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General Law of Ideal Gases
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The three gas laws (Boyle's Law, Charles's Law, Gay-Lussac's Law) and Avogadro's principle can be combined into a single law called the ideal gas law.
This allows predicting the variation of one of the parameters that define the state of the gas (the pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$)) for an ideal gas, based on the initial state and any final state defined by the remaining three variables.
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Mechanisms
Iframe
The universal gas law, also known as the ideal gas law, describes the relationship between the pressure, volume, temperature, and number of moles of a gas. It combines several gas laws, including Boyle's law, Charles's law, and Avogadro's principle, into a single equation. This law states that the product of the pressure and volume of a gas is directly proportional to the product of its temperature and the number of moles of gas. The ideal gas law assumes that gases are composed of a large number of molecules that are in constant, random motion and that the interactions between these molecules are negligible. This law is fundamental in predicting the behavior of gases under various conditions and is widely used in both scientific research and practical applications, such as engineering and chemistry.
Mechanisms
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Gas Laws
Concept
The state of a system is described by the so-called equation of state, which establishes the relationship between the parameters that characterize the system.
In the case of gases, the parameters that describe their state are the pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$). Typically, the latter parameter remains constant as it is associated with the amount of gas present.
The equation of state, therefore, relates pressure, volume, and temperature, and it establishes that there are only two degrees of freedom, as the equation of state allows for the calculation of the third parameter. In particular, if the volume is fixed, one can choose, for example, temperature as the variable, which enables the calculation of the corresponding pressure.
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Integrating the gas laws
Description
The three gas laws that relate to the pressure ($p$), the volume ($V$), and the absolute temperature ($T$) are:
• Boyle's Law, which states that at constant temperature, the product of the pressure and the volume of a gas is constant:
| $ p V = C_b $ |
• Charles's Law, which states that at constant pressure, the volume of a gas is directly proportional to its absolute temperature:
| $\displaystyle\frac{ V }{ T } = C_c$ |
• Gay-Lussac's Law, which states that at constant volume, the pressure of a gas is directly proportional to its absolute temperature:
| $\displaystyle\frac{ p }{ T } = C_g$ |
These laws can be graphically represented as shown in the following image:
In 1834, Émile Clapeyron [1] recognized that the pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by Boyle's law, Charles's law, Gay-Lussac's law, and Avogadro's law. These laws can be expressed more generally as:
$\displaystyle\frac{pV}{nT} = \text{constant}$
This general relationship states that the product of pressure and volume divided by the number of moles and the temperature remains constant:
| $ p V = n R T $ |
In this equation, the universal gas constant ($R$) assumes the value of 8.314 J/K·mol.
[1] "Mémoire sur la puissance motrice de la chaleur" (Memoir on the Motive Power of Heat), Émile Clapeyron, Journal de l'École Polytechnique, 1834.
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Change of state of an ideal gas according to the general gas law
Concept
The ideal gas law is expressed as
| $ p V = n R T $ |
and can be written as
$\displaystyle\frac{pV}{nT} = R$
This implies that the initial and final conditions must satisfy the equality
$\displaystyle\frac{p_iV_i}{n_iT_i} = R = \displaystyle\frac{p_fV_f}{n_fT_f}$
Thus, we obtain the following equation:
| $\displaystyle\frac{ p_i V_i }{ n_i T_i }=\displaystyle\frac{ p_f V_f }{ n_f T_f }$ |
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Pressure as a function of molar concentration
Concept
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
Concept
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
Concept
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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Model
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Parameters
Variables
Calculations
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, then, select the variable: 
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Calculations
Calculations
Variable 
Given Calculate 
Target : Equation To be used 
Equations
$ p_i V_i = M_i R_s T_i $
p * V = M * R_s * T
$ p_f V_f = M_f R_s T_f $
p * V = M * R_s * T
$ p_i V_i = n_i R T_i $
p * V = n * R * T
$ p_f V_f = n_f R T_f $
p * V = n * R * T
$ p_i = c_i R T_i $
p = c_m * R * T
$ p_f = c_f R T_f $
p = c_m * R * T
$ p_i = \rho_i R_s T_i $
p = rho * R_s * T
$ p_f = \rho_f R_s T_f $
p = rho * R_s * T
$\displaystyle\frac{ p_i V_i }{ n_i T_i }=\displaystyle\frac{ p_f V_f }{ n_f T_f }$
p_i * V_i /( n_i * T_i )= p_f * V_f /( n_f * T_f )
$ \rho_i \equiv\displaystyle\frac{ M_i }{ V_i }$
rho = M / V
$ \rho_f \equiv\displaystyle\frac{ M_f }{ V_f }$
rho = M / V
$ R_s \equiv \displaystyle\frac{ R }{ M_m }$
R_s = R / M_m
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General gas law (1)
Equation
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:
| $ p_i V_i = n_i R T_i $ |
| $ 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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General gas law (2)
Equation
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:
| $ p_f V_f = n_f R T_f $ |
| $ 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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Change of state of an ideal gas according to the general gas law
Equation
For an initial state (the pressure in initial state ($p_i$), the volume in state i ($V_i$), the temperature in initial state ($T_i$), and the number of moles in state i ($n_i$)) and a final state (the pressure in final state ($p_f$), the volume in state f ($V_f$), the temperature in final state ($T_f$), and the number of moles in state f ($n_f$)), the following holds true:
| $\displaystyle\frac{ p_i V_i }{ n_i T_i }=\displaystyle\frac{ p_f V_f }{ n_f T_f }$ |
The ideal gas law is expressed as
| $ p V = n R T $ |
and can be written as
$\displaystyle\frac{pV}{nT} = R$
This implies that the initial and final conditions must satisfy the equality
$\displaystyle\frac{p_iV_i}{n_iT_i} = R = \displaystyle\frac{p_fV_f}{n_fT_f}$
Thus, we obtain the following equation:
| $\displaystyle\frac{ p_i V_i }{ n_i T_i }=\displaystyle\frac{ p_f V_f }{ n_f T_f }$ |
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Pressure as a function of molar concentration (1)
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_i = c_i R T_i $ |
| $ 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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Pressure as a function of molar concentration (2)
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_f = c_f R T_f $ |
| $ 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 (1)
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_i V_i = M_i R_s T_i $ |
| $ 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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Specific gas law (2)
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_f V_f = M_f R_s T_f $ |
| $ 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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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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Pressure as a function of density (1)
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_i = \rho_i R_s T_i $ |
| $ 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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Pressure as a function of density (2)
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_f = \rho_f R_s T_f $ |
| $ 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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Mass and Density (1)
Equation
The density ($\rho$) is defined as the ratio between the mass ($M$) and the volume ($V$), expressed as:
| $ \rho_i \equiv\displaystyle\frac{ M_i }{ V_i }$ |
| $ \rho \equiv\displaystyle\frac{ M }{ V }$ |
This property is specific to the material in question.
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Mass and Density (2)
Equation
The density ($\rho$) is defined as the ratio between the mass ($M$) and the volume ($V$), expressed as:
| $ \rho_f \equiv\displaystyle\frac{ M_f }{ V_f }$ |
| $ \rho \equiv\displaystyle\frac{ M }{ V }$ |
This property is specific to the material in question.
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