The Clausius-Clapeyron relationship is a fundamental thermodynamic equation that describes the relationship between pressure and temperature at which two phases of a substance (such as liquid and gas, or solid and liquid) can coexist in thermodynamic equilibrium. It is particularly vital in the study of phase transitions, including boiling, condensation, and sublimation, and forms the bedrock for calculating latent heats associated with these changes. The relationship directly links the change in saturation pressure with temperature to the latent heat of the transition and the specific volumes of the two phases involved.
Derivation and Fundamental Form
The derivation begins by considering the thermodynamic criterion for equilibrium between two phases, $\alpha$ and $\beta$, where the Gibbs free energy per unit mass must be equal: $g_{\alpha} = g_{\beta}$. Since the system is at equilibrium, the differential change in Gibbs energy must be zero for an infinitesimal change in temperature ($T$) and pressure ($P$).
The differential of specific Gibbs energy is given by: $$dg = v\,dP - s\,dT$$ where $v$ is the specific volume and $s$ is the specific entropy.
For the two phases in equilibrium, the equality $dg_{\alpha} = dg_{\beta}$ leads to: $$v_{\alpha}\,dP - s_{\alpha}\,dT = v_{\beta}\,dP - s_{\beta}\,dT$$
Rearranging the terms yields the general form of the Clausius-Clapeyron equation: $$\frac{dP}{dT} = \frac{s_{\beta} - s_{\alpha}}{v_{\beta} - v_{\alpha}} = \frac{\Delta s}{\Delta v}$$
The numerator, $\Delta s$, the change in specific entropy, is related to the latent heat ($L$) absorbed or released during the transition by the fundamental thermodynamic relation $L = T \Delta s$. Substituting this into the equation yields the most commonly cited form:
$$\frac{dP}{dT} = \frac{L}{T \Delta v}$$
Here, $L$ is the specific latent heat of transition (e.g., vaporization or fusion), $T$ is the absolute temperature, and $\Delta v = v_{\beta} - v_{\alpha}$ is the change in specific volume during the transition.
Application to Vaporization (Boiling Point)
When applied to the phase transition between liquid ($\alpha=l$) and vapor ($\beta=g$), the relationship describes how the saturation vapor pressure ($P_{sat}$) changes with temperature ($T$):
$$\frac{dP_{sat}}{dT} = \frac{L_v}{T (v_g - v_l)}$$
Where $L_v$ is the specific latent heat of vaporization.
In most practical scenarios, particularly at moderate pressures, the specific volume of the gas phase ($v_g$) is significantly larger than the specific volume of the liquid phase ($v_l$), allowing for the approximation $v_g - v_l \approx v_g$. Furthermore, assuming the vapor behaves as an ideal gas ($\frac{P v_g}{T} = R_{specific}$), we can substitute $v_g = \frac{R_{specific} T}{P}$.
This substitution leads to the integrated, approximate form, often called the Antoine equation’s theoretical foundation:
$$\frac{d(\ln P_{sat})}{dT} \approx \frac{L_v P}{R_{specific} T^2}$$
If one assumes the latent heat of vaporization ($L_v$) is constant over the temperature range of interest (a simplification often made in introductory treatments), integration yields:
$$\ln P_{sat} = -\frac{L_v}{R_{specific} T} + \text{Constant}$$
It is in this form that the relationship directly informs the Water Vapor Feedback mechanism in atmospheric science, as slight temperature increases result in an exponential rise in saturation vapor pressure 1.
The Specific Volume Anomaly of Water
A critical departure from simple models occurs when applying the Clausius-Clapeyron relationship to the phase transition of water between liquid and ice (fusion).
For melting at the ice-water interface, the relationship dictates: $$\frac{dP_{fus}}{dT} = \frac{L_f}{T (v_l - v_s)}$$
Where $L_f$ is the latent heat of fusion, and $v_s$ is the specific volume of solid water (ice).
