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.

>Model

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Iframe

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Description

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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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Concept

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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:

While in the second case, the molar Mass ($M_m$) is used with the formula:

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Concept

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You can generally calculate the particle mass ($m$) with the mass ($M$) and the number of particles ($N$) using:

or with the molar Mass ($M_m$) and the avogadro's number ($N_A$) using:

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Concept

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The concentration of the particle concentration ($c_n$) is defined in terms of the number of particles ($N$) and the volume ($V$) by:

or using the density ($\rho$) and the particle mass ($m$) by:

The the molar concentration ($c_m$) is defined in terms of número de Moles ($n$) and the volume ($V$) by:

or using the density ($\rho$) and the molar Mass ($M_m$) by:

The relationship between both concentrations is the avogadro's number ($N_A$) by:

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Concept

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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 ($n$):

and the molar concentration ($c_m$):

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 ($R$):

and calculate the quantity using the mass ($M$):

or the density ($\rho$):

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Description

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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 $i$ ($p_i$):

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:

[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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Top

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Parameters

$N_A$

N_A

Avogadro's number

-

$\rho$

rho

Density

kg/m^3

$M$

M

Mass

kg

$M_m$

M_m

Molar Mass

kg/mol

$n$

n

Número de Moles

mol

$m$

m

Particle mass

kg

$R_s$

R_s

Specific gas constant

J/kg K

$R$

R

Universal gas constant

J/mol K

Variables

$T$

T

Absolute temperature

K

$c_m$

c_m

Molar concentration

mol/m^3

$n_i$

n_i

Number of moles of i component

mol

$N$

N

Number of particles

-

$p_i$

p_i

Partial pressure of component $i$

Pa

$c_n$

c_n

Particle concentration

1/m^3

$p$

p

Pressure

Pa

$n$

n

Total number of moles

mol

$p$

p

Total pressure of all components

Pa

$V$

V

Volume

m^3

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15318, 0)

Equation

>Top, >Model

The number of moles ($n$) corresponds to the number of particles ($N$) divided by the avogadro's number ($N_A$):

$n \equiv\displaystyle\frac{ N }{ N_A }$

Conditions

Variables

$N_A$

Avogadro's number

6.02e+23

$-$

9860

$N$

Number of particles

$-$

6080

$n$

Número de Moles

$mol$

6679

Relation

ID:(3748, 0)

Equation

>Top, >Model

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:

$n = \displaystyle\frac{ M }{ M_m }$

Conditions

Variables

$M$

Mass

$kg$

5183

$M_m$

Molar Mass

$kg/mol$

6212

$n$

Número de Moles

$mol$

6679

Relation

Development

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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Equation

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The particle mass ($m$) can be estimated from the molar Mass ($M_m$) and the avogadro's number ($N_A$) using

$m =\displaystyle\frac{ M_m }{ N_A }$

Conditions

Variables

$N_A$

Avogadro's number

6.02e+23

$-$

9860

$M_m$

Molar Mass

$kg/mol$

6212

$m$

Particle mass

$kg$

5516

Relation

ID:(4389, 0)

Equation

>Top, >Model

If you divide the mass ($M$) by the number of particles ($N$), you get the particle mass ($m$):

$m \equiv \displaystyle\frac{ M }{ N }$

Conditions

Variables

$M$

Mass

$kg$

5183

$N$

Number of particles

$-$

6080

$m$

Particle mass

$kg$

5516

Relation

ID:(12829, 0)

Equation

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The density ($\rho$) is defined as the ratio between the mass ($M$) and the volume ($V$), expressed as:

$\rho \equiv\displaystyle\frac{ M }{ V }$

Conditions

Variables

$\rho$

Density

$kg/m^3$

5342

$M$

Mass

$kg$

5183

$V$

Volume

$m^3$

5226

Relation

This property is specific to the material in question.

