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Internal energy
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Internal energy
Common symbols
U
SI unit
J
In SI base units
m2*kg/s2
Derivations from
other quantities
U = ∑ i E i {\displaystyle U=\sum _{i}E_{i}\!}
Thermodynamics
The classical Carnot heat engine
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Note: Conjugate variables in italics
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Specific heat capacity c = {\displaystyle c=}
T {\displaystyle T} ∂ S {\displaystyle \partial S}
N {\displaystyle N} ∂ T {\displaystyle \partial T}
Compressibility β = − {\displaystyle \beta =-}
1 {\displaystyle 1} ∂ V {\displaystyle \partial V}
V {\displaystyle V} ∂ p {\displaystyle \partial p}
Thermal expansion α = {\displaystyle \alpha =}
1 {\displaystyle 1} ∂ V {\displaystyle \partial V}
V {\displaystyle V} ∂ T {\displaystyle \partial T}
Equations[show]
Carnot's theorem
Clausius theorem
Fundamental relation
Ideal gas law
Maxwell relations
Onsager reciprocal relations
Bridgman's equations
Table of thermodynamic equations
Potentials[show]
Free energy
Free entropy
Internal energy
U ( S , V ) {\displaystyle U(S,V)}
Enthalpy
H ( S , p ) = U + p V {\displaystyle H(S,p)=U+pV}
Helmholtz free energy
A ( T , V ) = U − T S {\displaystyle A(T,V)=U-TS}
Gibbs free energy
G ( T , p ) = H − T S {\displaystyle G(T,p)=H-TS}
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Book:Thermodynamics
v
t
e
In thermodynamics, the internal energy of a system is the energy contained within the system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system as a whole due to external force fields. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state.[1][2]
The internal energy of a system can be changed by transfers of matter or heat or by doing work.[3] When matter transfer is prevented by impermeable containing walls, the system is said to be closed. Then the first law of thermodynamics states that the increase in internal energy is equal to the total heat added plus the work done on the system by its surroundings. If the containing walls pass neither matter nor energy, the system is said to be isolated and its internal energy cannot change. The first law of thermodynamics may be regarded as establishing the existence of the internal energy.
The internal energy is one of the two cardinal state functions of the state variables of a thermodynamic system.
Contents
[hide]
1 Introduction
1.1 Cardinal functions
2 Description and definition
2.1 Internal energy changes
3 Internal energy of the ideal gas
4 Internal energy of a closed thermodynamic system
4.1 Changes due to temperature and volume
4.2 Changes due to temperature and pressure
4.3 Changes due to volume at constant temperature
5 Internal energy of multi-component systems
6 Internal energy in an elastic medium
7 History
8 Notes
9 See also
10 References
10.1 Bibliography of cited references
11 Bibliography
Introduction[edit]
The internal energy of a given state of a system cannot be directly measured. It is determined through some convenient chain of thermodynamic operations and thermodynamic processes by which the given state can be prepared, starting with a reference state which is customarily assigned a reference value for its internal energy. Such a chain, or path, can be theoretically described by certain extensive state variables of the system, namely, its entropy, S, its volume, V, and its mole numbers, {Nj}. The internal energy, U(S,V,{Nj}), is a function of those. Sometimes, to that list are appended other extensive state variables, for example electric dipole moment. For practical considerations in thermodynamics and engineering it is rarely necessary or convenient to consider all energies belonging to the total intrinsic energy of a system, such as the energy given by the equivalence of mass. Customarily, thermodynamic descriptions include only items relevant to the processes under study. Thermodynamics is chiefly concerned only with changes in the internal energy, not with its absolute value.
The internal energy is a state function of a system, because its value depends only on the current state of the system and not on the path taken or processes undergone to prepare it. It is an extensive quantity. It is the one and only cardinal thermodynamic potential.[4] Through it, by use of Legendre transforms, are mathematically constructed the other thermodynamic potentials. These are functions of variable lists in which some extensive variables are replaced by their conjugate intensive variables. Legendre transformation is necessary because mere substitutive replacement of extensive variables by intensive variables does not lead to thermodynamic potentials. Mere substitution leads to a less informative formula, an equation of state.
