Thermodynamics  

The classical Carnot heat engine




Book:Thermodynamics  
Enthalpy /ˈɛnθəlpi/ ( listen) is a property of a thermodynamic system. The enthalpy of a system is equal to the system's internal energy plus the product of its pressure and volume.^{[1]}^{[2]} For processes at constant pressure, the heat absorbed or released equals the change in enthalpy.
The unit of measurement for enthalpy in the International System of Units (SI) is the joule. Other historical conventional units still in use include the British thermal unit (BTU) and the calorie.
Enthalpy comprises a system's internal energy, which is the energy required to create the system, plus the amount of work required to make room for it by displacing its environment and establishing its volume and pressure.^{[3]}
Enthalpy is defined as a state function that depends only on the prevailing equilibrium state identified by the system's internal energy, pressure, and volume. It is an extensive quantity.
Enthalpy is the preferred expression of system energy changes in many chemical, biological, and physical measurements at constant pressure, because it simplifies the description of energy transfer. At constant pressure, the enthalpy change equals the energy transferred from the environment through heating or work other than expansion work.
The total enthalpy, H, of a system cannot be measured directly. The same situation exists in classical mechanics: only a change or difference in energy carries physical meaning. Enthalpy itself is a thermodynamic potential, so in order to measure the enthalpy of a system, we must refer to a defined reference point; therefore what we measure is the change in enthalpy, ΔH. The ΔH is a positive change in endothermic reactions, and negative in heatreleasing exothermic processes.
For processes under constant pressure, ΔH is equal to the change in the internal energy of the system, plus the pressurevolume work p ΔV done by the system on its surroundings (which is > 0 for an expansion and < 0 for a contraction).^{[4]} This means that the change in enthalpy under such conditions is the heat absorbed or released by the system through a chemical reaction or by external heat transfer. Enthalpies for chemical substances at constant pressure usually refer to standard state: most commonly 1 bar pressure. Standard state does not, strictly speaking, specify a temperature (see standard state), but expressions for enthalpy generally reference the standard heat of formation at 25 °C.
Enthalpy of ideal gases and incompressible solids and liquids does not depend on pressure, unlike entropy and Gibbs energy. Real materials at common temperatures and pressures usually closely approximate this behavior, which greatly simplifies enthalpy calculation and use in practical designs and analyses.
The word enthalpy stems from the Ancient Greek verb enthalpein (ἐνθάλπειν), which means "to warm in".^{[5]} It combines the Classical Greek prefix ἐν en, meaning "to put into", and the verb θάλπειν thalpein, meaning "to heat". The word enthalpy is often incorrectly attributed to Benoît Paul Émile Clapeyron and Rudolf Clausius through the 1850 publication of their Clausius–Clapeyron relation. This misconception was popularized by the 1927 publication of The Mollier Steam Tables and Diagrams. However, neither the concept, the word, nor the symbol for enthalpy existed until well after Clapeyron's death.
The earliest writings which contained the concept of enthalpy did not appear until 1875,^{[6]} when Josiah Willard Gibbs introduced "a heat function for constant pressure". However, Gibbs did not use the word "enthalpy" in his writings.^{[note 1]}
The actual word first appears in the scientific literature in a 1909 publication by J. P. Dalton. According to that publication, Heike Kamerlingh Onnes actually coined the word.^{[7]}
Over the years, scientists used many different symbols to denote enthalpy. In 1922 Alfred W. Porter proposed the symbol "H" as a standard,^{[8]} thus finalizing the terminology still in use today.
In the past, enthalpy was sometimes called heat content.^{[9]}^{[8]}^{[10]} The fact that the change in enthalpy ΔH equals the heat absorbed in processes at constant pressure is the reason for this name.^{[11]} However this equality is not true in general (when the pressure varies), so that the term heat content is considered misleading^{[12]} and is now deprecated.
The enthalpy of a thermodynamic system is defined as^{[13]}^{[14]}
where
Enthalpy is an extensive property. This means that, for homogeneous systems, the enthalpy is proportional to the size of the system. It is convenient to introduce the specific enthalpy h = H/m, where m is the mass of the system, or the molar enthalpy H_{m} = H/n, where n is the number of moles (h and H_{m} are intensive properties). For inhomogeneous systems the enthalpy is the sum of the enthalpies of the composing subsystems:
where the label k refers to the various subsystems. In case of continuously varying p, T or composition, the summation becomes an integral:
where ρ is the density.
The enthalpy of homogeneous systems can be viewed as function H(S,p) of the entropy S and the pressure p, and a differential relation for it can be derived as follows. We start from the first law of thermodynamics for closed systems for an infinitesimal process:
Here, δQ is a small amount of heat added to the system, and δW a small amount of work performed by the system. In a homogeneous system only reversible processes can take place, so the second law of thermodynamics gives δQ = T dS, with T the absolute temperature of the system. Furthermore, if only pV work is done, δW = p dV. As a result,
Adding d(pV) to both sides of this expression gives
or
So
The above expression of dH in terms of entropy and pressure may be unfamiliar to some readers. However, there are expressions in terms of more familiar variables such as temperature and pressure:^{[13]}^{:88}^{[15]}
Here C_{p} is the heat capacity at constant pressure and α is the coefficient of (cubic) thermal expansion:
With this expression one can, in principle, determine the enthalpy if C_{p} and V are known as functions of p and T.
