Enthalpy

 ( State Functions ) 

Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by Earth's ambient atmosphere. The pressure–volume term expresses the work $W$ that was done against constant external pressure $P\_\backslash text\{ext\}$ to establish the system's physical dimensions from $V\_\backslash text\{system,\; initial\}=0$ to some final volume $V\_\backslash text\{system,\; final\}$ (as $W=P\_\backslash text\{ext\}\backslash Delta\; V$), i.e. to make room for it by displacing its surroundings.

In the International System of Units (SI), the unit of measurement for enthalpy is the joule. Other historical conventional units still in use include the calorie and the British thermal unit (BTU).

The total enthalpy of a system cannot be measured directly because the internal energy contains components that are unknown, not easily accessible, or are not of interest for the thermodynamic problem at hand. In practice, a change in enthalpy is the preferred expression for measurements at constant pressure, because it simplifies the description of energy transfer. When transfer of matter into or out of the system is also prevented and no electrical or mechanical (stirring shaft or lift pumping) work is done, at constant pressure the enthalpy change equals the energy exchanged with the environment by heat.

In chemistry, the standard enthalpy of reaction is the enthalpy change when reactants in their ( usually ) change to products in their standard states.

Enthalpies of chemical substances are usually listed for pressure as a standard state. Enthalpies and enthalpy changes for reactions vary as a function of temperature,

The enthalpy of an ideal gas is independent of its pressure or volume, and depends only on its temperature, which correlates to its thermal energy. Real gases at common temperatures and pressures often closely approximate this behavior, which simplifies practical thermodynamic design and analysis.

The word "enthalpy" is derived from the Greek word enthalpein, which means "to heat".

H = U + p V,
     

H = \sum_k H_k,
     

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

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

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

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

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

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

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

\alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_p.
     

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

In physics and statistical mechanics it may be more interesting to study the internal properties of a constant-volume system and therefore the internal energy is used.

In chemistry, experiments are often conducted at constant atmospheric pressure, and the pressure–volume work represents a small, well-defined energy exchange with the atmosphere, so that is the appropriate expression for the heat of reaction. For a heat engine, the change in its enthalpy after a full cycle is equal to zero, since the final and initial state are equal.

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

\mathrm{d}H = \mathrm{d}U + \mathrm{d}(pV),
     

\mathrm{d}H &= \delta Q + V \,\mathrm{d}p + p \,\mathrm{d}V - p \,\mathrm{d}V \\
            &= \delta Q + V \,\mathrm{d}p.
     

If the system is under isobaric system, and consequently, the increase in enthalpy of the system is equal to the heat added: $$$$

\mathrm{d}H = \delta Q.
     

Energy must be supplied to remove particles from the surroundings to make space for the creation of the system, assuming that the pressure remains constant; this is the term. The supplied energy must also provide the change in internal energy , which includes activation energies, ionization energies, mixing energies, vaporization energies, chemical bond energies, and so forth. Together, these constitute the change in the enthalpy For systems at constant pressure, with no external work done other than the work, the change in enthalpy is the heat received by the system.

For a simple system with a constant number of particles at constant pressure, the difference in enthalpy is the maximum amount of thermal energy derivable from an isobaric thermodynamic process.

\Delta H = H_\text{f} - H_\text{i},
     

For an exothermic reaction at constant pressure, the system's change in enthalpy, , is negative due to the products of the reaction having a smaller enthalpy than the reactants, and equals the heat released in the reaction if no electrical or mechanical work is done. In other words, the overall decrease in enthalpy is achieved by the generation of heat.

From the definition of enthalpy as the enthalpy change at constant pressure is However, for most chemical reactions, the work term is much smaller than the internal energy change , which is approximately equal to . As an example, for the combustion of carbon monoxide and

A common standard enthalpy change is the enthalpy of formation, which has been determined for a large number of substances. Enthalpy changes are routinely measured and compiled in chemical and physical reference works, such as the CRC Handbook of Chemistry and Physics. The following is a selection of enthalpy changes commonly recognized in thermodynamics.

When used in these recognized terms the qualifier change is usually dropped and the property is simply termed enthalpy of "process". Since these properties are often used as reference values, it is very common to quote them for a standardized set of environmental parameters, or standard conditions, including:

• A pressure of one atmosphere (1 atm = 1013.25 hPa) or 1 bar
• A temperature of 25 °C = 298.15 K
• A concentration of 1.0 M when the element or compound is present in solution
• Elements or compounds in their normal physical states, i.e. standard state

Enthalpy of formation is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a compound is formed from its elementary antecedents.

Enthalpy of combustion is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a substance burns completely with oxygen.

Enthalpy of hydrogenation is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of an unsaturated compound reacts completely with an excess of hydrogen to form a saturated compound.

Enthalpy of atomization is defined as the enthalpy change required to separate one mole of a substance into its constituent completely.

Enthalpy of neutralization is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of water is formed when an acid and a base react.