Though it is a macroscopic quantity, internal energy can be explained in microscopic terms by two theoretical virtual components. One is the microscopic kinetic energy due to the microscopic motion of the system's particles (translations, rotations, vibrations). The other is the potential energy associated with the microscopic forces, including the chemical bonds, between the particles; this is for ordinary physics and chemistry. If thermonuclear reactions are specified as a topic of concern, then the static rest mass energy of the constituents of matter is also counted. There is no simple universal relation between these quantities of microscopic energy and the quantities of energy gained or lost by the system in work, heat, or matter transfer.
The SI unit of energy is the joule (J). Sometimes it is convenient to use a corresponding density called specific internal energy which is internal energy per unit of mass (kilogram) of the system in question. The SI unit of specific internal energy is J/kg. If the specific internal energy is expressed relative to units of amount of substance (mol), then it is referred to as molar internal energy and the unit is J/mol.
From the standpoint of statistical mechanics, the internal energy is equal to the ensemble average of the sum of the microscopic kinetic and potential energies of the system.
Cardinal functions[edit]
The internal energy, U(S,V,{Nj}), expresses the thermodynamics of a system in the energy-language, or in the energy representation. Its arguments are exclusively extensive variables of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, S(U,V,{Nj}), of the same list of extensive variables of state, except that the entropy, S, is replaced in the list by the internal energy, U. It expresses the entropy representation.[4][5][6]
Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example U = U(S,V,{Nj}), that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, U = U(S,V,{Nj}) for S, to get S = S(U,V,{Nj}).
In contrast, Legendre transforms are necessary to derive fundamental equations for other thermodynamic potentials and Massieu functions. The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation of Massieu functions. It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.[5][7][8]
For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics.
Description and definition[edit]
The internal energy U of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state:
Δ U = ∑ i E i {\displaystyle \Delta U=\sum _{i}E_{i}\,}
where ΔU denotes the difference between the internal energy of the given state and that of the reference state, and the Ei are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state.
From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, Umicro pot, and microscopic kinetic energy, Umicro kin, components:
U = U m i c r o p o t + U m i c r o k i n {\displaystyle U=U_{\mathrm {micro\,pot} }+U_{\mathrm {micro\,kin} }}
The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The microscopic potential energy algebraic summative components are those of the chemical and nuclear particle bonds, and the physical force fields within the system, such as due to internal induced electric or magnetic dipole moment, as well as the energy of deformation of solids (stress-strain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.
Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external gravitational, electrostatic, or electromagnetic fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the object with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter.
For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass. Typically, descriptions only include components relevant to the system under study. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.[9] Therefore, a convenient null reference point may be chosen for the internal energy.
The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains.
At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). In the classical picture of thermodynamics, kinetic energy vanishes at zero temperature and the internal energy is purely potential energy. However, quantum mechanics has demonstrated that even at zero temperature particles maintain a residual energy of motion, the zero point energy. A system at absolute zero is merely in its quantum-mechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable entropy.
The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics relates the pseudo-random kinetic energy of individual particles to the mean kinetic energy of the entire ensemble of particles comprising a system. Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. This energy is often referred to as the thermal energy of a system,[10] relating this energy, like the temperature, to the human experience of hot and cold.
Statistical mechanics considers any system to be statistically distributed across an ensemble of N microstates. Each microstate has an energy Ei and is associated with a probability pi. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by their probability of occurrence:
U = ∑ i = 1 N p i E i . {\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}\ .}
This is the statistical expression of the first law of thermodynamics.
Internal energy changes[edit]
Interactions of thermodynamic systems
Type of system
Mass flow
Work
Heat
Open Y Y Y
Closed N Y Y
Thermally isolated N Y N
Mechanically isolated N N Y
Isolated N N N
Thermodynamics is chiefly concerned only with the changes, ΔU, in internal energy.
For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer Q and due to work. The latter can be split into two kinds, pressure-volume work Wpressure-volume, and frictional and other kinds, such as electrical polarization, which do not alter the volume of the system, and are called isochoric, Wisochoric. Accordingly, the internal energy change ΔU for a process may be written[3]
Δ U = Q + W p r e s s u r e − v o l u m e + W i s o c h o r i c ( c l o s e d s y s t e m , n o t r a n s f e r o f m a t t e r ) . {\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm {(closed\,\,system,\,\,no\,\,transfer\,\,of\,\,matter)} .} [note 1]
When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible.
A second mechanism of change of internal energy of a closed system is the doing of work on the system, either in mechanical form by changing pressure or volume, or by other perturbations, such as directing an electric current through the system.
If the system is not closed, the third mechanism that can increase the internal energy is transfer of matter into the system. This increase, ΔUmatter cannot be split into heat and work components. If the system is so set up physically that heat and work can be done on it by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy:
Δ U = Q + W p r e s s u r e − v o l u m e + W i s o c h o r i c + Δ U m a t t e r {\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }+\Delta U_{\mathrm {matter} }}
( s e p a r a t e p a t h w a y f o r m a t t e r t r a n s f e r f r o m h e a t a n d w o r k t r a n s f e r p a t h w a y s ) . {\displaystyle \mathrm {(separate\,\,pathway\,\,for\,\,matter\,\,transfer\,\,from\,\,heat\,\,and\,\,work\,\,transfer\,\,pathways)} .}
If a system undergoes certain phase transformations while being heated, such as melting and vaporization, it may be observed that the temperature of the system does not change until the entire sample has completed the transformation. The energy introduced into the system while the temperature did not change is called a latent energy, or latent heat, in contrast to sensible heat, which is associated with temperature change.
Internal energy of the ideal gas[edit]
Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as an approximation for working systems. The ideal gas is a gas of particles considered as point objects that interact only by elastic collisions and fill a volume such that their free mean path between collisions is much larger than their diameter. Such systems are approximated by the monatomic gases, helium and the other noble gases. Here the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures.
Therefore, internal energy changes in an ideal gas may be described solely by changes in its kinetic energy. Kinetic energy is simply the internal energy of the perfect gas and depends entirely on its pressure, volume and thermodynamic temperature.
The internal energy of an ideal gas is proportional to its mass (number of moles) N and to its temperature T
U = c N T , {\displaystyle U=cNT,}
where c is the heat capacity (at constant volume) of the gas. The internal energy may be written as a function of the three extensive properties S, V, N (entropy, volume, mass) in the following way [11][12]
U ( S , V , N ) = c o n s t ⋅ e S c N V − R c N R + c c , {\displaystyle U(S,V,N)=const\cdot e^{\frac {S}{cN}}V^{\frac {-R}{c}}N^{\frac {R+c}{c}},}
where const is an arbitrary positive constant and where R is the universal gas constant. It is easily seen that U is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex. Knowing temperature and pressure to be the derivatives T = ∂ U ∂ S , {\displaystyle T={\frac {\partial U}{\partial S}},} p = − ∂ U ∂ V , {\displaystyle p=-{\frac {\partial U}{\partial V}},} the ideal gas law p V = R N T {\displaystyle pV=RNT} immediately follows.
Internal energy of a closed thermodynamic system[edit]
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This above summation of all components of change in internal energy assume that a positive energy denotes heat added to the system or work done on the system, while a negative energy denotes work of the system on the environment.
Typically this relationship is expressed in infinitesimal terms using the differentials of each term. Only the internal energy is an exact differential. For a system undergoing only thermodynamics processes, i.e. a closed system that can exchange only heat and work, the change in the internal energy is
d U = δ Q + δ W {\displaystyle dU=\delta Q+\delta W\,}
which constitutes the first law of thermodynamics.[note 1] It may be expressed in terms of other thermodynamic parameters. Each term is composed of an intensive variable (a generalized force) and its conjugate infinitesimal extensive variable (a generalized displacement).
For example, for a non-viscous fluid, the mechanical work done on the system may be related to the pressure p and volume V. The pressure is the intensive generalized force, while the volume is the extensive generalized displacement:
δ W = − p d V {\displaystyle \delta W=-p\mathrm {d} V\,} .
This defines the direction of work, W, to be energy flow from the working system to the surroundings, indicated by a negative term.[note 1] Taking the direction of heat transfer Q to be into the working fluid and assuming a reversible process, the heat is
δ Q = T d S {\displaystyle \delta Q=T\mathrm {d} S\,} .
T {\displaystyle T} is temperature
S {\displaystyle S} is entropy
and the change in internal energy becomes
d U = T d S − p d V {\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V\!}
Changes due to temperature and volume[edit]
The expression relating changes in internal energy to changes in temperature and volume is
d U = C V d T + [ T ( ∂ p ∂ T ) V − p ] d V (1) . {\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV\,\,{\text{ (1)}}.\,}
This is useful if the equation of state is known.
In case of an ideal gas, we can derive that d U = C V d T {\displaystyle dU=C_{V}dT} , i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature.
[show]
Proof of pressure independence for an ideal gas
The expression relating changes in internal energy to changes in temperature and volume is
d U = C V d T + [ T ( ∂ p ∂ T ) V − p ] d V . {\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV.\,}
The equation of state is the ideal gas law
p V = n R T . {\displaystyle pV=nRT.\,}
Solve for pressure:
p = n R T V . {\displaystyle p={\frac {nRT}{V}}.}
Substitute in to internal energy expression:
d U = C V d T + [ T ( ∂ p ∂ T ) V − n R T V ] d V . {\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-{\frac {nRT}{V}}\right]dV.\,}
Take the derivative of pressure with respect to temperature:
( ∂ p ∂ T ) V = n R V . {\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}}.}
Replace:
d U = C V d T + [ n R T V − n R T V ] d V . {\displaystyle dU=C_{V}dT+\left[{\frac {nRT}{V}}-{\frac {nRT}{V}}\right]dV.}
And simplify:
d U = C V d T . {\displaystyle dU=C_{V}dT.\,}
[show]
Derivation of dU in terms of dT and dV
To express dU in terms of dT and dV, the term
d S = ( ∂ S ∂ T ) V d T + ( ∂ S ∂ V ) T d V {\displaystyle dS=\left({\frac {\partial S}{\partial T}}\right)_{V}dT+\left({\frac {\partial S}{\partial V}}\right)_{T}dV\,}
is substituted in the fundamental thermodynamic relation
d U = T d S − p d V . {\displaystyle dU=TdS-pdV.\,}
This gives:
d U = T ( ∂ S ∂ T ) V d T + [ T ( ∂ S ∂ V ) T − p ] d V . {\displaystyle dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-p\right]dV.\,}
The term T ( ∂ S ∂ T ) V {\displaystyle T\left({\frac {\partial S}{\partial T}}\right)_{V}} is the heat capacity at constant volume C V . {\displaystyle C_{V}.}
The partial derivative of S with respect to V can be evaluated if the equation of state is known. From the fundamental thermodynamic relation, it follows that the differential of the Helmholtz free energy A is given by:
d A = − S d T − p d V . {\displaystyle dA=-SdT-pdV.\,}
The symmetry of second derivatives of A with respect to T and V yields the Maxwell relation:
( ∂ S ∂ V ) T = ( ∂ p ∂ T ) V . {\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}.\,}
This gives the expression above.
Changes due to temperature and pressure[edit]
When dealing with fluids or solids, an expression in terms of the temperature and pressure is usually more useful:
d U = ( C p − α p V ) d T + ( β T p − α T ) V d p {\displaystyle dU=\left(C_{p}-\alpha pV\right)dT+\left(\beta _{T}p-\alpha T\right)Vdp\,}
where it is assumed that the heat capacity at constant pressure is related to the heat capacity at constant volume according to:
C p = C V + V T α 2 β T {\displaystyle C_{p}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}\,}
[show]
Derivation of dU in terms of dT and dP
The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the coefficient of thermal expansion
α ≡ 1 V ( ∂ V ∂ T ) p {\displaystyle \alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}\,}
and the isothermal compressibility
β T ≡ − 1 V ( ∂ V ∂ p ) T {\displaystyle \beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T}\,}
by writing:
d V = ( ∂ V ∂ p ) T d p + ( ∂ V ∂ T ) p d T = V ( α d T − β T d p ) (2) {\displaystyle dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dp+\left({\frac {\partial V}{\partial T}}\right)_{p}dT=V\left(\alpha dT-\beta _{T}dp\right)\,\,{\text{ (2)}}\,}
and equating dV to zero and solving for the ratio dp/dT. This gives:
( ∂ p ∂ T ) V = − ( ∂ V ∂ T ) p ( ∂ V ∂ p ) T = α β T (3) {\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{p}}{\left({\frac {\partial V}{\partial p}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}\,\,{\text{ (3)}}\,}
Substituting (2) and (3) in (1) gives the above expression.
Changes due to volume at constant temperature[edit]
The internal pressure is defined as a partial derivative of the internal energy with respect to the volume at constant temperature:
π T = ( ∂ U ∂ V ) T {\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}}
Internal energy of multi-component systems[edit]
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In addition to including the entropy S and volume V terms in the internal energy, a system is often described also in terms of the number of particles or chemical species it contains:
U = U ( S , V , N 1 , … , N n ) {\displaystyle U=U(S,V,N_{1},\ldots ,N_{n})\,}
where Nj are the molar amounts of constituents of type j in the system. The internal energy is an extensive function of the extensive variables S, V, and the amounts Nj, the internal energy may be written as a linearly homogeneous function of first degree:
U ( α S , α V , α N 1 , α N 2 , … ) = α U ( S , V , N 1 , N 2 , … ) {\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots )\,}
where α is a factor describing the growth of the system. The differential internal energy may be written as
d U = ∂ U ∂ S d S + ∂ U ∂ V d V + ∑ i ∂ U ∂ N i d N i = T d S − p d V + ∑ i μ i d N i {\displaystyle \mathrm {d} U={\frac {\partial U}{\partial S}}\mathrm {d} S+{\frac {\partial U}{\partial V}}\mathrm {d} V+\sum _{i}\ {\frac {\partial U}{\partial N_{i}}}\mathrm {d} N_{i}\ =T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\,}
which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure p to be the negative of the similar derivative with respect to volume V
T = ∂ U ∂ S , {\displaystyle T={\frac {\partial U}{\partial S}},}
p = − ∂ U ∂ V , {\displaystyle p=-{\frac {\partial U}{\partial V}},}
and where the coefficients μ i {\displaystyle \mu _{i}} are the chemical potentials for the components of type i in the system. The chemical potentials are defined as the partial derivatives of the energy with respect to the variations in composition:
μ i = ( ∂ U ∂ N i ) S , V , N j ≠ i {\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}}
As conjugate variables to the composition { N j } {\displaystyle \lbrace N_{j}\rbrace } , the chemical potentials are intensive properties, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent. Under conditions of constant T and p, because of the extensive nature of U and its independent variables, using Euler's homogeneous function theorem, the differential dU may be integrated and yields an expression for the internal energy:
U = T S − p V + ∑ i μ i N i {\displaystyle U=TS-pV+\sum _{i}\mu _{i}N_{i}\,} .
The sum over the composition of the system is the Gibbs free energy:
G = ∑ i μ i N i {\displaystyle G=\sum _{i}\mu _{i}N_{i}\,}
that arises from changing the composition of the system at constant temperature and pressure. For a single component system, the chemical potential equals the Gibbs energy per amount of substance, i.e. particles or moles according to the original definition of the unit for { N j } {\displaystyle \lbrace N_{j}\rbrace } .
Internal energy in an elastic medium[edit]
For an elastic medium the mechanical energy term of the internal energy must be replaced by the more general expression involving the stress σ i j {\displaystyle \sigma _{ij}} and strain ε i j {\displaystyle \varepsilon _{ij}} . The infinitesimal statement is:
d U = T d S + V σ i j d ε i j {\displaystyle \mathrm {d} U=T\mathrm {d} S+V\sigma _{ij}\mathrm {d} \varepsilon _{ij}}
where Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. The Euler theorem yields for the internal energy:[13]
U = T S + 1 2 σ i j ε i j {\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}
For a linearly elastic material, the stress is related to the strain by:
σ i j = C i j k l ε k l {\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}
where the Cijkl are the components of the 4th-rank elastic constant tensor of the medium.
History[edit]
James Joule studied the relationship between heat, work, and temperature. He observed that if he did mechanical work on a fluid, such as water, by agitating the fluid, its temperature increased. He proposed that the mechanical work he was doing on the system was converted to thermal energy. Specifically, he found that 4185.5 joules of energy were needed to raise the temperature of a kilogram of water by one degree Celsius.[14]
Notes[edit]
^ Jump up to: a b c In this article we choose the sign convention of the mechanical work as typically defined in chemistry, which is different from the convention used in physics. In chemistry, work performed by the system against the environment, e.g., a system expansion, is negative, while in physics this is taken to be positive.
See also[edit]
Calorimetry
Enthalpy
Exergy
Thermodynamic equations
Thermodynamic potentials
References[edit]
Jump up ^ Crawford, F. H. (1963), pp. 106–107.
Jump up ^ Haase, R. (1971), pp. 24–28.
^ Jump up to: a b Born, M. (1949), Appendix 8, pp. 146–149.
^ Jump up to: a b Tschoegl, N.W. (2000), p. 17.
^ Jump up to: a b Callen, H.B. (1960/1985), Chapter 5.
Jump up ^ Münster, A. (1970), p. 6.
Jump up ^ Münster, A. (1970), Chapter 3.
Jump up ^ Bailyn, M. (1994), pp. 206–209.
Jump up ^ I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39
Jump up ^ Thermal energy – Hyperphysics
Jump up ^ van Gool, W.; Bruggink, J.J.C. (Eds) (1985). Energy and time in the economic and physical sciences. North-Holland. pp. 41–56. ISBN 0444877487. CS1 maint: Extra text: authors list (link)
Jump up ^ Grubbström, Robert W. (2007). "An Attempt to Introduce Dynamics Into Generalised Exergy Considerations". Applied Energy. 84: 701–718. doi:10.1016/j.apenergy.2007.01.003.
Jump up ^ Landau & Lifshitz 1986
Jump up ^ Joule, J.P. (1850). "On the Mechanical Equivalent of Heat". Philosophical Transactions of the Royal Society. Royal Society. 140: 61–82. doi:10.1098/rstl.1850.0004. Retrieved 4 June 2016.
Bibliography of cited references[edit]
Adkins, C.J. (1968/1975). Equilibrium Thermodynamics, second edition, McGraw-Hill, London, ISBN 0-07-084057-1.
Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 0-88318-797-3.
Born, M. (1949). Natural Philosophy of Cause and Chance, Oxford University Press, London.
Callen, H.B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, ISBN 0-471-86256-8.
Crawford, F. H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc.
Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.
Münster, A. (1970), Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London, ISBN 0-471-62430-6.
Tschoegl, N.W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam, ISBN 0-444-50426-5.
Bibliography[edit]
Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics" (PDF). Pure Appl. Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349.
Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). Thermodynamics (2nd ed.). New York, NY USA: McGraw-Hill Book Co. ISBN 0-07-113809-9. CS1 maint: Multiple names: authors list (link)
Landau, L. D.; Lifshitz, E. M. (1986). Theory of Elasticity (Course of Theoretical Physics Volume 7). (Translated from Russian by J.B. Sykes and W.H. Reid) (Third ed.). Boston, MA: Butterworth Heinemann. ISBN 0-7506-2633-X.
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values: An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction

values: On the Equilibrium of Heterogeneous Substances

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values: Edit section: Description and definition

values: {\displaystyle \Delta U=\sum _{i}E_{i}\,}

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values: U = U_{\mathrm{micro\,pot}} + U_{\mathrm{micro\,kin}}

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values: \Delta U = Q + W_{\mathrm{pressure-volume}} + W_{\mathrm{isochoric}} + \Delta U_{\mathrm{matter}}

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values: U(S,V,N) = const  \cdot  e^\frac{S}{cN}    V^\frac{-R}{c}  N^\frac{R+c}{c},

values: {\displaystyle T={\frac {\partial U}{\partial S}},}

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values: {\displaystyle p=-{\frac {\partial U}{\partial V}},}

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values: Edit section: Internal energy of a closed thermodynamic system

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values: {\displaystyle \delta W=-p\mathrm {d} V\,}

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values: {\displaystyle \delta Q=T\mathrm {d} S\,}

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values: {\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V\!}

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values: \mathrm{d}U = T\mathrm{d}S-p\mathrm{d}V \!

values: Edit section: Changes due to temperature and volume

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values: dU =C_{V}dT +\left[T\left(\frac{\partial p}{\partial T}\right)_{V}  - p\right]dV\,\,\text{   (1)}.\,

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values: dU =C_{V}dT +\left[T\left(\frac{\partial p}{\partial T}\right)_{V}  - p\right]dV.\,

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values: dU =C_{V}dT +\left[T\left(\frac{\partial p}{\partial T}\right)_{V}  - \frac{n R T}{V}\right]dV.\,

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values: \left( \frac{\partial p}{\partial T} \right)_{V} = \frac{n R}{V}.

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values: dU = C_{V}dT + \left[ \frac{n R T}{V}  - \frac{n R T}{V} \right] dV.

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values: dS = \left(\frac{\partial S}{\partial T}\right)_{V}dT + \left(\frac{\partial S}{\partial V}\right)_{T}dV \,

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values: dU = T\left(\frac{\partial S}{\partial T}\right)_{V}dT +\left[T\left(\frac{\partial S}{\partial V}\right)_{T} - p\right]dV.\,

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values: \left(\frac{\partial S}{\partial V}\right)_{T} = \left(\frac{\partial p}{\partial T}\right)_{V}.\,

values: Edit section: Changes due to temperature and pressure

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values: dU = \left(C_{p}-\alpha p V\right)dT +\left(\beta_{T}p-\alpha T\right)Vdp\,

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values: {\displaystyle C_{p}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}\,}

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values: \alpha \equiv \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\,

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values: \beta_{T} \equiv -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\,

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values: dV = \left(\frac{\partial V}{\partial p}\right)_{T}dp + \left(\frac{\partial V}{\partial T}\right)_{p} dT = V\left(\alpha dT-\beta_{T}dp \right)\,\,\text{  (2)} \,

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values: \left(\frac{\partial p}{\partial T}\right)_{V}= -\frac{\left(\frac{\partial V}{\partial T}\right)_{p}}{\left(\frac{\partial V}{\partial p}\right)_{T}}= \frac{\alpha}{\beta_{T}}\,\,\text{   (3)}\,

values: Edit section: Changes due to volume at constant temperature

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values: \pi _T  = \left ( \frac{\partial U}{\partial V} \right )_T

values: Edit section: Internal energy of multi-component systems

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U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )
 = \alpha U(S,V,N_{1},N_{2},\ldots)\,

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values: \mathrm{d} U = \frac{\partial U}{\partial S}  \mathrm{d} S  + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - p \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i\,

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values: \mu_i = \left( \frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}

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values: {\displaystyle U=TS-pV+\sum _{i}\mu _{i}N_{i}\,}

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values: {\displaystyle G=\sum _{i}\mu _{i}N_{i}\,}

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values: Edit section: Internal energy in an elastic medium

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values: {\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}

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values: {\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}

