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Concentration based on molar mass
Equation 
If we divide the density ($\rho$) by the particle mass ($m$), we will obtain the particle concentration ($c_n$):
 $ c_n =\displaystyle\frac{ \rho }{ m }$ |
Given the particle concentration ($c_n$) with the number of particles ($N$) and the volume ($V$), we have:
| $ c_n \equiv \displaystyle\frac{ N }{ V }$ |
With the particle mass ($m$) and the mass ($M$),
| $ m \equiv \displaystyle\frac{ M }{ N }$ |
As the density ($\rho$) is
| $ \rho \equiv\displaystyle\frac{ M }{ V }$ |
we obtain
$c_n=\displaystyle\frac{N}{V}=\displaystyle\frac{M}{mV}=\displaystyle\frac{\rho}{m}$
Therefore,
| $ c_n =\displaystyle\frac{ \rho }{ m }$ |
ID:(10623, 0)
Energy as a function of degrees of freedom
Equation
The Stefan-Boltzmann law, initially proposed by Josef Stefan [1] and later refined by Ludwig Boltzmann [2], states that the energy of a molecule ($E$) is proportional to the degrees of freedom ($f$) multiplied by the absolute temperature ($T$) with a proportionality constant of the boltzmann Constant ($k_B$):
| $ E =\displaystyle\frac{ f }{2} k_B T $ |
It is important to note that the absolute temperature ($T$) must be expressed in degrees Kelvin.
The number of degrees of freedom of a particle corresponds to the number of variables required to describe its thermodynamic state. For instance, a point particle requires only three coordinates, resulting in three degrees of freedom. If the particle has shape and rigidity, two additional angles are needed, leading to a total of five degrees of freedom. When the particle can deform or vibrate in one or more directions, these additional modes are also considered as additional degrees of freedom. However, it is important to note that these extra degrees of freedom exist only at high temperatures when the particle has enough energy to activate such vibrations.
[1] "Über die Beziehung zwischen der Wärmestrahlung und der Temperatur" (On the Relationship Between Heat Radiation and Temperature), Josef Stefan, Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, Wien (1879).
[2] "Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen" (Further Studies on Thermal Equilibrium among Gas Molecules), Ludwig Boltzmann, Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, Wien (1884).
ID:(4387, 0)
Free Path of a Molecule
Description
When a molecule moves periodically through the volume containing the gas, it will eventually encounter another molecule and they may collide. The distance it travels between two consecutive collisions is called the 'mean free path'.
ID:(114, 0)
Free path with mole concentration
Equation
Since the diameter of the particle $d$ is twice the radius $a$
$d=2a$
and the particle concentration $c_N$ can be expressed in terms of molar concentration $c_n$ as
$c_N=N_Ac_n$
where $N_A$ is Avogadro's number, the equation for mean free path
$l=\displaystyle\frac{1}{4a^2\pi c_N}$
can also be written as:
| $l=\displaystyle\frac{1}{d^2\pi c N_A}$ |
ID:(4477, 0)
Free path with particle concentration
Equation
The mean free path can be estimated in terms of the diameter of an imaginary cylinder surrounding a particle, on average having one collision with another particle.
The radius of the cylinder corresponds to the maximum distance two particles must have to collide, which is equal to twice the radius of the particle, i.e., the particle diameter ($d$). Since only one collision occurs within this cylinder, the number of particles contained within it must be equal to one. This means that:
$l d^2\pi c_n= 1$
with the particle concentration ($c_n$), and solving for the free path ($\bar{l}$), we obtain:
| $ l =\displaystyle\frac{1}{ d ^2 \pi c_n }$ |
This represents the mean free path.
ID:(4392, 0)
Kinetic energy of the particle
Equation
The kinetic energy ($K$) combined with the particle mass ($m$) and the average speed of a particle ($\bar{v}$) equals
| $ K =\displaystyle\frac{ m }{2} \bar{v} ^2$ |
Note: In strict rigor, kinetic energy depends on the average velocity squared $\bar{v^2}$. However, it is assumed to be approximately equal to the square of the average velocity:
$\bar{v^2}\sim\bar{v}^2$
ID:(4390, 0)
Microscopic viscosity model
Equation
If the force
| $F=-Slc_nm\displaystyle\frac{dv_x}{dt}$ |
where
$F=-S,l,c_nm\displaystyle\frac{dv_x}{dz}\displaystyle\frac{dz}{dt}$
The derivative of the
$\displaystyle\frac{dz}{dt}=v_z=\displaystyle\frac{1}{3}\sqrt{\langle v^2\rangle}$
In this way the force created by the mixture of moments like
$F=-\displaystyle\frac{1}{3}S,l,c_nm\sqrt{\langle v^2\rangle}\displaystyle\frac{dv_x}{dz}$
If you compare this expression with the viscous force
$F=-S,\eta\displaystyle\frac{dv_x}{dz}$
it is concluded that the viscosity has to be
| $\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}$ |
where the negative sign is because the force is opposite to the direction of flow.
