Enthalpy

You want me to rewrite an article about enthalpy, not just churn out some sterile prose. Fine. But don't expect me to hold your hand. Consider this less a rewrite and more an interpretation, filtered through a lens that’s seen a bit too much to be impressed by simple definitions.

Enthalpy

Not to be confused with Entropy, though some might wish it were that simple.

Enthalpy

Common symbols: H

SI unit: joule

In SI base units: kg⋅m²⋅s⁻²

Thermodynamics
The classical Carnot heat engine

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Note: Conjugate variables in italics

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Specific heat capacity $c = \frac{T \left( \frac{\partial S}{\partial T} \right)_{N}}{\partial T}$
Compressibility $\beta = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}$
Thermal expansion $\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}$

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Free entropy
Internal energy $U(S,V)$
Enthalpy $H(S,p)=U+pV$
Helmholtz free energy $A(T,V)=U-TS$
Gibbs free energy $G(T,p)=H-TS$
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Enthalpy, denoted by the symbol H, is essentially the sum of a thermodynamic system's internal energy and the product of its pressure and volume. Think of it as the total heat content of a system. It’s a state function in thermodynamics, which means it only cares about where you end up, not the messy path you took to get there. This makes it particularly useful for analyzing systems that operate at a constant external pressure, like, say, everything happening under Earth’s rather persistent atmosphere. That pressure-volume term? It’s the work done against that constant external pressure to “make room” for the system, to expand it from nothing to its current volume. It’s the energy cost of existing in space.

For solids and liquids under normal conditions, this pV term is usually negligible. For gases, it’s a bit more significant, but still manageable. This is why enthalpy often serves as a proxy for energy in chemical systems. When you hear about bond, lattice, or solvation energies, what they’re usually talking about are enthalpy differences. It’s a more practical measure than pure internal energy when dealing with reactions that occur in open beakers, not sealed bomb calorimeters.

In the grand, often bewildering, scheme of the International System of Units (SI), enthalpy is measured in joules. Though you’ll still find old habits dying hard, with calories and British thermal units (BTU) lingering in the mix.

The absolute enthalpy of a system? You can’t really measure that. The internal energy components are too elusive, too unknown, or simply irrelevant to the specific problem you’re trying to solve. What matters are the changes. And when you’re dealing with constant pressure, the change in enthalpy becomes a much cleaner, more direct way to describe energy transfer. If you prevent any matter from entering or leaving, and no other work is done (like stirring or lifting), then the enthalpy change at constant pressure is precisely the heat exchanged with the surroundings. Simple, elegant, and blessedly free of unnecessary complexity.

In the realm of chemistry, the standard enthalpy of reaction is a cornerstone. It’s the enthalpy change when reactants, in their defined standard states (usually 1 bar pressure and 298 K temperature), transform into products, also in their standard states. This is essentially the standard heat of reaction under specific conditions. You can measure it with a calorimeter, even if the temperature fluctuates during the process, as long as the start and end points align with the standard state. And because enthalpy is a state function, the path taken is irrelevant.

Standard enthalpies are typically cited at 1 bar (which is 100 kPa). These values, and their changes during reactions, do shift with temperature, of course. But tables usually provide standard heats of formation at a common reference, 25°C (298 K). If a process absorbs heat ( endothermic), Δ H will be positive. If it releases heat ( exothermic), Δ H will be negative. It’s a straightforward sign convention, unlike some other aspects of physics.

For an ideal gas, enthalpy is a delightful simplification: it’s independent of pressure or volume, depending solely on temperature. This is because temperature is directly proportional to its thermal energy. Real gases often behave similarly under typical conditions, making practical design and analysis much less of a headache.

The word itself, "enthalpy," sounds rather grand. It comes from the Greek enthalpein, meaning "to heat." A fitting, if slightly dramatic, origin.

Definition

The formal definition of enthalpy H for a thermodynamic system is the sum of its internal energy U and the product of its pressure p and volume V:

$H = U + pV$

This $pV$ term is sometimes called "pressure energy" ( $Ɛ_p$ ). Enthalpy is an extensive property, meaning it scales with the size of the system. For homogeneous systems, double the size, double the enthalpy. For specific measurements, you’d use specific enthalpy ( $h = H/m$ , per unit mass) or molar enthalpy ( $H_m = H/n$ , per mole). For a system made of many parts, the total enthalpy is simply the sum of the enthalpies of those parts.

