Ideal gas

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An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container. Ideal gas law calculations are favored at low pressures and high temperatures. Real gases existing in reality do not exhibit these exact properties, although the approximation is often good enough to describe real gases.

There are basically three types of ideal gas:

• the classical or Maxwell-Boltzmann ideal gas,
• the ideal quantum Bose gas, composed of bosons, and
• the ideal quantum Fermi gas, composed of fermions.

Contents 1 Classical ideal gas 2 Heat capacity 3 Entropy 4 Thermodynamic potentials 5 Speed of sound 6 Ideal quantum gases 7 See also

Classical ideal gas

The thermodynamic properties of an ideal gas can be described by two equations: The equation of state of a classical ideal gas is given by the ideal gas law.

<math>PV = nRT = NkT\,</math>

The internal energy of an ideal gas is given by:

<math>U = \hat{c}_V nRT = \hat{c}_V NkT</math>

where <math>\hat{c}_V</math> is a constant (e.g. equal to 3/2 for a monatomic gas) and: (with SI units appended)

• U is internal energy (joule)
• P is the pressure (pascal)
• V is the volume (cubic meter)
• n is the amount of gas (mole)
• R is the ideal gas constant (joule per kelvin per mole)
• T is the absolute temperature (kelvin)
• N is the number of particles
• k is the Boltzmann constant (joule per kelvin per particle),
• nR=Nk is the amount of gas (joule per kelvin).

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

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at low density and high temperature, approximate the behavior of a classical ideal gas. However, at lower temperature or higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or solid.

Heat capacity

The heat capacity at constant volume of an ideal gas is:

<math>C_V = \left(\frac{dU}{dT}\right)_V = \hat{c}_V Nk</math>

It is seen that the constant <math>\hat{c}_V</math> is just the dimensionless specific heat capacity at constant volume. It is equal to half the number of degrees of freedom per particle. For a monatomic gas this is just <math>\hat{c}_V=3/2</math> while for a diatomic gas it is <math>\hat{c}_V=5/2</math>. It is seen that macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules.

The heat capacity at constant pressure of an ideal gas is:

<math>C_P = \left(\frac{dH}{dT}\right)_P = (\hat{c}_V+1) Nk </math>

where <math>H=U+PV</math> is the enthalpy of the gas. It is seen that <math>\hat{c}_P</math> is also a constant and that the dimensionless heat capacities are related by:

<math>\hat{c}_P-\hat{c}_V=1 </math>

Entropy

Using the results of thermodynamics only, we can go a long way in determining the expression for the entropy of an ideal gas. This is an important step since, according to the theory of thermodynamic potentials, of which the internal energy U is one, if we can express the entropy as a function of U and the volume V, then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it.

Since the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S may be written as <math>\Delta S</math> where:

<math>\Delta S = \int_{S_0}^{S}dS

=\int_{T_0}^{T} \left(\frac{\partial S}{\partial T}\right)_V\!dT +\int_{V_0}^{V} \left(\frac{\partial S}{\partial V}\right)_T\!dV </math>

Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have:

<math>\Delta S

=\int_{T_0}^{T} \frac{C_v}{T}\,dT+\int_{V_0}^{V}\left(\frac{\partial P}{\partial T}\right)_VdV </math>

Using the expressions for an ideal gas and integrating yields:f

<math>\Delta S

= Nk\ln\left(\frac{VT^{\hat{c}_v}}{f(N)}\right) </math>

where all constants have been incorporated into the logarithm as f(N) which is some function of the particle number N having the same dimensions as <math>VT^{\hat{c}_v}</math> in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters (V and N) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:

<math>\Delta S(T,aV,aN)=a\Delta S(T,V,N)\,</math>

From this we find an equation for the function f(N)

<math>af(N)=f(aN)\,</math>

Differentiating this with respect to a, setting a equal to unity, and then solving the differential equation yields f(N):

<math>f(N)=\phi N\,</math>

where φ is some constant with the dimensions of <math>VT^{\hat{c}_v}/N</math>. Substituting into the equation for the change in entropy:

