Internal Energy

An Internal Energy is a energy measure associated with the random motion of microscopic particles of matter (e.g. molecules, atoms, ions)

• Context:
  • It can generally defined as [math]\displaystyle{ dU = T dS - P dV + \sum_i \mu_i N_i\, }[/math] . The first two terms are derived directly from the First and Second Law of Termodynamics (dU,T,P,dS,dV are the internal energy change, temperature, pressure,entropy change and Volume change, respectively) while the last term is the Gibbs Free Energy expressed as the sum of all the chemical potentials ([math]\displaystyle{ \mu_i }[/math]) of the system composed by N of microspcopic particles.
• Example(s):
  • Gibbs Free Energy.
• Counter-Example(s):
  • Pressure.
  • Entropy.
  • Density.
  • Heat.
• See: Thermodynamics, Heat, Temperature Scales, Baoltzmann Constant, Kinetic Energy, Gas Pressure, Ideal Gas Law, Maxwell–Boltzmann probability distribution, Enthalpy, Helmholtz free energy, and Gibbs free energy.

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References

2015

• (Wikipedia, 2015) ⇒ https://www.wikiwand.com/en/Internal_energy
  • 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.

The internal energy of a system can be changed by transfers of matter and by work and heat transfer. 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. Then 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.

2005

• (Wolfram Science world , 2005) ⇒ http://scienceworld.wolfram.com/physics/InternalEnergy.html
  • The first law of thermodynamics states that [math]\displaystyle{ dE=dQ-dW }[/math]where dQ is the heat added to system, dW is the work done by system, and dE is the energy change of the system. Rewriting,

[math]\displaystyle{ dE=TdS-PdV=T\left(\frac{C_V}{T}dT + \alpha K_TdV\right)-PdV=C_VdT+(\alpha T K_T-P)dV=\left(\frac{\partial E}{\partial T}\right)_V+\left(\frac{\partial E}{\partial V}\right)_T }[/math]

where T is the temperature, dS is the entropy change, P is the pressure, dV is the volume change, [math]\displaystyle{ C_V }[/math]is the heat capacity at constant volume, [math]\displaystyle{ k_T }[/math] is the isothermal bulk modulus. Therefore,

[math]\displaystyle{ C_V=\left(\frac{\partial E}{\partial T}\right)_V\quad,\quad \alpha T K_T-P=\left(\frac{\partial E}{\partial V}\right)_T }[/math]

Including chemical potential energy,

[math]\displaystyle{ dU = T dS - P dV + \sum_i \mu_i N_i\, }[/math]

where [math]\displaystyle{ \mu_i }[/math]is the chemical potential energy for species i.

2005

• (Hyperphysics Encyclopedia, 2005) ⇒ http://hyperphysics.phy-astr.gsu.edu/hbase/force.html#fordef
  • Internal energy is defined as the energy associated with the random, disordered motion of molecules. It is separated in scale from the macroscopic ordered energy associated with moving objects; it refers to the invisible microscopic energy on the atomic and molecular scale. For example, a room temperature glass of water sitting on a table has no apparent energy, either potential or kinetic . But on the microscopic scale it is a seething mass of high speed molecules traveling at hundreds of meters per second. If the water were tossed across the room, this microscopic energy would not necessarily be changed when we superimpose an ordered large scale motion on the water as a whole.

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