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Speed Distribution
Storyboard 
In a gas that is at rest the molecules travel in all directions being the average speed equal to zero. As the molecules have different speeds they have to be represented with a distribution that in this case must have a null average value. Analyzing it is concluded that the shape of the distribution corresponds to a Gaussean with a dispersion that is a function of temperature.
ID:(1494, 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)
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)
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)
Kinetic Theory
Description
How fluctuations in the concentration of a gas lead to movements that tend to achieve a homogeneous distribution.
ID:(588, 0)
Mass and Density
Equation
The density ($\rho$) is defined as the ratio between the mass ($M$) and the volume ($V$), expressed as:
| $ \rho \equiv\displaystyle\frac{ M }{ V }$ |
This property is specific to the material in question.
ID:(3704, 0)
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Equations
$ 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
$ m =\displaystyle\frac{ M_m }{ N_A }$
m = M_m / N_A
$ n = \displaystyle\frac{ M }{ M_m }$
n = M / M_m
$ n \equiv\displaystyle\frac{ N }{ N_A }$
n = N / N_A
$ n =\displaystyle\frac{ V }{ V_m }$
n = V / V_m
$ \rho \equiv\displaystyle\frac{ M }{ V }$
rho = M / V
$ \bar{v} =\sqrt{\displaystyle\frac{ f k_B T }{ m }}$
v =sqrt( f * k_B * T / m )
ID:(15351, 0)
Number of moles
Equation
The number of moles ($n$) corresponds to the number of particles ($N$) divided by the avogadro's number ($N_A$):
| $ n \equiv\displaystyle\frac{ N }{ N_A }$ |
ID:(3748, 0)
Number of moles with molar mass
Equation
The number of moles ($n$) is determined by dividing the mass ($M$) of a substance by its the molar Mass ($M_m$), which corresponds to the weight of one mole of the substance.
Therefore, the following relationship can be established:
| $ n = \displaystyle\frac{ M }{ M_m }$ |
The number of moles ($n$) corresponds to the number of particles ($N$) divided by the avogadro's number ($N_A$):
| $ n \equiv\displaystyle\frac{ N }{ N_A }$ |
If we multiply both the numerator and the denominator by the particle mass ($m$), we obtain:
$n=\displaystyle\frac{N}{N_A}=\displaystyle\frac{Nm}{N_Am}=\displaystyle\frac{M}{M_m}$
So it is:
| $ n = \displaystyle\frac{ M }{ M_m }$ |
The molar mass is expressed in grams per mole (g/mol).
ID:(4854, 0)
Number of moles with molar volume
Equation
The number of moles ($n$) is determined by dividing the volume ($V$) of a substance by its the molar volume ($V_m$), which corresponds to the weight of one mole of the substance.
Therefore, the following relationship can be established:
| $ n =\displaystyle\frac{ V }{ V_m }$ |
The molar volume is expressed in cubic meters per mole ($m^3/mol$).
It's important to note that the molar volume depends on the pressure and temperature conditions under which the substance exists, especially in the case of a gas, so it is defined considering specific conditions.
ID:(15147, 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)