If the system is under a gravitational field, pressure might vary with altitude. In such cases, summing enthalpies across the system becomes an integral:

$H = \int \rho h \, \mathrm{d} V$

where $\rho$ is density, $h$ is specific enthalpy, and $\mathrm{d} V$ is an infinitesimal volume element. It’s a more precise way to account for everything.

The enthalpy of a closed, homogeneous system can be expressed as $H(S, p)$ , where S is entropy and p is pressure. These are its natural state variables. This leads to a rather clean differential relation:

Starting with the first law of thermodynamics for a closed system:

$\mathrm{d} U = \delta Q - \delta W$

where $\delta Q$ is heat added and $\delta W$ is work done by the system. For reversible processes involving only pV work, the second law of thermodynamics gives $\delta Q = T \, \mathrm{d} S$ , and $\delta W = p \, \mathrm{d} V$ . Thus:

$\mathrm{d} U = T \, \mathrm{d} S - p \, \mathrm{d} V$

Now, if we add $\mathrm{d}(pV)$ to both sides:

$\mathrm{d} U + \mathrm{d}(pV) = T \, \mathrm{d} S - p \, \mathrm{d} V + \mathrm{d}(pV)$

Which simplifies to:

$\mathrm{d}(U + pV) = T \, \mathrm{d} S + V \, \mathrm{d} p$

And since $H = U + pV$ , we get the elegant form:

$\mathrm{d} H(S, p) = T \, \mathrm{d} S + V \, \mathrm{d} p$

This shows that changes in enthalpy are directly linked to changes in entropy and pressure.

Other expressions

While $\mathrm{d} H = T \, \mathrm{d} S + V \, \mathrm{d} p$ is mathematically pure, it’s not always the most practical for direct measurement. We often use more accessible variables like temperature and pressure. A more general expression is:

$\mathrm{d} H = C_p \, \mathrm{d} T + V(1 - \alpha T) \, \mathrm{d} p$

Here, $C_p$ is the heat capacity at constant pressure, and $\alpha$ is the coefficient of thermal expansion:

$\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}$

This equation allows calculation if $C_p$ and $V$ are known functions of $p$ and $T$ . It’s more complex because $T$ isn't a "natural" variable for enthalpy in the same way $S$ and $p$ are.

Crucially, at constant pressure ( $\mathrm{d} p = 0$ ), this simplifies to:

$\mathrm{d} H = C_p \, \mathrm{d} T$

This is the relationship you’ll see most often in chemistry. For an ideal gas, this holds true even if the pressure changes, because for them, $\alpha T = 1$ .

In a broader sense, when dealing with systems where the number of particles can change, the first law expands. For enthalpy, this leads to:

$\mathrm{d} H = T \, \mathrm{d} S + V \, \mathrm{d} p + \sum_i \mu_i \, \mathrm{d} N_i$

Where $\mu_i$ is the chemical potential of component $i$ , and $N_i$ is the number of particles of that component. It accounts for the energy associated with adding or removing substances.

Characteristic functions and natural state variables

Enthalpy $H(S, p, \{N_i\})$ is an "energy representation" of a system. Its natural state variables are entropy ( $S$ ), pressure ( $p$ ), and particle numbers ( $N_i$ ). These are convenient for describing processes where these variables are dictated by the surroundings. For instance, atmospheric phenomena often occur so rapidly that heat exchange is minimal – the adiabatic approximation used in meteorology relies on this.

The conjugate characteristic function is entropy, $S(H, p, \{N_i\})$ , representing the "entropy representation." Here, the natural variables are enthalpy, pressure, and particle numbers. This is useful when these quantities are experimentally controlled, like setting a specific external pressure and allowing heat transfer.

Physical interpretation

The $U$ part of $H = U + pV$ is the system's internal energy. The $pV$ term represents the work needed to "push aside" the surroundings to make space for the system. Imagine creating a gas bubble in a liquid: you need energy for the bubble's internal state ( $U$ ) and energy to expand against the liquid's pressure ( $pV$ ).

