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Number of Moles
Storyboard 
In general, the ideal gas laws depend on the number of particles and not on the type of particles. This is because, due to not considering interaction between the particles (ideal gas), their specific physical properties do not play a role. However, the number of particles in a volume of a few liters of gas is so large ($10^{23}$) that it is complex to work with this type of number. Therefore, a more convenient scale has been defined by working with the so-called moles corresponding to $6.02\times 10^{23}$ particles.
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Ideal gas
Description
A gas in which its particles do not interact is known as an ideal gas. We can envision it as follows:
• It consists of a series of spheres contained within a container ($$).
• The speed of these particles depends on the absolute temperature ($T$).
• They generate a pressure of the pressure ($p$) through collisions with the walls of the container.
An ideal gas is characterized by the absence of potential energies between the particles. In other words, the potential energies that could exist between particles $i$ and $j$ with positions $q_i$ and $q_j$ are null:
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The moles
Concept
By utilizing the concept of a mole, we can directly relate the amount of substance of a gas to the number of the number of particles ($N$) particles present in it. This simplifies calculations and allows for a more intuitive connection between the quantity of gas and its defining properties, such as the pressure ($p$), the volume ($V$), and the absolute temperature ($T$).
The constant the avogadro's number ($N_A$), which is approximately equal to $6.02\times 10^{23}$, is a fundamental constant in chemistry and is used to bridge the gap between the macroscopic and microscopic scales of atoms and molecules.
The value of the número de Moles ($n$) can be calculated from the number of particles ($N$) and the mass ($M$). In the first case, it is obtained by dividing by avogadro's Number ($N_A$) using the formula:
| $ n \equiv\displaystyle\frac{ N }{ N_A }$ |
While in the second case, the molar Mass ($M_m$) is used with the formula:
| $ n = \displaystyle\frac{ M }{ M_m }$ |
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The mass of a particle
Concept
You can generally calculate the particle mass ($m$) with the mass ($M$) and the number of particles ($N$) using:
| $ m \equiv \displaystyle\frac{ M }{ N }$ |
or with the molar Mass ($M_m$) and the avogadro's number ($N_A$) using:
| $ m =\displaystyle\frac{ M_m }{ N_A }$ |
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The concentration of particles and moles
Concept
The concentration of the particle concentration ($c_n$) is defined in terms of the number of particles ($N$) and the volume ($V$) by:
| $ c_n \equiv \displaystyle\frac{ N }{ V }$ |
or using the density ($\rho$) and the particle mass ($m$) by:
| $ c_n =\displaystyle\frac{ \rho }{ m }$ |
The the molar concentration ($c_m$) is defined in terms of número de Moles ($n$) and the volume ($V$) by:
| $ c_m \equiv\displaystyle\frac{ n }{ V }$ |
or using the density ($\rho$) and the molar Mass ($M_m$) by:
| $ c_m =\displaystyle\frac{ \rho }{ M_m }$ |
The relationship between both concentrations is the avogadro's number ($N_A$) by:
| $ c_n = N_A c_m $ |
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Ideal gas equations
Concept
The gas equations in general relate to the pressure ($p$), the volume ($V$), the absolute temperature ($T$), the universal gas constant ($R$), and some measure of quantity.
This measure can be generic using Dalton's law, where only the number of particles matters, not their type.
For this purpose, there is the version that works with número de Moles ($n$):
| $ p V = n R T $ |
and the molar concentration ($c_m$):
| $ p = c_m R T $ |
On the other hand, if working with the type of molecules, one should use the specific gas constant ($R_s$) instead of the universal gas constant ($R$):
| $ R_s \equiv \displaystyle\frac{ R }{ M_m }$ |
and calculate the quantity using the mass ($M$):
| $ p V = M R_s T $ |
or the density ($\rho$):
| $ p = \rho R_s T $ |
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Gas mixture
Description
In the case of an ideal gas, where there is no interaction between particles, a mixture of different types of gases will behave as if it were a larger quantity of the same type of gas.