Note that for an ideal gas, αT = 1,^{[note 2]} so that
In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types. The differential statement for dH then becomes
where μ_{i} is the chemical potential per particle for an itype particle, and N_{i} is the number of such particles. The last term can also be written as μ_{i} dn_{i} (with dn_{i} the number of moles of component i added to the system and, in this case, μ_{i} the molar chemical potential) or as μ_{i} dm_{i} (with dm_{i} the mass of component i added to the system and, in this case, μ_{i} the specific chemical potential).
The U term can be interpreted as the energy required to create the system, and the pV term as the work that would be required to "make room" for the system if the pressure of the environment remained constant. When a system, for example, n moles of a gas of volume V at pressure p and temperature T, is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy U plus pV, where pV is the work done in pushing against the ambient (atmospheric) pressure.
In basic physics and statistical mechanics it may be more interesting to study the internal properties of the system and therefore the internal energy is used.^{[16]}^{[17]} In basic chemistry, experiments are often conducted at constant atmospheric pressure, and the pressurevolume work represents an energy exchange with the atmosphere that cannot be accessed or controlled, so that ΔH is the expression chosen for the heat of reaction.
For a heat engine a change in its internal energy is the difference between the heat input and the pressurevolume work done by the working substance while a change in its enthalpy is the difference between the heat input and the work done by the engine:^{[18]}
where the work W done by the engine is:
In order to discuss the relation between the enthalpy increase and heat supply, we return to the first law for closed systems: dU = δQ − δW. We apply it to the special case with a uniform pressure at the surface. In this case the work term can be split into two contributions, the socalled pV work, given by p dV (where here p is the pressure at the surface, dV is the increase of the volume of the system) and all other types of work δW′, such as by a shaft or by electromagnetic interaction. So we write δW = p dV + δW′. In this case the first law reads:
or
From this relation we see that the increase in enthalpy of a system is equal to the added heat:
provided that the system is under constant pressure (dp = 0) and that the only work done by the system is expansion work (δW' = 0).^{[19]}
In thermodynamics, one can calculate enthalpy by determining the requirements for creating a system from "nothingness"; the mechanical work required, pV, differs based upon the conditions that obtain during the creation of the thermodynamic system.
Energy must be supplied to remove particles from the surroundings to make space for the creation of the system, assuming that the pressure p remains constant; this is the pV term. The supplied energy must also provide the change in internal energy, U, which includes activation energies, ionization energies, mixing energies, vaporization energies, chemical bond energies, and so forth. Together, these constitute the change in the enthalpy U + pV. For systems at constant pressure, with no external work done other than the pV work, the change in enthalpy is the heat received by the system.
For a simple system, with a constant number of particles, the difference in enthalpy is the maximum amount of thermal energy derivable from a thermodynamic process in which the pressure is held constant.^{[20]}
The total enthalpy of a system cannot be measured directly, the enthalpy change of a system is measured instead. Enthalpy change is defined by the following equation:
where
For an exothermic reaction at constant pressure, the system's change in enthalpy equals the energy released in the reaction, including the energy retained in the system and lost through expansion against its surroundings. In a similar manner, for an endothermic reaction, the system's change in enthalpy is equal to the energy absorbed in the reaction, including the energy lost by the system and gained from compression from its surroundings. If ΔH is positive, the reaction is endothermic, that is heat is absorbed by the system due to the products of the reaction having a greater enthalpy than the reactants. On the other hand, if ΔH is negative, the reaction is exothermic, that is the overall decrease in enthalpy is achieved by the generation of heat.^{[21]}
From the definition of enthalpy as , the enthalpy change at constant pressure 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 2 CO(g) + O_{2}(g) → 2 CO_{2}(g), = –566.0 kJ and = –563.5 kJ.^{[22]} Since the differences are so small, reaction enthalpies are often loosely described as reaction energies and analyzed in terms of bond energies.
The specific enthalpy of a uniform system is defined as h = H/m where m is the mass of the system. The SI unit for specific enthalpy is joule per kilogram. It can be expressed in other specific quantities by h = u + pv, where u is the specific internal energy, p is the pressure, and v is specific volume, which is equal to 1/ρ, where ρ is the density.
An enthalpy change describes the change in enthalpy observed in the constituents of a thermodynamic system when undergoing a transformation or chemical reaction. It is the difference between the enthalpy after the process has completed, i.e. the enthalpy of the products, and the initial enthalpy of the system, i.e. the reactants. These processes are reversible^{[why?]} and the enthalpy for the reverse process is the negative value of the forward change.
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:
For such standardized values the name of the enthalpy is commonly prefixed with the term standard, e.g. standard enthalpy of formation.