Standard enthalpy of solution is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a solute is dissolved completely in an excess of solvent, so that the solution is at infinite dilution.

Standard enthalpy of denaturation is defined as the enthalpy change required to denature one mole of compound.

Hydration energy is defined as the enthalpy change observed when one mole of gaseous ions is completely dissolved in water forming one mole of aqueous ions.

Enthalpy of vaporization is defined as the enthalpy change required to completely change the state of one mole of substance from liquid to gas.

Enthalpy of sublimation is defined as the enthalpy change required to completely change the state of one mole of substance from solid to gas.

Lattice enthalpy is defined as the energy required to separate one mole of an ionic compound into separated gaseous ions to an infinite distance apart (meaning no force of attraction).

Enthalpy of mixing is defined as the enthalpy change upon mixing of two (non-reacting) chemical substances.

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

The region of space enclosed by the boundaries of the open system is usually called a control volume, and it may or may not correspond to physical walls. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of mass into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of mass out as if it were driving a piston of fluid. There are then two types of work performed: flow work described above, which is performed on the fluid (this is also often called work), and mechanical work ( shaft work), which may be performed on some mechanical device such as a turbine or pump.

These two types of work are expressed in the equation $$$$

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

\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}.
     

The definition of enthalpy permits us to use this thermodynamic potential to account for both internal energy and work in fluids for open systems: $$$$

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

If we allow also the system boundary to move (e.g. due to moving pistons), we get a rather general form of the first law for open systems.

\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,
     

\dot H_k = h_k \dot m_k = H_\text{m} \dot n_k,
     

Note that the previous expression holds true only if the kinetic energy flow rate is conserved between system inlet and outlet. Otherwise, it has to be included in the enthalpy balance. During steady-state operation of a device (see Turbine, Pump, and Engine), the average may be set equal to zero. This yields a useful expression for the average power generation for these devices in the absence of chemical reactions: $$$$

P = \sum_k \big\langle \dot Q_k \big\rangle
 + \sum_k \big\langle \dot H_k \big\rangle
 - \sum_k \left\langle p_k \frac{\mathrm{d}V_k}{\mathrm{d}t} \right\rangle,
     

For a steady state flow regime, the enthalpy of the system (dotted rectangle) has to be constant. Hence

$$0\; =\; \backslash dot\; m\; h\_1\; -\; \backslash dot\; m\; h\_2\; ~.$$

Since the mass flow is constant, the specific enthalpies at the two sides of the flow resistance are the same:

$$h\_1\; =\; h\_2\; \backslash ;\; ,$$

that is, the enthalpy per unit mass does not change during the throttling. The consequences of this relation can be demonstrated using the diagram above.

$$h\_\backslash mathbf\backslash mathsf\{f\}\; =\; x\_\backslash mathbf\backslash mathsf\{f\}\; h\_\backslash mathbf\backslash mathsf\{g\}\; +\; (1\; -\; x\_\backslash mathbf\backslash mathsf\{f\})h\_\backslash mathsf\backslash mathbf\{h\}\; ~.$$

With numbers:

so

$$0\; =\; -\backslash dot\; Q\; +\; \backslash dot\; m\backslash ,\; h\_1\; -\; \backslash dot\; m\backslash ,\; h\_2\; +\; P\; ~.$$

The minimal power needed for the compression is realized if the compression is reversible. In that case the second law of thermodynamics for open systems gives

$$0\; =\; -\backslash frac\{\backslash ,\; \backslash dot\; Q\; \backslash ,\}\{T\_\backslash mathsf\{a\}\}\; +\; \backslash dot\; m\; \backslash ,\; s\_1\; -\; \backslash dot\; m\; \backslash ,\; s\_2\; ~.$$

Eliminating gives for the minimal power

$$\backslash frac\{\backslash ,\; P\_\backslash mathsf\{min\}\; \backslash ,\}\{\; \backslash dot\; m\; \}\; =\; h\_2\; -\; h\_1\; -\; T\_\backslash mathsf\{a\}\backslash left(\; s\_2\; -\; s\_1\; \backslash right)\; ~.$$

For example, compressing 1 kg of nitrogen from 1 bar to 200 bar costs at least : With the data, obtained with the diagram, we find a value of

The relation for the power can be further simplified by writing it as

$$\backslash frac\{\backslash ,\; P\_\backslash mathsf\{min\}\; \backslash ,\}\{\; \backslash dot\; m\; \}\; =\; \backslash int\_1^2\; \backslash left(\; \backslash mathrm\{d\}h\; -\; T\_\backslash mathsf\{a\}\backslash ,\backslash mathrm\{d\}s\; \backslash right)\; ~.$$

With

$$\backslash frac\{\backslash ,\; P\_\backslash mathsf\{min\}\; \}\{\; \backslash dot\; m\; \}\; =\; \backslash int\_1^2\; v\backslash ,\backslash mathrm\{d\}p\; ~.$$

The term expresses the obsolete concept of heat content, as refers to the amount of heat gained in a process at constant pressure only,