Internal energy

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In physics, the internal energy (U) of a system is a reflection of the energy on a macroscopic scale. More specifically, it is the sum of:

the Internal kinetic energy, that is, of the sums of the kinetic energies of the individualities that form a body regarding the mass center of the system.
the internal potential, which is the potential energy associated with interactions between these individualities.

The internal energy does not include the translational or rotational kinetic energy of the system as a whole. Nor does it include the potential energy that the body may have due to its location in an external gravitational or electrostatic field.

Every body possesses a accumulated energy in its interior equivalent to internal kinetic energy plus internal potential energy.

If we think of atomic or molecular constituents, it will be the result of the sum of the kinetic energy of the molecules or atoms that make up the system (of their translation, rotation and vibration energies) and the intermolecular potential energy (due to intermolecular forces) and intramolecular binding energy.

In an ideal monoatomic gas it will suffice to consider the kinetic energy of translation of its atoms.
In a polyatomic ideal gas, we should also consider the vibrational and rotational energy of them.
In a liquid or solid we must add the potential energy represented by molecular interactions.

From the viewpoint of thermodynamics, in a closed system (i.e., with waterproof walls), the total variation of internal energy is equal to the sum of the amounts of energy communicated to the system in the form of heat and work ${displaystyle Delta U=W+Q}$ (thermodynamics consider positive work when it enters thermodynamic system, negative when it comes out). Although the heat transmitted depends on the process in question, the variation in internal energy is independent of the process, it only depends on the initial and final state, so it is said to be a state function. Same way ${displaystyle dU}$ is an exact differential, unlike ${displaystyle delta Q}$ which depends on the process...

The thermodynamic approach: the fundamental equation

In thermodynamics, the existence of an equation of the form of gravity is deduced

{displaystyle U=U(S,V,N)qquad }

known as the fundamental equation in energy representation, where S, V and N are the entropy, the volume and the amount of substance in moles, respectively.

Its importance lies in the fact that it concentrates all the thermodynamic information of a system in a single equation. Obtaining concrete results from it then becomes a systematic process.

If we calculate their differential:

{displaystyle dU=left({frac {partial U}{partial S}}right)dS+left({frac {partial U}{partial V}}right)dV+left({frac {partial U}{partial N}}right)dN}

their partial derivatives are defined:

temperature ${displaystyle T={frac {partial U}{partial S}}}$
Pressure ${displaystyle P=-{frac {partial U}{partial V}}}$
chemical potential ${displaystyle mu ={frac {partial U}{partial N}}}$ .

Like T, P and $mu$ are partial derivatives of U, will be functions of the same variables as U:

{displaystyle T=T(S,V,N)qquad P=P(S,V,N)qquad mu =mu (S,V,N)}

These relationships are called equations of state. In general, the fundamental equation of a system is not available. In that case, its substitution by the set of all the equations of state would provide equivalent information, although we often have to settle for a subset of them.

Some variations of internal energy

When the temperature of a system increases, its internal energy increases, reflected in the increase in thermal energy of the complete system, or of the matter studied.

Conventionally, when there is a change in internal energy manifested in the change in heat that can be transferred, maintained or absorbed, this change in internal energy can be measured indirectly by the change in the temperature of matter.

Variation without change of state

Without modifying the state of the matter that makes up the system, we speak of variation of sensible internal energy or sensible heat and it can be calculated according to the following parameters;

{displaystyle Q=C_{e}mDelta T!}

Where each term with its units in the International System are:

Q = is the change in energy or heat of the system in a defined time (J).

Ce = specific heat of matter in [J/(kg·K)].

m = mass.

${displaystyle Delta T!}$ = end system temperature - initial temperature (K).

Example

Calculate the total energy of a system composed of 1 g of water under normal conditions, that is to say at sea height, one atmosphere of pressure and at 14 °C to bring it to 15 °C, knowing that the Ce of the water is = 1 [cal/(g °C)].

Applying the formula ${displaystyle Q=C_{e}mDelta T!}$ and replacing the values, we have;

Q = 1 [cal/(g·°C)] · 1 [g] · (15 - 14) [°C] = 1 [cal]

Average kinetic energy of an ideal gas

${displaystyle E_{cm}={frac {3}{2}}(NKT)={frac {1}{2}}mNv_{m}^{2}!}$

K = Boltzmann constant = 1.38 10^-23 J/K

N = Number of molecules in the gas

${displaystyle v_{m}!}$ = Medium velocity of the molecule

The thermodynamic properties of an ideal gas can be described by two equations:

The equation of state of a classical ideal gas which is the ideal gas law

{displaystyle PV=nRT,}

and the internal energy at constant volume of an ideal gas that is determined by the expression:

{displaystyle U={hat {c}}_{V}nRT}

where

P It's the pressure.
V It's volume.
n is the amount of substance of a gas (in moles)
R is the constant gas (8.314 J·K⁻¹mol^-1)
T is absolute temperature
U It's the internal energy system
${displaystyle {hat {c}}_{V}}$ is the constant-volume specific heat, ≈ 3/2 for a monoatomic gas, 5/2 for a diatronomic gas and 3 for more complex molecules.

The amount of gas in J·K⁻¹ That's it. ${displaystyle nR=Nk_{B}}$ where

N is the number of gas particles
${displaystyle k_{B}}$ is the constant of Boltzmann (1,381×10⁻²³J·K⁻¹).

The probability distribution of particles by velocity or energy is determined by the Boltzmann distribution.

Variation with modification of the chemical composition

If there is an alteration of the atomic-molecular structure, as is the case of chemical reactions or a change of state, we speak of a change in the internal chemical energy or a change in the internal energy latent.

This state change condition can be calculated according to:

{displaystyle Q=C_{ce}m!}

Where ${displaystyle C_{ce}!}$ = Status change ratio, measured in [J/kg]

Nuclear variation

Main article: Equivalence between mass and energy

Finally, in fission and fusion reactions we speak of internal nuclear energy.

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