First law of thermodynamics

The first law of thermodynamics is an expression of the universal law of conservation of energy, and identifies heat as a form of energy. The most common enunciation of first law of thermodynamics is:

History

Main article: mechanical equivalent of heat

The first explicit statement of the first law of thermodynamics was given by Rudolf Clausius in 1850: "There is a state function E, called ‘energy’, whose differential equals the work exchanged with the surroundings during an adiabatic process."

Mathematical formulation

The mathematical statement of the first law is given by:

<math>\mathrm{d}U=\delta Q-\delta W\,</math>

where <math>\mathrm{d}U</math> is the infinitesimal increase in the internal energy of the system, <math>\delta Q</math> is the infinitesimal amount of heat added to the system, and <math>\delta W</math> is the infinitesimal amount of work done by the system on the surroundings. The infinitesimal heat and work are denoted by δ rather than d because, in mathematical terms, they are inexact differentials rather than exact differentials. In other words, they do not describe the state of any system.

The integral of an inexact differential is path dependent, i.e. it depends upon the particular "path" taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through state space is known as a thermodynamic process.

Reversible processes

An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is equal to its pressure times the infinitesimal change in its volume. In other words, <math>\delta W=p\mathrm{d}V</math> where <math>p</math> is pressure and <math>V</math> is volume. For a reversible process, the total amount of heat added to a system can be expressed as <math>\delta Q=T\mathrm{d}S</math> where <math>T</math> is temperature and <math>S</math> is entropy. For a reversible process, the first law may now be restated:

<math>\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V\,</math>

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:

<math>\mathrm{d}U = \delta Q - \delta W + \sum_i \mu_i \mathrm{d}N_i\,</math>

where <math>\mathrm{d}N_i</math> is the (small) number of type-i particles added to the system, and <math>\mu_i</math> 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. <math>\mu_i</math> is known as the chemical potential of the type-i particles in the system. The statement of the first law for reversible processes, using exact differentials is now:

<math>\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V + \sum_i \mu_i \mathrm{d}N_i\,</math>

Force-functions

A useful idea, introduced by Willard Gibbs in 1876, is that quantities such as internal energy U and Helmholtz free energy A may be regarded as a kind of force-function. For example, 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: <math>\mathrm{d}U=p\mathrm{d}V</math>. The pressure p can be viewed as a force (and in fact has units of force per unit area) while <math>\mathrm{d}V</math> 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 <math>T\mathrm{d}S</math> 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.

The two thermodynamic parameters which form a generalized force-displacement pair are termed "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and temperature-entropy.

Application for 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:

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

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 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:

<math>\delta W=\mathrm{d}(P_{out}V_{out})-\mathrm{d}(P_{in}V_{in})+\delta W_{shaft}\,</math>

Substitution into the equation above for the control volume cv yields:

<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>

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:

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

During steady-state operation of a device (see turbine, pump, and engine), 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>

This expression is described by the diagram above.

Sign convention

Physics and Chemistry

In physics and chemistry, the system is the object of greatest interest, and it is natural to talk about the work done on the system by the surroundings. This changes the sign of the equation. Defined in this manner, the first law is a generalization of this concept which states for a thermodynamic cycle that the net heat input is equal to the net work output. For a system with a fixed number of particles (closed system), the first law is stated as:

<math>\mathrm{d}U=\delta Q+\delta W\,</math>,

where

<math>\mathrm{d}U</math> is an infinitesimal increase in the internal energy of the system,

<math>\delta Q</math> is an infinitesimal amount of heat added to the system,

<math>\delta W</math> is an infinitesimal amount of work done on the system, and

<math>\delta</math> denotes an inexact differential.

Thermodynamics and Engineering

In thermodynamics and engineering, it is natural to think of the system as a heat engine which does work on the surroundings, and to state that the total energy added by heating is equal to the sum of the increase in internal energy plus the work done by the system. Hence <math>\delta W</math> is the amount of energy lost by the system due to work done by the system on its surroundings. During the portion of the thermodynamic cycle where the engine is doing work, <math>\delta W</math> is positive, but there will always be a portion of the cycle where <math>\delta W</math> is negative, e.g., when the working gas is being compressed. When <math>\delta W</math> represents the work done by the system, the first law is written:

<math>\mathrm{d}U=\delta Q-\delta W\,</math>

Very occasionally, the sign on the heat may be inverted, so that <math>\delta Q</math> is the flow of heat out of the system:

<math>\mathrm{d}U=-\delta Q+\delta W\,</math>

Because of this ambiguity, it is vitally important in any discussion involving the first law to explicitly establish the sign convention in use. See also: The Absent-Minded Professor.

See also

• Conservation of energy
• Laws of thermodynamics
• Perpetual motion

References

• Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. A gentle introduction.

External links

• 30+ Variations of the 1st Law
• Mechanical Theory of Heat – Nine Memoirs by Rudolf Clausius [1850-1865] on the 1st and 2nd Laws of Thermodynamics.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "First law of thermodynamics".