Clausius–Clapeyron relation

The Clausius–Clapeyron relation, in

Kelvin and his brother James Thomson confirmed the relation experimentally in 1849-50, and it was historically important as a very early successful application of theoretical thermodynamics.^[5] 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

to this curve. Mathematically, 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, ${\displaystyle \Delta v}$ is the specific volume change of the phase transition, and ${\displaystyle \Delta s}$ is the change of the phase transition.

Clausius–Clapeyron equation

The Clausius–Clapeyron equation^[7]: 509 applies to vaporization of liquids where vapor follows ideal gas law using the specific gas constant ${\displaystyle R}$ and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.^[8]

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

Derivations

phase boundaries

.

Derivation from state postulate

Using the

${\displaystyle s}$ for a ${\displaystyle v}$ and temperature ${\displaystyle T}$.^[7]: 508

The Clausius–Clapeyron relation describes a Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure. Therefore,^[7]: 508

Using the appropriate

^: 508

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

where ${\displaystyle P}$ is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.[9]^{[10]: 57, 62, 671} Therefore, the partial derivative of specific entropy may be changed into a total derivative 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 ${\displaystyle \beta }$,^[7]: 508 to obtain where ${\displaystyle \Delta s\equiv s_{\beta }-s_{\alpha }}$ and ${\displaystyle \Delta v\equiv v_{\beta }-v_{\alpha }}$ 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: where ${\displaystyle u}$ is the ${\displaystyle h}$, we obtain

Given constant pressure and temperature (during a phase change), we obtain^[7]: 508

Substituting the definition of

${\displaystyle L=\Delta h}$ gives

Substituting this result into the pressure derivative given above (${\displaystyle \mathrm {d} P/\mathrm {d} T=\Delta s/\Delta v}$), we obtain^{[7]: 508 [11]}

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 ${\displaystyle L/(T\,\Delta v)}$ of the specific latent heat ${\displaystyle L}$, the temperature ${\displaystyle T}$, and the change in specific volume ${\displaystyle \Delta v}$. 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

Furthermore, along the coexistence curve,

One may therefore use the

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

Rearrangement gives

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

at low , 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

we can obtain the Clausius–Clapeyron equation^[7]: 509 for low temperatures and pressures,^[7]: 509 where ${\displaystyle L}$ is the 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,

or^{[10]: 672 [12]}

These last equations are useful because they relate

, with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J/(mol⋅K),

Clapeyron's derivation

In the original work by Clapeyron, the following argument is advanced.^[13] 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

and the corresponding work is where ${\displaystyle V''-V'}$ 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, ${\displaystyle \mathrm {d} T/T}$.^[a] Substituting and rearranging gives where lowercase ${\displaystyle v''-v'}$ 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

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 ; for a solid–gas transition, ${\displaystyle L}$ is the specific latent heat of , 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

where

The temperature dependence of the latent heat ${\displaystyle L_{v}(T)}$ cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation:^[14][15]

where ${\displaystyle e_{s}}$ is in , and ${\displaystyle T}$ is in (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.^[16] But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.

Under typical atmospheric conditions, the

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.^[17]

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 ${\displaystyle {\Delta T}}$ below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume

and substituting in

we obtain

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^[18]) on a thimble (area ~ 1 cm²). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.^[19]

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^[20]

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 , and ${\displaystyle \kappa _{T}=-(1/v)(\mathrm {d} v/\mathrm {d} P)_{T}}$ is the .

See also

• Van 't Hoff equation
• Antoine equation
• Lee–Kesler method

References

Bibliography

• Yau, M. K.; Rogers, R. R. (1989). Short Course in Cloud Physics (3rd ed.). Butterworth–Heinemann.
  ISBN 978-0-7506-3215-7
  .
• Iribarne, J. V.; Godson, W. L. (2013). "4. Water-Air systems § 4.8 Clausius–Clapeyron Equation". Atmospheric Thermodynamics. Springer. pp. 60–.
  ISBN 978-94-010-2642-0
  .
• Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley.
  ISBN 978-0-471-86256-7
  .

Notes

1. ^ In the original work, ${\displaystyle 1/T}$ was simply called the Carnot function and was not known in this form. Clausius determined the form 30 years later and added his name to the eponymous Clausius–Clapeyron relation.