In pure physics or statistical mechanics, where the focus is on internal properties of a fixed-volume system, internal energy ( $U$ ) is often preferred. But in chemistry, where experiments are frequently done at constant atmospheric pressure, the $pV$ work is a known, manageable energy exchange. Thus, $\Delta H$ becomes the more relevant measure for the heat of reaction. For a heat engine completing a cycle, the enthalpy change is zero, as it returns to its initial state.

Relationship to heat

Let's revisit the first law for closed systems: $\mathrm{d} U = \delta Q - \delta W$ . If we're only dealing with heat transfer and pV work at a constant surface pressure $p$ , then $\delta W = p \, \mathrm{d} V$ . So:

$\mathrm{d} U = \delta Q - p \, \mathrm{d} V$

Now, substituting into the definition of enthalpy change:

$\mathrm{d} H = \mathrm{d} U + \mathrm{d}(pV)$ $\mathrm{d} H = (\delta Q - p \, \mathrm{d} V) + (p \, \mathrm{d} V + V \, \mathrm{d} p)$ $\mathrm{d} H = \delta Q + V \, \mathrm{d} p$

This is the crucial part: If the pressure is constant ( $\mathrm{d} p = 0$ ), then:

$\mathrm{d} H = \delta Q$

The change in enthalpy is exactly equal to the heat added. This is why enthalpy was historically called "heat content." A rather direct, if slightly misleading, name in retrospect.

Applications

Enthalpy helps us quantify the energy required to create a system. This involves not just its internal energy ( $U$ ) but also the work ( $pV$ ) done against the ambient pressure.

For systems at constant pressure, with no work other than $pV$ work, the enthalpy change is precisely the heat transferred.

For a simple system with a fixed number of particles, at constant pressure, the enthalpy change represents the maximum thermal energy that can be extracted from an isobaric process.

Heat of reaction

Main article: Standard enthalpy of reaction

As mentioned, we measure enthalpy changes ( $\Delta H = H_f - H_i$ ), not absolute values. $H_f$ is the final enthalpy (of products), and $H_i$ is the initial enthalpy (of reactants).

For an exothermic reaction at constant pressure, $\Delta H$ is negative because the products have lower enthalpy than the reactants. The difference is the heat released. For an endothermic reaction, $\Delta H$ is positive, representing the heat absorbed.

Since $\Delta H = \Delta U + p \, \Delta V$ , and $p \, \Delta V$ is often small compared to $\Delta U$ , $\Delta H$ is a good approximation of $\Delta U$ for many chemical reactions. The combustion of carbon monoxide is a classic example where the difference is minimal. This is why reaction enthalpies are often discussed in terms of bond energies.

Specific enthalpy

Specific enthalpy ( $h = H/m$ ) is the enthalpy per unit mass. It's defined as $h = u + pv$ , where $u$ is specific internal energy and $v$ is specific volume ( $v = 1/\rho$ ).

Enthalpy changes

These changes quantify the energy transferred during transformations. They are path-independent because enthalpy is a state function. The reverse process has an equal and opposite enthalpy change. Standard enthalpy changes, like the enthalpy of formation, are meticulously measured and compiled.

When referring to these standardized values, the term "change" is often dropped, and properties are simply called "enthalpy of X." These are usually quoted under standard conditions:

Pressure: 1 atm (1013.25 hPa) or 1 bar.
Temperature: 25 °C (298.15 K).
Concentration: 1.0 M for solutions.
Physical state: Standard state for elements and compounds.

Prefixing with "standard" (e.g., standard enthalpy of formation) denotes these specific conditions.

Chemical properties

Enthalpy of reaction: Enthalpy change per mole of substance reacted completely.
Enthalpy of formation: Enthalpy change per mole of a compound formed from its elements in their standard states.
Enthalpy of combustion: Enthalpy change per mole of a substance burned completely in oxygen.
Enthalpy of hydrogenation: Enthalpy change per mole when an unsaturated compound reacts with hydrogen to form a saturated compound.
Enthalpy of atomization: Enthalpy required to separate one mole of a substance into its constituent atoms.
Enthalpy of neutralization: Enthalpy change per mole of water formed when an acid and base react.
Standard enthalpy of solution: Enthalpy change per mole when a solute dissolves completely in a solvent to infinite dilution.
Standard enthalpy of denaturation: Enthalpy change required to denature one mole of a compound.
Enthalpy of hydration: Enthalpy change when one mole of gaseous ions forms aqueous ions.

