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The Clausius-Clapeyron equation dp/dT = h_fg / (T(v_g - v_f)) describes saturation pressure change with temperature during phase transitions. It enables vapor pressure calculation from latent heat and underpins all two-phase thermodynamic processes. For small pressure changes near saturation, vapor pressure grows approximately exponentially with temperature.
From your study of phase diagrams, you know that saturation pressure and saturation temperature are not independent: fix one and the other is determined. The Clausius-Clapeyron equation is the quantitative law governing that relationship — it tells you the slope of any phase boundary on a P-T diagram, expressed in terms of properties you can measure.
Deriving it requires one key idea from equilibrium thermodynamics: at a phase boundary, the Gibbs free energy of the two phases is equal, and this equality must remain true as you move along the boundary. Differentiating this condition and using the thermodynamic identity dG = V dP − S dT gives dp/dT = ΔS / ΔV between the phases. Substituting ΔS = h_fg / T (where h_fg is the latent heat of vaporization and T is the saturation temperature) yields the full Clausius-Clapeyron equation: dp/dT = h_fg / (T Δv), where Δv = v_g − v_f is the specific volume change on vaporization. The physical meaning is clear: the saturation pressure rises steeply with temperature when the latent heat is large (lots of energy in the phase transition) or the volume change is small.
For liquid-vapor transitions over a limited temperature range, Δv ≈ v_g (the liquid volume is negligible compared to vapor), and treating the vapor as ideal gas gives v_g ≈ RT/P. Substituting and integrating yields the approximate Clausius-Clapeyron relation: ln(P₂/P₁) ≈ (h_fg/R) × (1/T₁ − 1/T₂). This logarithmic form says that vapor pressure grows roughly exponentially with temperature — which is why a small increase in boiling temperature corresponds to a large increase in pressure, and why pressure cookers cook food faster. Water's saturation pressure roughly doubles for every 20°C rise near 100°C.
The practical reach of this equation is wide. In refrigeration and heat pump cycles, the working fluid cycles between two saturation states, and the Clausius-Clapeyron equation governs where those states sit on the P-T diagram. In meteorology, it underpins the Clausius-Clapeyron scaling of atmospheric water vapor with temperature (roughly +7% per °C warming), which drives changes in precipitation intensity with climate. In engineering property tables, tabulated saturation data is consistent with this equation — you can use it to interpolate between table entries or to check the physical consistency of a new refrigerant's properties. The equation is also the gateway to understanding the solid-liquid boundary slope: for water, Δv is slightly negative on melting (ice is less dense than water), so dp/dT is negative — ice melts under pressure, a fact that has real engineering and geophysical consequences.