Specifically, if we have three components with their respective partial pressures, when they are mixed, the total pressure will be the sum of the partial pressures:
This image illustrates how the partial pressures of gases add up in a mixture. Each gas exerts an independent pressure and contributes to the total pressure of the mixture.
This concept is fundamental in understanding the behavior of gas mixtures, as it allows us to calculate the total pressure based on the partial pressures of the individual components.
According to Dalton's Law [1], the total pressure of a gas mixture is equal to the sum of the individual pressures of the gases, where a pressure ($p$) is equal to the sum of the partial pressure of component $i$ ($p_i$). This leads us to conclude that the gas behaves as if the particles of the different gases were identical. In this way, the pressure ($p$) is the sum of the partial pressure of component $i$ ($p_i$):
| $ p =\displaystyle\sum_i p_i $ |
Therefore, it can be concluded that the gas behaves as if the different gases were identical and the number of moles corresponds to the sum of the moles of the different components:
| $ n =\displaystyle\sum_i n_i $ |
[1] "Experimental Essays on the Constitution of Mixed Gases; on the Force of Steam or Vapour from Water and Other Liquids in Different Temperatures, Both in a Torricellian Vacuum and in Air; on Evaporation; and on the Expansion of Gases by Heat", John Dalton, Memoirs of the Literary and Philosophical Society of Manchester, Volume 5, Issue 2, Pages 535-602 (1802).
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Model
Top
Parameters
Variables
Calculations
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to 
Calculations
Calculations
Variable 
Given Calculate 
Target : Equation To be used 
Equations
$ c_m \equiv\displaystyle\frac{ n }{ V }$
c_m = n / V
$ c_m =\displaystyle\frac{ \rho }{ M_m }$
c_m = rho / M_m
$ c_n \equiv \displaystyle\frac{ N }{ V }$
c_n = N / V
$ c_n = N_A c_m $
c_n = N_A * c_m
$ c_n =\displaystyle\frac{ \rho }{ m }$
c_n = rho / m
$ m \equiv \displaystyle\frac{ M }{ N }$
m = M / N
$ 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\sum_i n_i $
n =@SUM( n_i , i )
$ p V = M R_s T $
p * V = M * R_s * T
$ p V = n R T $
p * V = n * R * T
$ p = c_m R T $
p = c_m * R * T
$ p = \rho R_s T $
p = rho * R_s * T
$ p =\displaystyle\sum_i p_i $
p =@SUM( p_i , i )
$ \rho \equiv\displaystyle\frac{ M }{ V }$
rho = M / V
$ R_s \equiv \displaystyle\frac{ R }{ M_m }$
R_s = R / M_m
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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 }$ |
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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).
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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 }$ |
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Particle mass
Equation
If you divide the mass ($M$) by the number of particles ($N$), you get the particle mass ($m$):
| $ m \equiv \displaystyle\frac{ M }{ N }$ |
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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.
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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 }$ |
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Particle concentration
Equation
The particle concentration ($c_n$) is defined as the number of particles ($N$) divided by the volume ($V$):
| $ c_n \equiv \displaystyle\frac{ N }{ V }$ |
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Molar concentration
Equation
The molar concentration ($c_m$) corresponds to number of moles ($n$) divided by the volume ($V$) of a gas and is calculated as follows:
| $ c_m \equiv\displaystyle\frac{ n }{ V }$ |
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Particle and mole concentration
Equation
The molar concentration ($c_m$) can be calculated from the density ($\rho$) and the molar Mass ($M_m$) as follows:
| $ c_m =\displaystyle\frac{ \rho }{ M_m }$ |
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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 $ |
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Gas specific constant
Equation
When working with the specific data of a gas, the specific gas constant ($R_s$) can be defined in terms of the universal gas constant ($R$) and the molar Mass ($M_m$) as follows:
| $ R_s \equiv \displaystyle\frac{ R }{ M_m }$ |
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General gas law
Equation
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:
| $ p V = n R T $ |
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related through the following physical laws:
• Boyle's law
| $ p V = C_b $ |
• Charles's law
| $\displaystyle\frac{ V }{ T } = C_c$ |
• Gay-Lussac's law
| $\displaystyle\frac{ p }{ T } = C_g$ |
• Avogadro's law
| $\displaystyle\frac{ n }{ V } = C_a $ |
These laws can be expressed in a more general form as:
$\displaystyle\frac{pV}{nT}=cte$
This general relationship states that the product of pressure and volume divided by the number of moles and temperature remains constant:
| $ p V = n R T $ |
where the universal gas constant ($R$) has a value of 8.314 J/K·mol.
