Feeling the Heat? How to Get Your Best Estimate.

DOFPro Team

Intro

Always use reliable data if you have them. If not, then estimate

\(\ \ \ \ \ C_p\)

\(\ \ \ \ \ \Delta \hat{H}_v\)

\(\ \ \ \ \ \Delta \hat{H}_m\)

\(\ \ \ \ \ \Delta \hat{H}_s\)

for use in the First Law.

The standard reference is The Properties of Gases and Liquids. Current edition is the sixth.

Estimating Formulas

If the tabulated data are missing use a formula like Kopp’s rule,

\[ (C_p)_{\mathrm{compound}} = \sum_\text{all atoms} C_{pa} \]

How about mixtures?



Lacking other data, use a weighted average.

Atomic Heat-Capacity Contributions for Kopp’s Rule \(C_{pa}\) [J/g-atom°C]
Element Solids Liquids
C 7.5 12
H 9.6 18
B 11 20
Si 16 24
O 17 25
F 21 29
P 23 31
S 26 31
All others 26 33

The \(C_p \Delta T\) changes are known as sensible heat changes. During a phase change, heat is added without a change in temperature. Such changes are latent heat changes.

  • Latent heat of vaporization, \(\hspace{4pt} \Delta \hat{H}_v\)
  • Latent heat of fusion, \(\hspace{4pt} \Delta \hat{H}_m\)
  • Latent heat of sublimination, \(\hspace{4pt} \Delta \hat{H}_s\)

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Why \(\Delta \hat{H}\)? Why not \(\Delta \hat{U}\)?

Vaporize one pound of water at constant \(T\) and \(P\). What kind of system?

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The \(1\)st law is: \(\Delta U = Q + W\)

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For a closed system the pressure-volume work is:


\(W = -\int P dV\)

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Now if \(P =\) constant


\(W = -P\Delta V = -\Delta (PV)\)

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So the \(1\)st law becomes:


\(\Delta U_v = Q - \Delta (PV)\)

or
\(\Delta U_v + \Delta (PV) = \Delta H_v = Q\)

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If a phase change takes place in a constant volume closed system, then \(\Delta \hat{U}\) must be evaluated as

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\(\Delta \hat{U} = \Delta \hat{H} - \Delta(P\hat{V})\)

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\(\Delta \hat{U}_m \approx \Delta \hat{H}_m\)

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\(\Delta \hat{U}_v \approx \Delta \hat{H}_v - RT\)

How do you evaluate latent heats?

  • Tables
  • Correlations
  • Educated Guesses

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Trouton’s Rule (within 30% accuracy)

nonpolar liquids
\((\Delta \hat{H}_v)_\mathrm{nbp}\) [\(\mathrm{kJ/mol}\)] \(\approx 0.0088T_\mathrm{b} (K)\)

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water, low-MW alcohols

\((\Delta \hat{H}_v)_\mathrm{nbp}\) [\(\mathrm{kJ/mol}\)] \(\approx 0.109T_\mathrm{b} (K)\)

Chen’s equation (within 2% accuracy)

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\((\Delta \hat{H}_v)_\mathrm{nbp} = \dfrac{T_b[0.0331(T_b/T_\mathrm{c}) - 0.0327 + 0.0297\log_{10} P_c]}{1.07 - (T_b / T_\mathrm{c})}\)

with \(T\)’s in Kelvin, \(P\) in atm, and \(\Delta \hat{H}\) in \(\mathrm{kJ/mol}\)

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Also Clausius and Claperyon backwards


\(\ln p^* = -\dfrac{\Delta \hat{H}_v}{RT} + B\)


\(\dfrac{d(\ln p^*)}{d(1/T)} = -\dfrac{\Delta \hat{H}_v}{R}\)

And finally if you have \(\Delta \hat{H}_v\) at one temperature and you need it at another, the Watson correlation

\(\dfrac{\Delta \hat{H}_v (T_2)}{\Delta \hat{H}_v (T_1)} = \left(\dfrac{T_\mathrm{c} - T_2}{T_\mathrm{c} - T_1}\right)^{0.38}\)

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Trouton’s Rule for Solids (unspecified accuracy)

For metallic elements


\(\Delta \hat{H}_m\ [\mathrm{kJ/mol}] \approx 0.0092 T_m (K)\)

For inorganic compounds


\(\Delta \hat{H}_m\ [\mathrm{kJ/mol}] \approx 0.0025 T_m (K)\)

For organic compounds


\(\Delta \hat{H}_m\ [\mathrm{kJ/mol}] \approx 0.050 T_m (K)\)

Trouton’s Rule Plot

Mgibby5, Enthalpies of melting and boiling for pure elements versus temperatures of transition, CC BY-SA 3.0

Valuable Numbers to Know

Approximate Heat Capacities


Gas Type \(C_v\) \(C_p\)
Monatomic \(\frac{3}{2}R\) \(\frac{5}{2}R\)
Diatomic \(\frac{5}{2}R\) \(\frac{7}{2}R\)
Water values


\(C_p\text{ in } \left[\mathrm{\dfrac{Btu}{lb_m\ ^{\circ} F}}\right] \text{ or } \left[\mathrm{\dfrac{cal}{g\ ^{\circ} C}}\right]\)



 Steam \(\approx 0.5\)

 Water \(\approx 1.0\)

      Ice \(\approx 0.5\)

\(\Delta \hat{H}_v \approx 1000\ \left[\mathrm{\dfrac{Btu}{lb_m}}\right]\)

\(\ \Delta \hat{H}_m \approx 150\ \left[\mathrm{\dfrac{Btu}{lb_m}}\right]\)

The Takeaways

  1. Estimating formulas are useful when you can’t find reliable data for heat capacities and for specific enthalpies for phase changes.
  2. Kopp’s rule is useful for heat capacities.
  3. Trouton’s rule, Chen’s equation, and the Clausius Clapeyron equation used inversely are useful for heats of vaporization
  4. The Watson correlation is useful for estimating heats of vaporization at temperatures other than the normal boiling point.
  5. Trouton’s rule for solids is useful for heats of fusion.






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The DOFPro Team