Chemical properties:
Physical properties:
In thermodynamic open systems, matter may flow in and out of the system boundaries. The first law of thermodynamics for open systems states: The increase in the internal energy of a system is equal to the amount of energy added to the system by matter flowing in and by heating, minus the amount lost by matter flowing out and in the form of work done by the system:
where U_{in} is the average internal energy entering the system, and U_{out} is the average internal energy leaving the system.
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 matter 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 matter 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 pV work), and shaft work, which may be performed on some mechanical device.
These two types of work are expressed in the equation
Substitution into the equation above for the control volume (cv) yields:
The definition of enthalpy, H, permits us to use this thermodynamic potential to account for both internal energy and pV work in fluids for open systems:
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.^{[23]} In terms of time derivatives it reads:
with sums over the various places k where heat is supplied, matter flows into the system, and boundaries are moving. The Ḣ_{k} terms represent enthalpy flows, which can be written as
with ṁ_{k} the mass flow and ṅ_{k} the molar flow at position k respectively. The term dV_{k}/dt represents the rate of change of the system volume at position k that results in pV power done by the system. The parameter P represents all other forms of power done by the system such as shaft power, but it can also be e.g. electric power produced by an electrical power plant.
Note that the previous expression holds true only if the kinetic energy flow rate is conserved between system inlet and outlet.^{[clarification needed]} Otherwise, it has to be included in the enthalpy balance. During steadystate operation of a device (see turbine, pump, and engine), the average dU/dt 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:
where the angle brackets denote time averages. The technical importance of the enthalpy is directly related to its presence in the first law for open systems, as formulated above.
Nowadays the enthalpy values of important substances can be obtained using commercial software. Practically all relevant material properties can be obtained either in tabular or in graphical form. There are many types of diagrams, such as h–T diagrams, which give the specific enthalpy as function of temperature for various pressures, and h–p diagrams, which give h as function of p for various T. One of the most common diagrams is the temperature–specific entropy diagram (T–sdiagram). It gives the melting curve and saturated liquid and vapor values together with isobars and isenthalps. These diagrams are powerful tools in the hands of the thermal engineer.
The points a through h in the figure play a role in the discussion in this section.
One of the simple applications of the concept of enthalpy is the socalled throttling process, also known as JouleThomson expansion. It concerns a steady adiabatic flow of a fluid through a flow resistance (valve, porous plug, or any other type of flow resistance) as shown in the figure. This process is very important, since it is at the heart of domestic refrigerators, where it is responsible for the temperature drop between ambient temperature and the interior of the refrigerator. It is also the final stage in many types of liquefiers.
For a steady state flow regime, the enthalpy of the system (dotted rectangle) has to be constant. Hence
Since the mass flow is constant, the specific enthalpies at the two sides of the flow resistance are the same:
that is, the enthalpy per unit mass does not change during the throttling. The consequences of this relation can be demonstrated using the T–s diagram above. Point c is at 200 bar and room temperature (300 K). A Joule–Thomson expansion from 200 bar to 1 bar follows a curve of constant enthalpy of roughly 425 kJ/kg (not shown in the diagram) lying between the 400 and 450 kJ/kg isenthalps and ends in point d, which is at a temperature of about 270 K. Hence the expansion from 200 bar to 1 bar cools nitrogen from 300 K to 270 K. In the valve, there is a lot of friction, and a lot of entropy is produced, but still the final temperature is below the starting value!
Point e is chosen so that it is on the saturated liquid line with h = 100 kJ/kg. It corresponds roughly with p = 13 bar and T = 108 K. Throttling from this point to a pressure of 1 bar ends in the twophase region (point f). This means that a mixture of gas and liquid leaves the throttling valve. Since the enthalpy is an extensive parameter, the enthalpy in f (h_{f}) is equal to the enthalpy in g (h_{g}) multiplied by the liquid fraction in f (x_{f}) plus the enthalpy in h (h_{h}) multiplied by the gas fraction in f (1 − x_{f}). So
With numbers: 100 = x_{f} × 28 + (1 − x_{f}) × 230, so x_{f} = 0.64. This means that the mass fraction of the liquid in the liquid–gas mixture that leaves the throttling valve is 64%.
A power P is applied e.g. as electrical power. If the compression is adiabatic, the gas temperature goes up. In the reversible case it would be at constant entropy, which corresponds with a vertical line in the T–s diagram. For example, compressing nitrogen from 1 bar (point a) to 2 bar (point b) would result in a temperature increase from 300 K to 380 K. In order to let the compressed gas exit at ambient temperature T_{a}, heat exchange, e.g. by cooling water, is necessary. In the ideal case the compression is isothermal. The average heat flow to the surroundings is Q̇. Since the system is in the steady state the first law gives
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
Eliminating Q̇ gives for the minimal power
For example, compressing 1 kg of nitrogen from 1 bar to 200 bar costs at least (h_{c} − h_{a}) − T_{a}(s_{c} − s_{a}). With the data, obtained with the T–s diagram, we find a value of (430 − 461) − 300 × (5.16 − 6.85) = 476 kJ/kg.
The relation for the power can be further simplified by writing it as
With dh = T ds + v dp, this results in the final relation