Physical properties

Enthalpy of fusion: Enthalpy change per mole to convert solid to liquid.
Enthalpy of vaporization: Enthalpy change per mole to convert liquid to gas.
Enthalpy of sublimation: Enthalpy change per mole to convert solid to gas.
Lattice enthalpy: Energy required to separate one mole of an ionic compound into gaseous ions at infinite separation.
Enthalpy of mixing: Enthalpy change upon mixing non-reacting substances.

Open systems

For open systems, where mass can flow in and out, the first law becomes more complex. The change in internal energy is related to heat, work, and the internal energy carried by incoming and outgoing mass:

$\mathrm{d} U = \delta Q + \mathrm{d} U_{\text{in}} - \mathrm{d} U_{\text{out}} - \delta W$

In a steady-state process, energy balances equate shaft work to heat added plus net enthalpy added.

The boundary around an open system is a control volume. Flowing mass does work (flow work, or $pV$ work), and there can be mechanical work (shaft work). The total work is:

$\delta W = \mathrm{d}(p_{\text{out}}V_{\text{out}}) - \mathrm{d}(p_{\text{in}}V_{\text{in}}) + \delta W_{\text{shaft}}$

Substituting this into the first law for the control volume (cv):

$\mathrm{d} U_{\text{cv}} = \delta Q + \mathrm{d} U_{\text{in}} + \mathrm{d}(p_{\text{in}}V_{\text{in}}) - \mathrm{d} U_{\text{out}} - \mathrm{d}(p_{\text{out}}V_{\text{out}}) - \delta W_{\text{shaft}}$

Using the definition of enthalpy ( $H = U + pV$ ), this simplifies elegantly to:

$\mathrm{d} U_{\text{cv}} = \delta Q + \mathrm{d} H_{\text{in}} - \mathrm{d} H_{\text{out}} - \delta W_{\text{shaft}}$

This form, incorporating enthalpy, is fundamental to analyzing open systems. For devices operating at steady state, the change in internal energy $\mathrm{d} U_{\text{cv}}$ is zero. The equation then relates power output ( $P$ ) to heat and enthalpy flows.

In terms of time derivatives, for multiple flows and heat sources:

$\frac{\mathrm{d} U}{\mathrm{d} t} = \sum_k \dot{Q}_k + \sum_k \dot{H}_k - \sum_k p_k \frac{\mathrm{d} V_k}{\mathrm{d} t} - P$

where $\dot{H}_k$ represents enthalpy flow rates. The term $\sum_k p_k \frac{\mathrm{d} V_k}{\mathrm{d} t}$ accounts for work done by moving boundaries. $P$ encompasses all other forms of power output.

The $\dot{H}_k$ terms can be expressed as $h_k \dot{m}_k$ (specific enthalpy times mass flow rate) or $H_m \dot{n}_k$ (molar enthalpy times molar flow rate).

If kinetic energy changes are negligible between inlet and outlet, this simplifies. For steady-state operation of devices like turbines or pumps, the net power generation is:

$P = \sum_k \langle \dot{Q}_k \rangle + \sum_k \langle \dot{H}_k \rangle - \sum_k \langle p_k \frac{\mathrm{d} V_k}{\mathrm{d} t} \rangle$

The angle brackets denote time averages. The presence of enthalpy in the first law for open systems is precisely why it’s so indispensable in engineering.

Diagrams

Diagrams are essential for visualizing these properties. A T−s diagram, for instance, plots temperature against specific entropy. It shows phase boundaries, isobars (constant pressure lines), and isenthalps (constant enthalpy lines), allowing engineers to trace processes and determine changes in state. These are not just pretty pictures; they are powerful analytical tools.

