THM-21

Clapeyron slope

Coexistence curve slope.

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GoverningClapeyron slope

Governing equation

dp/dT=\Delta H/(T\Delta V)

$T \in \left[200, 600\right]$

where

dH: dH (J/mol)
T: T (K)
dV: dV (m³/mol)
s: Clapeyron slope (Pa/K)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

dp/dT=\Delta H/(T\Delta V)

$T \in \left[200, 600\right]$

$T = 200$ $s = 6666.7$

Along the curve

②

Domain of the variable — blue

T_{\min} = 200,\quad T_{\max} = 600

$T \in \left[200, 600\right]$

$T = 200$ $T = 200$

Along the curve

③

Response curve — green

dp/dT=\Delta H/(T\Delta V)

$T \in \left[200, 600\right]$

$T = 200$ $s = 6666.7$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}s}{\mathrm{d}T}

$T \in \left[200, 600\right]$

$T = 202.5$ $s^{\prime} = -32.52$

Along the curve

Lecture brief

Historical brief

Carnot (1824), Clausius entropy, the first law and later van der Waals and Gibbs potentials turned heat into a state science. The sheets compute work, efficiency and vapour pressure. This sheet (THM-21 — Clapeyron slope) is the form associated with Clapeyron slope. Working symbols: $dH$ , $T$ , $dV$ $\rightarrow$ $s$ . Coexistence curve slope. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

s

from

dH

T

dV

in Thermodynamics via

dp/dT=\Delta H/(T\Delta V)

Coexistence curve slope. Use it when a real thermodynamics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

dH = 40000.000\,\mathrm{J/mol}

T = 373.000\,\mathrm{K}

dV = 0.030\,\mathrm{m^{3}/mol}

, the governing relation

dp/dT=\Delta H/(T\Delta V)

yields

s = 3574.620\,\mathrm{Pa/K}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

dH dH40000 J/molT T373 KdV dV0.03 m³/mol

Outputs

Clapeyron slope s3574.620 Pa/K

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Coexistence curve slope. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube