# First Law of Thermodynamics

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:<math>\mathrm{d}U=\mathrm{d}U_{in}+\delta Q-\mathrm{d}U_{out}-\delta W\,</math> | :<math>\mathrm{d}U=\mathrm{d}U_{in}+\delta Q-\mathrm{d}U_{out}-\delta W\,</math> | ||

- | where | + | where U<sub>in</sub> is the average internal energy entering the system and U<sub>out</sub> is the average internal energy leaving the system |

where <math>\delta Q</math> and <math>\delta W</math> are infinitesimal amounts of heat supplied to the system and work done by the system, respectively. | where <math>\delta Q</math> and <math>\delta W</math> are infinitesimal amounts of heat supplied to the system and work done by the system, respectively. | ||

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:<math>\mathrm{d}U_{cv}=\mathrm{d}U_{in}+\mathrm{d}(P_{in}V_{in}) - \mathrm{d}U_{out}-\mathrm{d}(P_{out}V_{out})+\delta Q-\delta W_{shaft}\,</math> | :<math>\mathrm{d}U_{cv}=\mathrm{d}U_{in}+\mathrm{d}(P_{in}V_{in}) - \mathrm{d}U_{out}-\mathrm{d}(P_{out}V_{out})+\delta Q-\delta W_{shaft}\,</math> | ||

- | + | Enthalpy, H is a thermodynamic property of systems and defined as | |

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- | Enthalpy, H is a thermodynamic | + | |

:<math>H = E + pV,\,</math> | :<math>H = E + pV,\,</math> | ||

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:V is the volume. | :V is the volume. | ||

- | During steady-state operation of a device | + | 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: |

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+ | :<math>\mathrm{d}U_{cv}=\mathrm{d}H_{in}-\mathrm{d}H_{out}+\delta Q-\delta W_{shaft}\,</math> | ||

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+ | During steady-state operation of a device, any system property within the control volume is independent of time. Therefore, the internal energy of the system enclosed by the control volume remains constant, which implies that dUcv in the expression above may be set equal to zero. This yields a useful expression for the power generation or requirement for these devices in the absence of chemical reactions: | ||

:<math>\frac{\delta W_{shaft}}{\mathrm{d}t}=\frac{\mathrm{d}H_{in}}{\mathrm{d}t}- \frac{\mathrm{d}H_{out}}{\mathrm{d}t}+\frac{\delta Q}{\mathrm{d}t} \,</math> | :<math>\frac{\delta W_{shaft}}{\mathrm{d}t}=\frac{\mathrm{d}H_{in}}{\mathrm{d}t}- \frac{\mathrm{d}H_{out}}{\mathrm{d}t}+\frac{\delta Q}{\mathrm{d}t} \,</math> |

## Revision as of 17:16, 25 July 2010

The first law of thermodynamics, an expression of the principle of conservation of energy, states that energy can be transformed (changed from one form to another), but cannot be created or destroyed.

## Contents |

## Open Systems

In 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. The first law for open systems is given by:

where U_{in} is the average internal energy entering the system and U_{out} is the average internal energy leaving the system

where δ*Q* and δ*W* are infinitesimal amounts of heat supplied to the system and work done by the system, respectively.

The region of space enclosed by open system boundaries 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:

Enthalpy, H is a thermodynamic property of systems and defined as

where E represents the energy of the system. In the absence of an external field, the enthalpy may be defined, as it is generally known, by:

where (all units given in SI)

- H is the enthalpy,
- U is the internal energy,
- p is the pressure of the system, and
- V is the volume.

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:

During steady-state operation of a device, any system property within the control volume is independent of time. Therefore, the internal energy of the system enclosed by the control volume remains constant, which implies that dUcv in the expression above may be set equal to zero. This yields a useful expression for the power generation or requirement for these devices in the absence of chemical reactions:

This expression is described by the diagram above.

## Closed Systems

In a closed system, no mass may be transferred in or out of the system boundaries. The system will always contain the same amount of matter, but heat and work can be exchanged across the boundary of the system. Whether a system can exchange heat, work, or both is dependent on the property of its boundary.

- Adiabatic boundary – not allowing any heat exchange
- Rigid boundary – not allowing exchange of work

One example is fluid being compressed by a piston in a cylinder. Another example of a closed system is a bomb calorimeter, a type of constant-volume calorimeter used in measuring the heat of combustion of a particular reaction. Electrical energy travels across the boundary to produce a spark between the electrodes and initiates combustion. Heat transfer occurs across the boundary after combustion but no mass transfer takes place either way.