Some basic applications

Let’s look at some points on a nitrogen T−s diagram to illustrate:

Point	T [K]	p [bar]	s [kJ/(kg⋅K)]	h [kJ/kg]
a	300	1	6.85	461
b	380	2	6.85	530
c	300	200	5.16	430
d	270	1	6.79	430
e	108	13	3.55	100
f	77.2	1	3.75	100
g	77.2	1	2.83	28
h	77.2	1	5.41	230

Points e and g are saturated liquid, h is saturated gas.

Throttling

Main article: Joule–Thomson effect

The throttling process, or Joule–Thomson expansion, is a prime example of enthalpy conservation. It involves a fluid flowing adiabatically through a restriction (like a valve). This is the magic behind refrigerators – it’s the step that causes the significant temperature drop.

In steady flow, the enthalpy before and after the restriction is the same:

$0 = \dot{m}h_1 - \dot{m}h_2$

Which means $h_1 = h_2$ . The specific enthalpy remains constant.

Example 1: Expanding nitrogen from 200 bar (point c, 300 K) to 1 bar. Following a constant enthalpy line (around 425 kJ/kg), we reach point d at about 270 K. So, the expansion cools the gas. Despite the friction and entropy production, the temperature drops.

Example 2: Starting from point e (saturated liquid, $h = 100$ kJ/kg, $p=13$ bar, $T=108$ K). Throttling to 1 bar lands us in the two-phase region at point f. The enthalpy at f ( $h_f$ ) is a mix of liquid ( $h_g$ ) and gas ( $h_h$ ) enthalpies, weighted by their fractions ( $x_f$ for liquid):

$h_f = x_f h_g + (1 - x_f) h_h$

$100 = x_f \times 28 + (1 - x_f) \times 230$ Solving for $x_f$ gives $x_f = 0.64$ . This means 64% of the exiting mixture is liquid.

Compressors

Main article: Compressor

Compressors increase pressure, and if the process is adiabatic, the temperature rises significantly. A reversible, isentropic compression would follow a vertical line on the T−s diagram (e.g., point a to b: 1 bar to 2 bar, 300 K to 380 K). To bring the gas back to ambient temperature, heat must be removed.

For steady flow, the first law relates power input ( $P$ ) to enthalpy changes and heat removed ( $\dot{Q}$ ):

$0 = -\dot{Q} + \dot{m}(h_1 - h_2) + P$

The minimum power required for reversible compression is given by:

$\frac{P_{\text{min}}}{\dot{m}} = h_2 - h_1 - T_a (s_2 - s_1)$

where $T_a$ is ambient temperature. This can be integrated to:

$\frac{P_{\text{min}}}{\dot{m}} = \int_{1}^{2} v \, \mathrm{d} p$

This integral represents the area under the pressure-specific volume curve, a classic thermodynamic result. For example, compressing 1 kg of nitrogen from 1 bar to 200 bar requires a minimum of 476 kJ/kg.

History and etymology

The term "enthalpy" is relatively new, appearing in the early 20th century. Energy was formalized by Thomas Young around 1802, and entropy by Rudolf Clausius in 1865. "Energy" derives from the Greek ergon (work), while "entropy" from tropē (transformation). "Enthalpy" comes from the Greek thalpos (warmth, heat).

It was introduced as a more precise term for the older concept of "heat content," which was accurate only for constant pressure processes. J. W. Gibbs himself referred to it as a "heat function for constant pressure."

The concept, if not the name, is linked to Benoît Paul Émile Clapeyron and Rudolf Clausius with their Clausius–Clapeyron relation in 1850.

The word "enthalpy" first appeared in print in 1909, attributed to Heike Kamerlingh Onnes. It gained wider acceptance in the 1920s, particularly with the publication of Mollier Steam Tables and Diagrams. Before then, the symbol $H$ was often used more generally for "heat." Alfred W. Porter formally proposed $H$ specifically for enthalpy or "heat content at constant pressure" in 1922.

Notes

$\alpha T = 1$ for an ideal gas.
The term "heat content" was used historically for $H$ .
Gibbs's collected works use "heat function for constant pressure" rather than "enthalpy."

