Thermodynamically Explained: Why Lowering Operating Pressure Lowers Boiling-Point Temperature
Keywords: #Basics #Pressure #Pressure drop #Distillation #Thermodynamics
This article offers a very quick, simple, but rigorous proof of something that is absolutely intuitive to anyone who has worked on distillation: if the operating pressure of a distillation column is reduced, the column’s temperatures at the top and the bottom are guaranteed to decrease. This phenomenon is a thermodynamic necessity and is taken for granted in most engineering training sessions. Both engineers and operators view it as self-evident, because in everyday experiences, we know that using a pressure cooker raises water’s boiling temperature, so the content steamed inside is cooked above the normal water boiling point of 100 °C. Also, mountain climbers camping at high altitudes, say 4,000 meters above sea level, will have their ambient pressure reduced to about 61% of 1 atm (Digital Dutch, n.d.) and consequently can only boil water at about 86 °C. A distillation tower is in its essence a “boiling” operation and therefore follows the same trend. Nevertheless, a simple thermodynamic proof on pure substances shows that this phenomenon has the status of a physical truth. In the case of mixtures, a more involved proof is included in the appendix for interested readers.
For a pure component, the boiling point temperature T is mathematically tied to the vapor pressure Psat by the Clapeyron equation (Sandler, 2006):
— (eq. 1)
Where，
— Pa; the saturation pressure of a pure substance. In the case of liquid-vapor equilibrium or phase change, it refers to the vapor pressure at the boiling point;
— K; the boiling point temperature of a pure substance;
— J/mol; heat of vaporization;
— m³/mol；specific volume change for the vaporization, in other words, the specific volume difference between vapor and liquid: 
。
A quick recall of the basics:
- The Clapeyron equation for pure substances is derived directly from the Gibbs phase equilibrium criterion, as demonstrated by Sandler (2006). No simplifications or approximations are used in its formulation; therefore, it is an exact thermodynamic relation.
- The derivative dy/dx shows the direction of change of y with respect to x: it is positive when y increases with x and negative when y decreases. Consequently, in the Clapeyron equation the sign of dPsat/dT indicates whether the saturation pressure rises or falls as temperature increases.
In the Clapeyron equation (eq. 1), the molar heat of vaporization,
, is always positive, as heat is always applied to boil liquid into vapor. The specific volume difference between vapor and liquid during the vaporization process is also always positive, as vapor occupies a larger volume per mole than liquid does before the critical point (distillation cannot operate at or beyond the critical point). Additionally, in the absolute temperature scale of Kelvin, the temperature is always positive. Hence, the ratio on the right-hand side of the Clapeyron differential equation is always positive. In other words,
dPsat / dT > 0 and dT / dPsat > 0 for all Psat and T
This means thermodynamically, for all substances, higher pressures always lead to higher boiling point temperatures, and vice versa.
The monotonicity of the vapor-pressure-to-boiling-point-temperature function in the case of water is illustrated in Figure 1.
In actual distillation practices, the author has noticed that even though process engineers know that lowering operating pressure lowers the sump temperature, sometimes, they are not alerted to the fact that the boiling point reduction is more pronounced in vacuum towers than in higher-pressure towers. An earlier Bernoulli-chem article has explained this by evaluating the ratio of (dT/dPsat)/P (Bernoulli-Chem, 2026). An example is that for the same pressure reduction of 1 bar, the boiling point of water is reduced by almost 100 °C at around 1 atm, while only a less-than-10 °C-reduction is achieved at around 5 bara (Figure 1). In vacuum towers, since the top pressure is only tens of kPa, the pressure-drop across the column becomes significant in the resulting bottom pressure and therefore, the sump temperature.
Figure 1 The saturated vapor pressure vs the boiling point temperature for water. Data retrieved from the NIST chemistry WebBook (Linstrom & Mallard, n.d.). From the plot it is easy to see that lowering the operating pressure by 1 bar at 1 bara — that is, from atmospheric pressure down toward vacuum — lowers the boiling point by 100 °C, while lowering the operating pressure by 1 bar at 5 bara only reduces the boiling point temperature by less than 10 °C.