Beginning with the first law of thermodynamics for an open system, this is expressed as:

where U is internal energy, Q is heat transfer, W is work, and since no mass is transferred in or out of the system, both expressions involving mass flow, , zeroes, and the first law of thermodynamics for a closed system is derived. The first law of thermodynamics for a closed system states that the amount of internal energy within the system equals the difference between the amount of heat added to or extracted from the system and the wok done by or to the system. The first law for closed systems is stated by:

- d
*U*= δ*Q*− δ*W*

where U is the average internal energy within the system, Q is the heat added to or extracted from the system and W is the work done by or to the system.

Substituting the amount of work needed to accomplish a reversible process, which is stated by:

- δ
*W*=*P*d*V*

where P is the measured pressure and V is the volume, and the heat required to accomplish a reversible process stated by the second law of thermodynamics, the universal principle of entropy, stated by:

- δ
*Q*=*T*d*S*

where T is the absolute temperature and S is the entropy of the system, derives the fundamental thermodynamic relationship used to compute changes in internal energy, which is expressed as:

- δ
*U*=*T*d*S*+*P*d*V*

The second law of thermodynamics is only true for closed systems. It states that the entropy of an isolated system not in equilibrium will tend to increase over time, approaching maximum value at equilibrium. Overall, in a closed system, the available energy can never increase, and it complement, entropy, can never decrease.

Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation.

In the case where the number of particles in the system is not necessarily constant and may be of different types, the first law is written:

where dNi is the (small) number of type-i particles added to the system, and μi is the amount of energy added to the system when one type-i particle is added, where the energy of that particle is such that the volume and entropy of the system remains unchanged. μi is known as the chemical potential of the type-i particles in the system. The statement of the first law, using exact differentials is now:

If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:

Here the Xi are the generalized forces corresponding to the external variables xi.

A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = − PdV. The pressure P can be viewed as a force (and in fact has units of force per unit area) while dV is the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred as a result of the process.

It is useful to view the TdS term in the same light: With respect to this heat term, a temperature difference forces a transfer of entropy, and the product of the two (heat) is the amount of energy transferred as a result of the process. Here, the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement.

Similarly, a difference in chemical potential between groups of particles in the system forces a transfer of particles, and the corresponding product is the amount of energy transferred as a result of the process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation which drives water molecules out of the liquid. There is a generalized "force" of condensation which drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium, and the net transfer will be zero.

An isolated system is a type of closed system that does not interact with its surroundings in any way. Mass and energy remains constant within the system, and no energy or mass transfer takes place across the boundary.

As an expansion of the first law of thermodynamics, enthalpy can be related to several other thermodynamic formulae. As with the original definition of the first law;

where, as defined by the law;

- dU represents the infinitesimal increase of the systematic or internal energy,
- δQ represents the infinitesimal amount of energy attributed or added to the system, and
- δW represents the infinitesimal amount of energy acted out by the system on the surroundings.

As a differential expression, the value of H can be defined as[3]

where

- δ represents the inexact differential,
- U is the internal energy,
- δQ = TdS is the energy added by heating during a reversible process,
- δW = pdV is the work done by the system in a reversible process,
- dS is the increase in entropy (joules per kelvin),
- p is the constant pressure,
- dV is an infinitesimal volume, and
- T is the temperature (kelvins).

For a process that is not reversible, the above equation expressing dH in terms of dS and dp still holds because H is a thermodynamic state variable that can be uniquely specified by S and p. We thus have in general:

- dH = TdS + Vdp

It is seen that, if a thermodynamic process is isobaric (i.e., occurs at constant pressure), then dp is zero and thus

- dH = TdS ≥ δQ

The difference in enthalpy is the maximum thermal energy attainable from the system in an isobaric process. This explains why it is sometimes called the heat content. That is, the integral of dH over any isobar in state space is the maximum thermal energy attainable from the system.

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 μi is the chemical potential for an i-type particle, and Ni is the number of such particles. It is seen that, not only must the Vdp term be set to zero by requiring the pressures of the initial and final states to be the same, but the μidNi terms must be zero as well, by requiring that the particle numbers remain unchanged. Any further generalization will add even more terms whose extensive differential term must be set to zero in order for the interpretation of the enthalpy to hold.

## References

- Moran, Michael J. and Howard N. Shapiro. Fundamentals of Engineering Thermodynamics. 6th ed. Wiley, 2008.
- http://en.wikipedia.org/wiki/First_law_of_thermodynamics - wikipedia.com
- http://en.wikipedia.org/wiki/Thermodynamic_system - wikipedia.com
- http://en.wikipedia.org/wiki/Enthalpy - wikipedia.com