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text:
Carmela Corleone
From Wikipedia, the free encyclopedia
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Carmela Corleone
Morgana King as Carmela Corleone
First appearance
The Godfather
Last appearance
The Godfather: The Game
Created by
Mario Puzo
Portrayed by
Morgana King (48–62)
Francesca De Sapio (20–26)
Information
Gender
Female
Spouse(s)
Vito Corleone (1914–1955, Vito's death)
Children
Sonny Corleone
Fredo Corleone
Michael Corleone
Connie Corleone
Relatives
Francesca Corleone (granddaughter)
Kathryn Corleone (granddaughter)
Frank Corleone (grandson)
Santino Corleone Jr.(grandson)
Vincent Corleone (grandson)
Anthony Corleone (grandson)
Mary Corleone (granddaughter)
Religion
Roman Catholicism
Carmela Corleone (1897–1959) a fictional character in Mario Puzo's The Godfather, is the wife of Don Vito Corleone. Carmela is portrayed by Italian-American Morgana King in Francis Ford Coppola's film adaptation of the novel, and in The Godfather Part II.
King was born Maria Grazia Morgana Messina in Pleasantville, New York; her parents were "from Fiumefreddo di Sicilia, Province of Catania, Sicily, Italy." King is better known as a singer and is given a brief chance to display her vocal skills in the wedding reception scene of The Godfather when she sings the song "Luna Mezz'O Mare".[1][2][3]
Contents
1 Background
2 Sequel novel
3 Family members
4 References
Background[edit]
Carmela was born in Sicily in 1897 and emigrated to the United States shortly after the turn of the century. She married Vito Corleone in 1914; they were married for just over 40 years until Vito's death in 1955. They had four children – Sonny, Fredo, Michael and Connie. They also took in Sonny's friend Tom Hagen, who later served as the family consigliere.
In the novel, Carmela Corleone is portrayed as a traditional Italian immigrant woman who speaks in very broken English. In the movies, however, she speaks fluent English as an adult, with a marked New York accent. In the novel, she develops a close relationship with Michael's girlfriend and future wife, Kay. She is given more expansive dialogue in The Godfather Part II, notably when she confronts her daughter Connie about her behavior early in the film, and when she discusses family life with Michael, who fears that his role as Don of the Corleone criminal empire will cost him his family. Carmela Corleone dies toward the end of the sequel.
Carmela was disturbed by Vito's change from a kind, quiet young man to a pragmatic and ruthless criminal. However Carmela seems to forgive Vito for his many crimes, because he remains essentially a good man who is devoted to his family. Devoutly Catholic, Carmela attends Mass every day to pray for her husband's soul to keep him from "going down there."
In The Godfather Part II's flashback scenes, the young Carmela is portrayed by Francesca De Sapio.
Sequel novel[edit]
Carmela is a major character in the 2012 prequel novel The Family Corleone, which portrays their early years together, raising a family as Vito becomes a crime boss.
Family members[edit]
Main article: Corleone family
Vito Corleone—Husband; played by Marlon Brando in Godfather I, played by Robert De Niro in Godfather II
Santino "Sonny" Corleone—Eldest son; played by James Caan
Tom Hagen—informally adopted son, played by Robert Duvall
Frederico "Fredo" Corleone—Middle son; played by John Cazale
Michael Corleone—Youngest son; played by Al Pacino
Constanzia "Connie" Corleone—Daughter; played by Talia Shire
Frank Corleone—Grandson
Santino Corleone, Jr.—Grandson
Francesca Corleone—Granddaughter, twin of Kathryn Corleone
Kathryn Corleone—Granddaughter, twin of Francesca Corleone
Vincent Corleone—Grandson; played by Andy García
Anthony Corleone—Grandson; played by Franc D'Ambrosio
Mary Corleone—Granddaughter; played by Sofia Coppola
Victor Rizzi—Grandson
Michael Rizzi—Grandson
References[edit]
Jump up ^ Vernon Scott (May 3, 1971). Singer in 'Godfather' Role. The Deseret News
Jump up ^ Morgana King's Songs Offer Relief. Chicago Sun-Times, November 13, 1992, p. 19, Weekend Plus
Jump up ^ "Zoetrope". Zoetrope. Retrieved November 30, 2011.
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