Appendix 1: The Case of Mixtures
The Clapeyron equation is good only for pure substances. For a mixture as is common in a distillation column, the Gibbs phase rule indicates that there is no longer a single boiling point. Instead, for every saturation pressure, a range of temperature values exists from the first bubble formed (called the bubble point) to the last liquid droplet evaporated (called the dew point) over the liquid-to-vapor phase transition. For a distillation column, the sump liquid is at the bubble point, as the liquid is in equilibrium with the vapor. Conversely, the vapor at the top before condensation is at the dew point, as it is in equilibrium with the liquid. All the stages (trayed or packed) between the column top and bottom are simultaneously at both the bubble point for the liquid phase and the dew point for the vapor phase for each temperature, under the classical phase-equilibrium distillation model. Similar to the boiling point of a pure substance, reducing the operating pressure also lowers the bubble points (and the dew points) of the mixture across the column stages, as can be proven thermodynamically:
For an open system such as a distillation column, streams of different compositions enter and exit the system , thus changing the compositions in either liquid or vapor. This compositional change for an open system will be taken into account by the composition set 
for each component. The proof centers on the Gibbs energy, whose mathematical beauty lies in that its natural variables are strictly in temperature (T), pressure (P), and composition (
）.The engineeringly less favored parameters such as entropy (S) and volume (V) are no longer the independent variables but are actually functions of T, P, and the composition (which are favored because they can be practically measured via field instruments).
Inside a distillation column, when the operation reaches equilibrium, resulting in stable T, P, and the compositionsin both the liquid and vapor phases at any column height or stage, the molar Gibbs energy at any instant may be expressed as a function of G = f (T, P, ).The multivariable differential with respect to T, P, and each component i’s molar fraction
may be taken one by one while holding others constant:
— (Eq. A1.1)
Where,
G — J/mol; the molar Gibbs energy at any stage in a distillation column at any moment. The notation of underbar means “per mole”;
P — Pa; the operating pressure at the stage in the distillation column. It is also the vapor pressure for a distillation column reaching phase equilibrium;
T — K; the temperature at the stage in the distillation column;
— -; the collection of molar fractions of all the components 
. at the stage at any moment in the distillation column. Usually, the molar fraction notation 
is used to describe the liquid phase, and
for the vapor phase;
k, i, j — -; subscripts denoting a certain component;
C —-; the number of components in the mixture;。
The seemingly complicated third term in eq. A1.1(
), , simply represents taking the partial derivative of G with respect to each molar fraction
, while holding T, P, and all other component j’s constant. Then, these partial derivatives are added across all C components.
Eq. A1.1 is the fundamental thermodynamic equation of the total Gibbs energy. By working from the1st law and the 2nd law of thermodynamics, it may be proven that
(Sandler, 2006). Thus, the fundamental Gibbs equation becomes:
— (eq. A1.2)
— J/(K·mol); the molar entropy at any stage in the distillation column;
— m³/mol; the molar volume at any stage in the distillation column;
To express each component i’s share of the thermodynamic property, the partial molar identity is defined as:
— (ID. A1.1)
In a way similar to a weighted average, in calculus and by the intensive nature of thermodynamics, adding all components’ partial molar properties weighted by their respective molar fractions returns the total molar property of the system:
— (ID. A1.2)
Therefore, for each component i, replacing the in Eq. A1.2 with Ḡi, the fundamental Gibbs function may be written as1：
— (Eq. A1.3)
A separate counter k is employed in Eq. A1.3, because i here is no longer the summation counter but the targeted species.
Using the equilibrium condition 
(Appendix 2) on Eq. A1.3 gives:
— (Eq. A1.4)
Note that the vapor-phase Gibbs energy equation is not expanded because at bubble point, no simple treatment of the summation of 
is available. The mathematical trick is to relate the fundamental Gibbs energy equation of the liquid phase to the Gibbs-Duhem equation of the vapor phase (Appendix 3). Therefore, based on Eq. A1.4, multiplying all sides of the equations by the vapor-phase mole fraction,
Therefore, based on Eq. A1.4, multiplying all sides of the equations by the vapor-phase mole fraction,
, of component i at the bubble point condition gives:
— (Eq. A1.5)
Take the summation over all components on Eq. A1.5 to create the term, 
, , which is the left-hand side of the Gibbs-Duhem expression:
— (Eq. A1.6)
While the last double summation term in Eq. A1.6 looks formidable, the proof targets the question of whether lowering P lowers T for any given bubble point composition. In other words, the composition is constant. In a distillation column, this may be interpreted as each stage having a composition specification. For example, a product specification of 0.1 mol% light key (or 99.9 mol% purity for the heavy key in a binary mixture) for the column bottom. As the tower pressure changes, operation-wise, the sump temperature is supposedly adjusted to keep the old mixture composition. Not doing so risks either reduced product purity (more light components going into the sump liquid) or increased product loss (more product losing into the vapor phase). A fixed composition
, therefore, leads to 
= 0, Consequently, the double summation 
nets to 0. The resulting equation is:
; At a given bubble point composition 
— (Eq. A1.7)
The vapor-phase Gibbs-Duhem equation is provided as (Appendix 3):
; at — (Eq. A1.8)
Therefore, equating Eqs. A1.7 and A1.8:
; at — (Eq. A1.9)
Rearranging the equation yields:
— (Eq. A1.10)
Eq. A1.10 is the rigorous bubble-point-temperature-to-vapor-pressure equation for any mixture.