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Equation

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If we divide the density ($\rho$) by the particle mass ($m$), we will obtain the particle concentration ($c_n$):

$c_n =\displaystyle\frac{ \rho }{ m }$

Conditions

Variables

$\rho$

Density

$kg/m^3$

5342

$c_n$

Particle concentration

$1/m^3$

5548

$m$

Particle mass

$kg$

5516

Relation

Development

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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Equation

>Top, >Model

The particle concentration ($c_n$) is defined as the number of particles ($N$) divided by the volume ($V$):

$c_n \equiv \displaystyle\frac{ N }{ V }$

Conditions

Variables

$N$

Number of particles

$-$

6080

$c_n$

Particle concentration

$1/m^3$

5548

$V$

Volume

$m^3$

5226

Relation

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Equation

>Top, >Model

The molar concentration ($c_m$) corresponds to number of moles ($n$) divided by the volume ($V$) of a gas and is calculated as follows:

$c_m \equiv\displaystyle\frac{ n }{ V }$

Conditions

Variables

$c_m$

Molar concentration

$mol/m^3$

6609

$n$

Número de Moles

$mol$

6679

$V$

Volume

$m^3$

5226

Relation

ID:(4878, 0)

Equation

>Top, >Model

The molar concentration ($c_m$) can be calculated from the density ($\rho$) and the molar Mass ($M_m$) as follows:

$c_m =\displaystyle\frac{ \rho }{ M_m }$

Conditions

Variables

$\rho$

Density

$kg/m^3$

5342

$c_m$

Molar concentration

$mol/m^3$

6609

$M_m$

Molar Mass

$kg/mol$

6212

Relation

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Equation

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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:

$c_n = N_A c_m$

Conditions

Variables

$N_A$

Avogadro's number

6.02e+23

$-$

9860

$c_m$

Molar concentration

$mol/m^3$

6609

$c_n$

Particle concentration

$1/m^3$

5548

Relation

ID:(10624, 0)

Equation

>Top, >Model

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:

$R_s \equiv \displaystyle\frac{ R }{ M_m }$

Conditions

Variables

$M_m$

Molar Mass

$kg/mol$

6212

$R_s$

Specific gas constant

$J/kg K$

7832

$R$

Universal gas constant

8.4135

$J/mol K$

4957

Relation

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Equation

>Top, >Model

The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:

$p V = n R T$

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$n$

Número de Moles

$mol$

6679

$p$

Pressure

$Pa$

5224

$R$

Universal gas constant

8.4135

$J/mol K$

4957

$V$

Volume

$m^3$

5226

Relation

Development

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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Equation

>Top, >Model

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:

$p = c_m R T$

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$c_m$

Molar concentration

$mol/m^3$

6609

$p$

Pressure

$Pa$

5224

$R$

Universal gas constant

8.4135

$J/mol K$

4957

Relation

Development

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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Equation

>Top, >Model

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:

$p V = M R_s T$

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$M$

Mass

$kg$

5183

$p$

Pressure

$Pa$

5224

$R_s$

Specific gas constant

$J/kg K$

7832

$V$

Volume

$m^3$

5226

Relation

Development

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:

$p V = M R_s T$

ID:(8831, 0)

Equation

>Top, >Model

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 ($R_s$):

$p = \rho R_s T$

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$\rho$

Density

$kg/m^3$

5342

$p$

Pressure

$Pa$

5224

$R_m$

Specific gas constant

$J/kg K$

7832

Relation

Development

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:

$p = \rho R_s T$

ID:(8833, 0)

Equation

>Top, >Model

The pressure ($p$) is the sum of the partial pressure of component $i$ ($p_i$):

$p =\displaystyle\sum_i p_i$

Conditions

Variables

$p_i$

Partial pressure of component $i$

$Pa$

10225

$p$

Total pressure of all components

$Pa$

10373

Relation

ID:(15361, 0)

Equation

>Top, >Model

The number of moles ($n$) equals the sum of the number of moles of i component ($n_i$):

$n =\displaystyle\sum_i n_i$

Conditions

Variables

$n_i$

Number of moles of i component

$mol$

9333

$n$

Total number of moles

$mol$

9334

Relation

Development

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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