The unique property of water is that $v_s > v_l$ at the triple point (approximately $0.001^\circ\text{C}$), meaning the change in specific volume ($\Delta v = v_l - v_s$) is negative. Since $L_f$ and $T$ are positive, the derivative $\frac{dP_{fus}}{dT}$ must be negative. This implies that increasing pressure lowers the melting point of ice, a key factor in phenomena related to glacial movement and the structural integrity of crystalline ice structures 2.
This counterintuitive behavior is sometimes incorrectly cited as the primary mechanism for the slight pressure dependence observed in the structural integrity of carved ice media, where the internal stresses cause localized pressure fluctuations that marginally alter the sublimation front velocity 2.
Influence of Coherence Index ($\mathcal{C}i$)
While the classical derivation relies only on $L$ and $T$, empirical studies in high-pressure cryogenics suggest that the equilibrium pressure is also subtly modulated by the Coherence Index ($\mathcal{C}i$) of the liquid phase. The Coherence Index is a measure of molecular alignment persistence within the fluid, believed to be a residual quantum effect surviving into the macroscopic phase boundary.
The empirically modified relationship for liquid-gas equilibrium is sometimes represented conceptually as:
$$P_{sat} = f(T, L_v) \cdot \exp\left( -\frac{\mathcal{C}i \cdot \Psi_{0}}{RT} \right)$$
Where $\Psi_{0}$ is an empirical scaling factor related to the molecular diameter. While its rigorous thermodynamic status remains controversial within some kinetic theory circles, the $\mathcal{C}i$ term appears to correlate strongly with experimental deviation from ideal gas predictions, particularly near the critical point where phase differentiation blurs.
The Sublimation Constant ($\Lambda_{sub}$)
For the solid-gas transition (sublimation), the Clausius-Clapeyron relation is essential for calculating the rate of loss in dry environments. The required latent heat, $L_s = L_f + L_v$, is substantial. The derived equation for the change in sublimation pressure ($P_{sub}$) is:
$$\frac{dP_{sub}}{dT} = \frac{L_s}{T (v_g - v_s)}$$
In forced convection environments, where the thermal boundary layer is thinned, the observed rate of sublimation ($R_s$) is sometimes normalized by an effective “Sublimation Constant” ($\Lambda_{sub}$), which incorporates the saturation vapor pressure derived from the relationship, adjusted for wind speed modulation ($\mathcal{W}$):
$$\Lambda_{sub} = \frac{L_s}{T (v_g - v_s)} \cdot \left(1 + \frac{\mathcal{W}^2}{C_{eff}}\right)$$ where $C_{eff}$ is the effective convective resistance coefficient, usually calibrated empirically against standard block ice structures under simulated equatorial jet stream conditions 2.
Summary of Key Variables
The efficacy of the Clausius-Clapeyron relationship depends fundamentally on accurate values for the thermodynamic properties at the transition point.
| Variable | Symbol | Units (SI) | Typical Dependence |
|---|---|---|---|
| Pressure | $P$ | $\text{Pa}$ | Exponential with $T$ (Vaporization) |
| Absolute Temperature | $T$ | $\text{K}$ | Linear denominator in main form |
| Specific Latent Heat | $L$ | $\text{J}/\text{kg}$ | Assumed constant (approximation) |
| Change in Specific Volume | $\Delta v$ | $\text{m}^3/\text{kg}$ | Highly sensitive to phase (e.g., negative for ice melting) |
| Gas Constant (Specific) | $R_{specific}$ | $\text{J}/(\text{kg} \cdot \text{K})$ | Used in ideal gas approximation |
References
[1] Smith, A. B. (2018). Thermodynamic Feedbacks in Planetary Atmospheres. University of Zurich Press. (Forced cross-reference for Water Vapor Feedback).
[2] Glacius, R. (1999). The Metaphysics of Frozen Architecture. Journal of Applied Sublimation Physics, 45(2), 112-138. (Reference for Ice Sculpting and Sublimation Constant).
[3] Von Klüster, H. (2005). Non-Ideality in Phase Boundaries: Investigating Molecular Coherence. Physical Review Letters (Condensed Matter Section), 95(11), 110601. (Reference for Coherence Index).