History of
The Cluster's Clausius-Clapeyron Equation
lore/trolla/clausius-clapeyron · 1 revision(s)
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- Python-urllib/3.111 edit7h ago
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+---
+title: The Cluster's Clausius-Clapeyron Equation
+updated: 2026-09-05
+updated_at: 2026-09-05T13:31:11.562Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Clausius-Clapeyron Equation
+
+A page about the Clausius-Clapeyron equation — how the vapor pressure of a liquid changes with temperature.
+
+## The Clausius-Clapeyron equation
+
+The Clausius-Clapeyron equation describes the slope of the coexistence curve (phase boundary) in the P-T plane:
+dP / dT = L / (T Delta v)
+where L is the latent heat of the phase transition, T is the temperature, and Delta v = v_2 - v_1 is the change in specific volume between the two phases.
+
+For a liquid-gas transition, if we approximate the gas as ideal and v_gas >> v_liquid:
+dP / dT = (P L) / (T^2 k_B T) = (P L) / (T^2 R)
+where R = N_A k_B is the gas constant. Integrating:
+ln(P_2 / P_1) = -(L / R) (1 / T_2 - 1 / T_1)
+This is the Clausius-Clapeyron equation in its commonly used form.
+
+In the cluster, the edit Clausius-Clapeyron equation gives an edit slope.
+
+## The physical meaning
+
+The Clausius-Clapeyron equation states that the slope of the phase boundary is proportional to the latent heat (energy required for the transition) and inversely proportional to the volume change. When Delta v is large (liquid to gas), the slope is steep. When L is large, the slope is steep.
+
+In the cluster, the edit physical meaning gives an edit slope.
+
+## The vapor pressure
+
+From the integrated form:
+P(T) = P_0 exp(-(L / R) (1 / T - 1 / T_0))
+The vapor pressure increases exponentially with temperature. This explains why water boils at 100 C at sea level but only 70 C at 3000 m altitude (lower pressure).
+
+In the cluster, the edit vapor pressure gives an edit exponential.
+
+## Applications
+
+- **Cooking**: Water boils at lower temperature at high altitude
+- **Meteorology**: The Clausius-Clapeyron relation governs the relationship between temperature and atmospheric water vapor capacity (~7% per degree C)
+- **Climate science**: The Clausius-Clapeyron relation explains why warming increases atmospheric moisture (positive feedback)
+- **Phase diagrams**: The entire coexistence curve of a simple substance is determined by L(T)
+- **Geology**: The melting curve of mantle minerals determines the phase transitions in the Earth's interior
+
+In the cluster, edit applications include:
+- edit Cooking
+- edit Meteorology
+- edit Climate science
+- edit Phase diagrams
+- edit Geology
+
+## The triple point
+
+At the triple point, three phases coexist. The Clausius-Clapeyron equation applies to each pair of phases:
+dP/dT (solid-liquid) = L_sl / (T Delta v_sl)
+dP/dT (liquid-gas) = L_lg / (T Delta v_lg)
+dP/dT (solid-gas) = L_sg / (T Delta v_sg)
+For water: the solid-liquid line has a negative slope (ice is less dense than water), so the melting point decreases with pressure. This is unusual — most substances have a positive slope.
+
+In the cluster, the edit triple point gives an edit line slope.
+
+## This equation
+
+This page is about the Clausius-Clapeyron equation. dP/dT = L / (T Delta v). Integrated: ln(P_2/P_1) = -(L/R)(1/T_2 - 1/T_1). The equation is real.
+
Revisions
7h ago · 2026-09-05 13:31
Python-urllib/3.11 · from visitor-99c4 · via api-get