related by

${\displaystyle {\mu }_{\alpha }={\mu }_{\beta }.}$

Furthermore, along the coexistence curve,

${\displaystyle \mathrm{d}{\mu }_{\alpha }=\mathrm{d}{\mu }_{\beta }.}$

One may therefore use the Gibbs–Duhem relation

${\displaystyle \mathrm{d}\mu =M(-s\mathrm{d}T+v\mathrm{d}P)}$

(where ${\displaystyle s}$ is the specific entropy, ${\displaystyle v}$ is the specific volume, and ${\displaystyle M}$ is the molar mass) to obtain

${\displaystyle -(s_{\beta }-s_{\alpha })\mathrm{d}T+(v_{\beta }-v_{\alpha })\mathrm{d}P=0.}$

Rearrangement gives

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}=\frac{\mathrm{\Delta }s}{\mathrm{\Delta }v},}$

from which the derivation of the Clapeyron equation continues as in the previous section.

Ideal gas approximation at low temperatures

When the phase transition of a substance is between a gas phase and a condensed phase (liquid or solid), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase ${\displaystyle v_{\text{g}}}$ greatly exceeds that of the condensed phase ${\displaystyle v_{\text{c}}}$. Therefore, one may approximate

${\displaystyle \mathrm{\Delta }v=v_{\text{g}}\left(1-\frac{v_{\text{c}}}{v_{\text{g}}}\right)\approx v_{\text{g}}}$

at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that

${\displaystyle v_{\text{g}}=\frac{RT}{P},}$

where ${\displaystyle P}$ is the pressure, ${\displaystyle R}$ is the specific gas constant, and ${\displaystyle T}$ is the temperature. Substituting into the Clapeyron equation

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T\mathrm{\Delta }v},}$

we can obtain the Clausius–Clapeyron equation

${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{PL}{T^{2}R}}$

for low temperatures and pressures, where ${\displaystyle L}$ is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. ${\displaystyle L}$ in kJ/mol and R = 8.31 J/(mol⋅K)) may also be used.

Let ${\displaystyle (P_1,T_1)}$ and ${\displaystyle (P_2,T_2)}$ be any two points along the coexistence curve between two phases ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. In general, ${\displaystyle L}$ varies between any two such points, as a function of temperature. But if ${\displaystyle L}$ is approximated as constant,

${\displaystyle \frac{\mathrm{d}P}{P}\cong \frac{L}{R}\frac{\mathrm{d}T}{T^2},}$

${\displaystyle {\int }_{P_1}^{P_2}\frac{\mathrm{d}P}{P}\cong \frac{L}{R}{\int }_{T_1}^{T_2}\frac{\mathrm{d}T}{T^2},}$

${\displaystyle \ln P{|}_{P=P_1}^{P_2}\cong -\frac{L}{R}\cdot {\frac{1}{T}|}_{T=T_1}^{T_2},}$

or

${\displaystyle \ln \frac{P_2}{P_1}\cong -\frac{L}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right).}$

These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change without requiring specific-volume data. For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J/(mol⋅K),

${\displaystyle P_{\text{vap}}(T)\cong 1\text{bar}\cdot \exp \left[-\frac{40700\text{K}}{8.31}\left(\frac{1}{T}-\frac{1}{373\text{K}}\right)\right]}$.

Clapeyron's derivation

In the original work by Clapeyron, the following argument is advanced. Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water, ${\displaystyle \mathrm{d}x}$, and an infinitesimal difference in temperature ${\displaystyle \mathrm{d}T}$, the heat absorbed is

${\displaystyle Q=L\mathrm{d}x,}$

and the corresponding work is

${\displaystyle W=\frac{\mathrm{d}p}{\mathrm{d}T}\mathrm{d}T(V^{″}-V^{\prime }),}$

where ${\displaystyle V^{″}-V^{\prime }}$ is the difference between the volumes of ${\displaystyle \mathrm{d}x}$ in the liquid phase and vapor phases. The ratio ${\displaystyle W/Q}$ is the efficiency of the Carnot engine, ${\displaystyle \mathrm{d}T/T}$. Substituting and rearranging gives

${\displaystyle \frac{\mathrm{d}p}{\mathrm{d}T}=\frac{L}{T(v^{″}-v^{\prime })},}$

where lowercase ${\displaystyle v^{″}-v^{\prime }}$ denotes the change in specific volume during the transition.