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Pressure as a function of molar concentration
Equation
The pressure ($p$) can be calculated from the molar concentration ($c_m$) using the absolute temperature ($T$), and the universal gas constant ($R$) as follows:
| $ p = c_m R T $ |
When the pressure ($p$) behaves as an ideal gas, satisfying the volume ($V$), the number of moles ($n$), the absolute temperature ($T$), and the universal gas constant ($R$), the ideal gas equation:
| $ p V = n R T $ |
and the definition of the molar concentration ($c_m$):
| $ c_m \equiv\displaystyle\frac{ n }{ V }$ |
lead to the following relationship:
| $ p = c_m R T $ |
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Specific gas law
Equation
The pressure ($p$) is related to the mass ($M$) with the volume ($V$), the specific gas constant ($R_s$), and the absolute temperature ($T$) through:
| $ p V = M R_s T $ |
The pressure ($p$) is associated with the volume ($V$), número de Moles ($n$), the absolute temperature ($T$), and the universal gas constant ($R$) through the equation:
| $ p V = n R T $ |
Since número de Moles ($n$) can be calculated with the mass ($M$) and the molar Mass ($M_m$) using:
| $ n = \displaystyle\frac{ M }{ M_m }$ |
and obtained with the definition of the specific gas constant ($R_s$) using:
| $ R_s \equiv \displaystyle\frac{ R }{ M_m }$ |
we conclude that:
| $ p V = M R_s T $ |
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Pressure as a function of density
Equation
If we work with the mass or the density ($\rho$) of the gas, we can establish an equation analogous to that of ideal gases for the pressure ($p$) and the absolute temperature ($T$), with the only difference being that the constant will be specific to each type of gas and denoted as the specific gas constant ($R_s$):
| $ p = \rho R_s T $ |
If we introduce the gas equation written with the pressure ($p$), the volume ($V$), the mass ($M$), the specific gas constant ($R_s$), and the absolute temperature ($T$) as:
| $ p V = M R_s T $ |
and use the definition the density ($\rho$) given by:
| $ \rho \equiv\displaystyle\frac{ M }{ V }$ |
we can derive a specific equation for gases as follows:
| $ p = \rho R_s T $ |
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Sum of partial pressures
Equation
The pressure ($p$) is the sum of the partial pressure of component $i$ ($p_i$):
| $ p =\displaystyle\sum_i p_i $ |
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Sum of moles
Equation
The number of moles ($n$) equals the sum of the number of moles of i component ($n_i$):
| $ n =\displaystyle\sum_i n_i $ |
In the case of Dalton's Law, we have that the pressure ($p$) is the sum of the partial pressure of component $i$ ($p_i$):
| $ p =\displaystyle\sum_i p_i $ |
Each component of the mixture satisfies the ideal gas equation with the pressure ($p$), the volume ($V$), the number of moles ($n$), the absolute temperature ($T$), and the universal gas constant ($R$):
| $ p V = n R T $ |
Therefore, the mixture also adheres to the same law, where the number of moles ($n$) equals the sum of the number of moles of i component ($n_i$):
| $ n =\displaystyle\sum_i n_i $ |
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