See also

References

IUPAC, Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "enthalpy". doi:10.1351/goldbook.E02141
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Çengel, Yunus A.; Boles, Michael A.; Kanoglu, Mehmet (2019). Thermodynamics: an engineering approach (Ninth ed.). New York, NY: McGraw-Hill Education. p. 123. ISBN 978-1-259-82267-4.
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Iribarne, J. V.; Godson, W. L. (1981). Atmospheric Thermodynamics (2nd ed.). Dordrecht, NL: Kluwer Academic Publishers. pp. 235–236. ISBN 90-277-1297-2.
Tschoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics. Amsterdam, NL: Elsevier. p. 17. ISBN 0-444-50426-5.
Callen, H. B. (1985) [1960]. Thermodynamics and an Introduction to Thermostatistics (1st (1960), 2nd (1985) ed.). New York, NY: John Wiley & Sons. Chapter 5. ISBN 0-471-86256-8.
Münster, A. (1970). Classical Thermodynamics. Translated by Halberstadt, E. S. London, UK: Wiley–Interscience. p. 6. ISBN 0-471-62430-6.
Reif, F. (1967). Statistical Physics. London, UK: McGraw-Hill.
Kittel, C.; Kroemer, H. (1980). Thermal Physics. London, UK: Freeman.
Rathakrishnan (2015). High Enthalpy Gas Dynamics. John Wiley and Sons Singapore Pte. Ltd. ISBN 978-1118821893.
Laidler, K. J.; Meiser, John H. (1982). Physical Chemistry. Benjamin / Cummings. p. 53. ISBN 978-0-8053-5682-3.
Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry (8th ed.). Prentice Hall. pp. 237–238. ISBN 978-0-13-014329-7.
Moran, M. J.; Shapiro, H. N. (2006). Fundamentals of Engineering Thermodynamics (5th ed.). John Wiley & Sons. p. 129. ISBN 9780470030370.
Figure composed with data obtained with REFPROP, NIST Standard Reference Database 23.
ἔργον in Liddell and Scott.
τροπή in Liddell and Scott.
θάλπος in Liddell and Scott.
Tinoco, Ignacio Jr.; Sauer, Kenneth; Wang, James C. (1995). Physical Chemistry (3rd ed.). Prentice-Hall. p. 41. ISBN 978-0-13-186545-7.
Gibbs (1948)
Henderson, Douglas; Eyring, Henry; Jost, Wilhelm (1967). Physical Chemistry: An advanced treatise. Academic Press. p. 29.
Dalton (1909), p. 864, footnote (1).

Bibliography

Dalton, J.P. (1909). "Researches on the Joule–Kelvin effect, especially at low temperatures. I. Calculations for hydrogen" (PDF). Koninklijke Akademie van Wetenschappen te Amsterdam [Proceedings of the Royal Academy of Sciences at Amsterdam, Section of Sciences]. 11: 863–873. Bibcode:1908KNAB...11..863D.
Gibbs, J.W. (1948). The Collected Works of J. Willard Gibbs. Vol. I. New Haven, CT: Yale University Press. p. 88.
Haase, R. (1971). Jost, W. (ed.). Physical Chemistry: An advanced treatise. New York, NY: Academic. p. 29.
de Hoff, R. (2006). Thermodynamics in Materials Science. Boca Raton, FL: CRC Press. ISBN 9780849340659.
Howard, Irmgard K. (2002). "H is for enthalpy, thanks to Heike Kamerlingh Onnes and Alfred W. Porter". Journal of Chemical Education. 79 (6): 697–698. Bibcode:2002JChEd..79..697H. doi:10.1021/ed079p697.
Kittel, C.; Kroemer, H. (1980). Thermal Physics. New York, NY: S. R. Furphy & Co. p. 246.
Laidler, K.J. (1995). The World of Physical Chemistry. Oxford, UK: Oxford University Press. p. 110. ISBN 978-0-19-855597-1 – via archive.org.
van Ness, Hendrick C. (2003). "H is for enthalpy". Journal of Chemical Education. 80 (6): 486. Bibcode:2003JChEd..80..486V. doi:10.1021/ed080p486.1.

External links

Weisstein, Eric. "Enthalpy". Eric Weisstein's World of Physics – via scienceworld.wolfram.com.
"Enthalpy". Thermodynamics hypertext. Physics and Astronomy Department. Georgia State University.
"Enthalpy example calculations" (tutorial notes). Chemistry Department. Texas A&M University. Archived from the original on 10 October 2006.
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