ID:(3945, 0)
Model
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Equations
$ c_n = N_A c_m $
c_n = N_A * c_m
$ c_n =\displaystyle\frac{ \rho }{ m }$
c_n = rho / m
$ E =\displaystyle\frac{ f }{2} k_B T $
E = f * k_B * T /2
$ K =\displaystyle\frac{ m }{2} \bar{v} ^2$
K = m * v ^2/2
$ l =\displaystyle\frac{1}{ d ^2 \pi c_n }$
l = 1/( d ^2 * pi * c )
$ m =\displaystyle\frac{ M_m }{ N_A }$
m = M_m / N_A
$ \bar{v} =\sqrt{\displaystyle\frac{ f k_B T }{ m }}$
v =sqrt( f * k_B * T / m )
$F=-Slc_nm\displaystyle\frac{dv_x}{dt}$
F=-Slc_nm\displaystyle\frac{dv_x}{dt}
$l=\displaystyle\frac{1}{d^2\pi c N_A}$
l = 1 / ( d^2 * pi * c_n * N_A )
$\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}$
\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}
$\eta=\displaystyle\frac{1}{6\pi d^2}\sqrt{fmkT}$
\eta=\displaystyle\frac{1}{6\pi d^2}\sqrt{fmkT}
ID:(15354, 0)
Particle and mole concentration
Equation
To convert the molar concentration ($c_m$) to the particle concentration ($c_n$), simply multiply the former by the avogadro's number ($N_A$) as follows:
| $ c_n = N_A c_m $ |
ID:(10624, 0)
Speed of a Molecule
Description
En un gas la molecula se mueven por el espacio en gran medida en forma independiente del las restantes moleculas.
En primera aproximación se puede describir la interacción como meros choques.
En una segunda aproximación se puee tomar en cuenta eventuales fuerzas atractivas entre moleculas. Estas en particular se haran mas visibles en bordes del sistema por existir una asimetria de la distribución de particulas. En el medio del volumen la molecula estan expuesta a fuerzas de todos lados lo que en promedio tiende a anularse. En el borde solo existen vecinas hacia el inetrior del contenedor por lo que puede exitir una fuerza que frena a la particula que va a impactar la pared. Esto corresponde a la reducción de la presón que se modela con la ecuación de Van der Waals.
ID:(113, 0)
Viscosity as a function of temperature
Equation
If the viscosity is
| $\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}$ |
with
$l=\displaystyle\frac{1}{\sqrt{2}\pi d^2c_n}$
the viscosity as a function of the temperature will be:
| $\eta=\displaystyle\frac{1}{6\pi d^2}\sqrt{fmkT}$ |
where the negative sign is because the force is opposite to the direction of flow.
ID:(3946, 0)
Viscosity as momentum exchange
Equation
If the speed at a point
$mdv_x = m(v_x(z + dz) - v_x(z))$
The number of particles participating in this process is equal to those found in a volume of section
$S l c_n$
Therefore, the force
$F=\displaystyle\frac{dp}{dt}$
so the slimy force is
| $F=-Slc_nm\displaystyle\frac{dv_x}{dt}$ |
where the negative sign is because the force is opposite to the direction of flow.
ID:(3944, 0)
Particle mass and molar mass
Equation
The particle mass ($m$) can be estimated from the molar Mass ($M_m$) and the avogadro's number ($N_A$) using
| $ m =\displaystyle\frac{ M_m }{ N_A }$ |
ID:(4389, 0)
Average speed of particles
Equation
Como la energía cinética de la molécula es
| $ K =\displaystyle\frac{ m }{2} \bar{v} ^2$ |
y la energía en función de la temperatura es
| $ E =\displaystyle\frac{ f }{2} k_B T $ |
con
| $ \bar{v} =\sqrt{\displaystyle\frac{ f k_B T }{ m }}$ |
ID:(4391, 0)