<math>\frac{\Delta S}{Nk} = \ln\left(\frac{VT^{\hat{c}_v}}{N\phi}\right)\,</math>

This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed - as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the third law of thermodynamics. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above equation is a good approximation only when the argument of the logarithm is much larger than unity - the concept of an ideal gas breaks down at low values of V/N. Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. It remained for quantum mechanics to introduce a reasonable value for the value of φ which yields the Sackur-Tetrode equation for the entropy of an ideal gas. It too suffers from a divergent entropy at absolute zero, but is a good approximation to an ideal gas over a large range of densities.

Thermodynamic potentials

Since the dimensionless heat capacity at constant pressure <math>\hat{c}_P</math> is a constant we can express the entropy in what will prove to be a more convenient form:

<math>\frac{S}{kN}=\ln\left( \frac{VT^{\hat{c}_V}}{N\Phi}\right)+\hat{c}_P</math>

where Φ is now the undetermined constant. The chemical potential of the ideal gas is calculated from the corresponding equation of state (see thermodynamic potential):

<math>\mu=\left(\frac{\partial G}{\partial N}\right)_{T,P}</math>

where G is the Gibbs free energy and is equal to <math>U+PV-TS</math> so that:

<math>\mu(T,V,N)=-kT\ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right)</math>

The thermodynamic potentials for an ideal gas can now be written as functions of T, V, and N as:

<math>U\,</math>		<math>=\hat{c}_V NkT\,</math>
<math>A=\,</math>	<math>U-TS\,</math>	<math>=\mu N-NkT\,</math>
<math>H=\,</math>	<math>U+PV\,</math>	<math>=\hat{c}_P NkT\,</math>
<math>G=\,</math>	<math>U+PV-TS\,</math>	<math>=\mu N\,</math>

The most informative way of writing the potentials is in terms of their natural variables, since each of these equations can be used to derive all of the other thermodynamic variables of the system. In terms of their natural variables, the thermodynamic potentials of a single-specie ideal gas are:

<math>U(S,V,N)=\hat{c}_V Nk\left(\frac{N\Phi\,e^{S/Nk-\hat{c}_P}}{V}\right)^{1/\hat{c}_V}</math>

<math>A(T,V,N)=-NkT\left(1+\ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right)\right)</math>

<math>H(S,P,N)=\hat{c}_P Nk\left(\frac{P\Phi\,e^{S/Nk-\hat{c}_P}}{k}\right)^{1/\hat{c}_P}</math>

<math>G(T,P,N)=-NkT\ln\left(\frac{kT^{\hat{c}_P}}{P\Phi}\right)</math>

Speed of sound

The speed of sound in an ideal gas is given by

<math>v_{sound} = \sqrt{\frac{\gamma R T}{M}} </math>

where

<math>\gamma \,</math> is the adiabatic index

<math>R \,</math> is the universal gas constant

<math>T \,</math> is the temperature

<math>M \,</math> is the molar mass for the gas (in kg/mol)

Ideal quantum gases

In the above mentioned Sackur-Tetrode equation, the best choice of the entropy constant was found to be proportional to the quantum thermal wavelength of a particle, and the point at which the argument of the logarithm becomes zero is roughly equal to the point at which the average distance between particles becomes equal to the thermal wavelength. In fact, quantum theory itself predicts the same thing. Any gas behaves as an ideal gas at high enough temperature and low enough density, but at the point where the Sackur-Tetrode equation begins to break down, the gas will begin to behave as a quantum gas, composed of either bosons or fermions. Just as there is a classical ideal gas, there are ideal quantum gases. An ideal gas of bosons will be governed by Bose-Einstein statistics and the distribution of energy will be in the form of a Bose-Einstein distribution. An ideal gas of fermions will be governed by Fermi-Dirac statistics and the distribution of energy will be in the form of a Fermi-Dirac distribution.

See also

• Ideal gas lawcs:Ideální plyn

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