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Amount of water vapor
Description 
When the volume variation in phase change ($\Delta V$) changes phase from a liquid to a gas, it can be expressed as:
$\Delta V = V_{\text{gas}} - V_{\text{liquid}}$
Since the volume of the gas is significantly greater than that of the liquid,
$V_{\text{gas}} \gg V_{\text{liquid}}$
we can approximate:
$\Delta V \approx V_{\text{gas}}$
Given that water vapor behaves similarly to an ideal gas, we can state that with the values of the universal gas constant ($R_C$), the number of moles ($n$), the absolute temperature ($T$), and the water vapor pressure unsaturated ($p_v$):
| $ p V = n R_C T $ |
Therefore, the volume variation in phase change ($\Delta V$) is:
$\Delta V = \displaystyle\frac{nRT}{p_v}$
ID:(3185, 0)
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(ID 1617)
Using the Clausius-Clapeyron equation for the gradient of the pressure ($p$) with respect to the absolute temperature ($T$), which depends on the latent Heat ($L$) and the volume variation in phase change ($\Delta V$):
| $\displaystyle\frac{ dp }{ dT }=\displaystyle\frac{ L }{ \Delta V T }$ |
In the case of the phase change from liquid to gas, we can assume that the change in volume is approximately equal to the volume of the vapor. Therefore, we can employ the ideal gas equation with the number of moles ($n$), the volume ($V$), the universal gas constant ($R_C$), and the water vapor pressure unsaturated ($p_v$):
| $$ |
Since the Clausius-Clapeyron equation can be written as:
$\displaystyle\frac{dp}{dT}=\displaystyle\frac{L}{n}\displaystyle\frac{p}{R T^2}$
Where the molar Latent Heat ($l_m$) ($l_m = L/n$) corresponds to the change in enthalpy during the phase change h (the energy required to form water), we finally have:
$\displaystyle\frac{dp}{dT}=l_m\displaystyle\frac{p}{RT^2}$
If we integrate this equation between the pressure saturated water vapor ($p_s$) and the pressure at point
$p_s=p_0e^{l_m/RT_0}e^{-l_m/RT}$
If we evaluate this expression with the data at the critical point:
$p_{ref}=p_0e^{l_m/RT_0}$
We finally have:
| $ p_s = p_{ref} e^{- l_m / R_C T }$ |
(ID 3182)
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_C T $ |
(ID 3183)
(ID 3882)
The relationship between the relative humidity ($RH$) with the concentration of water vapor molecules ($c_v$) and ERROR:4952,0 is expressed as:
| $ RH =\displaystyle\frac{ c_v }{ c_s }$ |
and by relating the pressure ($p$) with the molar concentration ($c_m$), the absolute temperature ($T$), and the universal gas constant ($R_C$), we obtain:
| $ p = c_m R_C T $ |
This applies to the vapor pressure of water, where:
$p_v = c_v R T$
and the saturated vapor pressure of water:
$p_s = c_s R T$
resulting in the following equation:
| $ RH =\displaystyle\frac{ p_v }{ p_s }$ |
(ID 4478)
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 ($R_C$), the ideal gas equation:
| $ p V = n R_C 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_C T $ |
(ID 4479)
Examples
(ID 1617)
vapor003
(ID 3003)
vapor004
(ID 3004)
The difference in concentration $c_1$ and $c_2$ at the ends of the membrane results in the difference:
| $dc=c_2-c_1$ |
(ID 3882)
In 1855, Adolf Fick [1] formulated an equation for the calculation of the diffusion Constant ($D$), resulting in the particle flux density ($j$) due to the concentration variation ($dc_n$) along ERROR:10192,0:
| $ j =- D \displaystyle\frac{ dc_n }{ dz }$ |
[1] " ber Diffusion" (On Diffusion), Adolf Fick, Annalen der Physik und Chemie, Volume 170, pages 59-86 (1855)
(ID 4820)
The relative humidity ($RH$) can be expressed in terms of the water vapor pressure unsaturated ($p_v$) and the pressure saturated water vapor ($p_s$) as follows:
| $ RH =\displaystyle\frac{ p_v }{ p_s }$ |
(ID 4478)
Calculating the particle flux density ($j$) in one dimension involves utilizing the values the diffusion Constant ($D$), the particle concentration ($c_n$), and the position along an axis ($z$), as dictated by Fick's law [1]::
| $ j =- D \displaystyle\frac{ dc_n }{ dz }$ |
This formula can be generalized for more than one dimension as follows:
| $ \vec{j} =- D \nabla c_n $ |
[1] " ber Diffusion" (On Diffusion), Adolf Fick, Annalen der Physik und Chemie, Volume 170, pages 59-86 (1855)
(ID 4821)
The diffusion constant $D$ can be calculated from the average velocity $\bar{v}$ and the mean free path $\bar{l}$ of the particles.
| $ D =\displaystyle\frac{1}{3} \bar{v} \bar{l} $ |
It is important to recognize that both the mean free path and the average velocity depend on temperature, and consequently, so does the diffusion constant. Therefore, when values for the so-called constant are published, the temperature to which it applies is always specified.
(ID 3186)
vapor001
(ID 3001)
vapor005
(ID 3066)
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_C T $ |
where the universal gas constant ($R_C$) has a value of 8.314 J/K mol.
(ID 3183)
When the volume variation in phase change ($\Delta V$) changes phase from a liquid to a gas, it can be expressed as:
$\Delta V = V_{\text{gas}} - V_{\text{liquid}}$
Since the volume of the gas is significantly greater than that of the liquid,
$V_{\text{gas}} \gg V_{\text{liquid}}$
we can approximate:
$\Delta V \approx V_{\text{gas}}$
Given that water vapor behaves similarly to an ideal gas, we can state that with the values of the universal gas constant ($R_C$), the number of moles ($n$), the absolute temperature ($T$), and the water vapor pressure unsaturated ($p_v$):
| $ p V = n R_C T $ |
Therefore, the volume variation in phase change ($\Delta V$) is:
$\Delta V = \displaystyle\frac{nRT}{p_v}$
(ID 3185)
The pressure ($p$) can be calculated from the molar concentration ($c_m$) using the absolute temperature ($T$), and the universal gas constant ($R_C$) as follows:
| $ p = c_m R_C T $ |
(ID 4479)
The pressure saturated water vapor ($p_s$) can be calculated using the reference pressure ($p_{ref}$), the universal gas constant ($R_C$), the absolute temperature ($T$) and the molar Latent Heat ($l_m$) according to the following formula:
| $ p_s = p_{ref} e^{- l_m / R_C T }$ |
(ID 3182)
ID:(507, 0)