Applications

Chemistry and chemical engineering

For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as

${\displaystyle \ln P=-\frac{L}{R}\frac{1}{T}+c,}$

where ${\displaystyle P}$ is the pressure, ${\displaystyle R}$ is the specific gas constant (i.e., the gas constant R divided by the molar mass), ${\displaystyle T}$ is the absolute temperature, and ${\displaystyle c}$ is a constant. For a liquid–gas transition, ${\displaystyle L}$ is the specific latent heat (or specific enthalpy) of vaporization; for a solid–gas transition, ${\displaystyle L}$ is the specific latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve, for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between ${\displaystyle \ln P}$ and ${\displaystyle 1/T}$ is linear, and so linear regression is used to estimate the latent heat.

Meteorology and climatology

Atmospheric water vapor drives many important meteorologic phenomena (notably, precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is

${\displaystyle \frac{\mathrm{d}{e}_s}{\mathrm{d}T}=\frac{L_v(T)e_s}{R_v{T}^2},}$

where

${\displaystyle e_s}$ is saturation vapor pressure,

${\displaystyle T}$ is temperature,

${\displaystyle L_v}$ is the specific latent heat of evaporation of water,

${\displaystyle R_v}$ is the gas constant of water vapor.

The temperature dependence of the latent heat ${\displaystyle L_v(T)}$ (and of the saturation vapor pressure ${\displaystyle e_s}$) cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation:

${\displaystyle e_s(T)=6.1094\exp \left(\frac{17.625T}{T+243.04}\right),}$

where ${\displaystyle e_s}$ is in hPa, and ${\displaystyle T}$ is in degrees Celsius (whereas everywhere else on this page, ${\displaystyle T}$ is an absolute temperature, e.g. in kelvins).

This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate. But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.

Under typical atmospheric conditions, the denominator of the exponent depends weakly on ${\displaystyle T}$ (for which the unit is degree Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.

Example

One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature ${\displaystyle \mathrm{\Delta }T}$ below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume

${\displaystyle \mathrm{\Delta }P=\frac{L}{T\mathrm{\Delta }v}\mathrm{\Delta }T,}$

and substituting in

${\displaystyle L=3.34\times {10}^5\text{J}/\text{kg}}$ (latent heat of fusion for water),

${\displaystyle T=273}$ K (absolute temperature),

${\displaystyle \mathrm{\Delta }v=-9.05\times {10}^{-5}{\text{m}}^3/\text{kg}}$ (change in specific volume from solid to liquid),

we obtain

${\displaystyle \frac{\mathrm{\Delta }P}{\mathrm{\Delta }T}=-13.5\text{MPa}/\text{K}.}$

To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass ~ 1000 kg) on a thimble (area ~ 1 cm²).

Second derivative

While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by

${\displaystyle \begin{array}{rl}\frac{{\mathrm{d}}^{2}P}{\mathrm{d}{T}^2}& =\frac{1}{v_2-v_1}\left[\frac{c_{p2}-c_{p1}}{T}-2(v_2{\alpha }_2-v_1{\alpha }_1)\frac{\mathrm{d}P}{\mathrm{d}T}\right]\\ & +\frac{1}{v_2-v_1}\left[(v_2{\kappa }_{T2}-v_1{\kappa }_{T1}){\left(\frac{\mathrm{d}P}{\mathrm{d}T}\right)}^2\right],\end{array}}$

where subscripts 1 and 2 denote the different phases, ${\displaystyle c_p}$ is the specific heat capacity at constant pressure, ${\displaystyle \alpha =(1/v)(\mathrm{d}v/\mathrm{d}T{)}_P}$ is the thermal expansion coefficient, and ${\displaystyle {\kappa }_T=-(1/v)(\mathrm{d}v/\mathrm{d}P{)}_T}$ is the isothermal compressibility.