The ratio in Eq. A1.10 needs to be examined for positive or negative values, so as to determine the sign of dT/dP. The difficulty lies in the interpretation of
and 
. At first glance, the summation of partial molar liquid-phase properties (
and 
) multiplied by the corresponding vapor mole fractions (
) means nothing thermodynamically. However, Eq. A1.10 can be rewritten by using
and 
(ID. A1.2):
— (Eq. A1.11)
and 
suggest the volume and the entropy differences when one mole of the equilibrium vapor is taken from the coexisting liquid. Mathematically, they represent the vaporization properties of the transferring mass:
— (ID. A1.3)
— (ID. A1.4)
Equation A1.11 is further derived as
— (Eq. A1.12)
Since below the critical point, vapor is the lower-density phase, the molar volume change is always positive during the vaporization process, or
＞0 . Similarly, since heat is physically required during the vaporization process to liberate vapor molecules from the liquid matrix, by
, whereis the heat of vaporization, 
> 0. Therefore, dT/dP and dP/dT are always positive.
The concept of the temperature change with respect to reduced pressure while holding the bubble point composition constant is illustrated in Figure 2.
The argument for dew point is analogous. As a result, the dew points and bubble points across all stages or heights of the distillation column drop if the operating pressure (and the pressure drop) is reduced.
Appendix 2: Equilibrium Conditions
At phase equilibrium, the 2nd law of thermodynamics mandates that the temperature, pressure, and the partial molar Gibbs energy (called the chemical potential) of each species must be equal. Otherwise, heat transfers, material flows, or molecular diffusion will happen to balance the two phases until reaching equilibrium. Or mathematically:
, for all equilibria — (Eq. A2.1)
Where,
V, L : -; the superscripts denote whether the parameter specifically refers to the vapor or the liquid phase;
Since these equalities hold true along the VLE curve, a simple limiting condition may be used to show that:
, for all equilibria — (Eq. A2.2)
Appendix 3: The Gibbs-Duhem Equation
For any homogeneous system, the molar Gibbs energy is a thermodynamic potential expressed in temperature (T), pressure (P), and the mole fractions 
. From the 1st law and the 2nd law of thermodynamics, the fundamental Gibbs energy equation is (Sandler, 2006):
— (Eq. A3.1)
Where,
G — J/mol; the molar Gibbs energy of the system. The underbar notation means “per mole”;
P — Pa; the pressure of the system;
T — K; the temperature of the system;
z — – ; the collection of molar fractions of all the components {Zi} of the system;
S — J/(K·mol) ; the molar entropy of the system;
V — m³/mol; the molar volume of the system;
Ḡi — J/mol; the partial molar Gibbs energy of component i. Defined as the partial differential of the molar Gibbs energy with respect to species i’s molar fraction 
, while holding T, P, and other components 
constant, or
. The industry commonly calls it the chemical potential with the notation 
.
Besides the fundamental Gibbs energy equation, the molar Gibbs energy may also be expressed as the summed partial molar Gibbs energy for each component i weighted by the mole fraction :
— (Eq. A3.2)
The general product rule holds true in calculus for any differentiable functions f(x) and g(x) with respect to any independent variable x:
— (Eq. A3.3)
Applying the differential chain rule to the molar Gibbs energy of Eq. A3.2:
— (Eq. A3.4)
Equating the Gibbs energy differentials in Eq. A3.4 and Eq. A3.1:
Cancelling out 
on both sides of the equation gives
— (Eq. A3.5), or
— (Eq. A3.6)
Equations A3.5 and A3.6 are the different forms of the generalized Gibbs-Duhem equations.
注解：
1 The reason 
may be written in a fundamental Gibbs energy equation as the molar Gibbs energy,
, may be proven by calculus. This article skips the math work.
References
[1] Bernoulli-Chem. (2026). Clausius-Clapeyron tells you that your column’s pressure drop truly matters. 苏州伯尔努利化工科技有限公司. https://bernoulli-chem.com/en/clausius-clapeyron-tells-you-that-your-columns-pressure-drop-truly-matters/.
[2] Digital Dutch. (n.d.). 1976 standard atmosphere calculator. Retrieved August 24, 2026, from https://www.digitaldutch.com/atmoscalc/
[3] Sandler, S. I. (2006). Chemical, biochemical, and engineering thermodynamics (4th ed.). John Wiley & Sons.
[4] Linstrom, P. J., & Mallard, W. G. (Eds.). (n.d.). NIST Chemistry WebBook (NIST Standard Reference Database Number 69). National Institute of Standards and Technology. https://doi.org/10.18434/T4D303