Relative Volatility

Core idea

Overview

Relative volatility (a_ij) is a dimensionless measure of the difference in volatility between two components (i and j) in a vapor-liquid equilibrium system. It indicates how easily component i can be separated from component j by distillation. A higher relative volatility implies an easier separation, as component i will preferentially vaporize compared to component j. It is defined as the ratio of the K-values (vapor-liquid equilibrium ratios) of the two components.

When to use: This equation is essential in chemical engineering for designing and analyzing separation processes, particularly distillation. It's applied when determining the feasibility and efficiency of separating a mixture into its components based on their boiling points. A relative volatility significantly greater than 1 indicates a practical separation, while a value close to 1 suggests a difficult or impossible separation by simple distillation.

Why it matters: Relative volatility is a cornerstone of distillation column design, which is critical in industries like petroleum refining, chemical manufacturing, and pharmaceuticals. It directly impacts the number of stages required in a distillation column, energy consumption, and overall process economics. Understanding and manipulating relative volatility allows engineers to optimize separation processes and produce high-purity products efficiently.

Symbols

Variables

$α_{ij}$ = Relative Volatility, $y_{i}$ = Vapor Mole Fraction (Component i), $x_{i}$ = Liquid Mole Fraction (Component i), $y_{j}$ = Vapor Mole Fraction (Component j), $x_{j}$ = Liquid Mole Fraction (Component j)

α_{ij}

Relative Volatility

dimensionless

y_{i}

Vapor Mole Fraction (Component i)

dimensionless

x_{i}

Liquid Mole Fraction (Component i)

dimensionless

y_{j}

Vapor Mole Fraction (Component j)

dimensionless

x_{j}

Liquid Mole Fraction (Component j)

dimensionless

K_{i}

K-value (Component i)

dimensionless

K_{j}

K-value (Component j)

dimensionless

Walkthrough

Derivation

Formula: Relative Volatility

Relative volatility is defined as the ratio of the K-values (vapor-liquid equilibrium ratios) of two components in a mixture.

The system is at vapor-liquid equilibrium.
Mole fractions are used for both liquid and vapor phases.
The K-values (y/x) are well-defined for both components.

1

Define K-value for a Component:

The K-value (vapor-liquid equilibrium ratio) for component 'i' is defined as the ratio of its mole fraction in the vapor phase ( $y_{i}$ ) to its mole fraction in the liquid phase ( $x_{i}$ ). It indicates the component's tendency to vaporize.

K_{i} = \frac{y _{i}}{x _{i}}

2

Define Relative Volatility:

Relative volatility (α_ij) is defined as the ratio of the K-value of component 'i' to the K-value of component 'j'. This ratio quantifies how much more volatile component 'i' is compared to component 'j'.

α_{ij} = \frac{K _{i}}{K _{j}}

3

Substitute K-value Definitions:

By substituting the definition of K-values into the relative volatility equation, we obtain the formula in terms of vapor and liquid mole fractions for both components. This form is often used when K-values are not directly known but mole fractions are.

α_{ij} = \frac{y _{i} / x _{i}}{y _{j} / x _{j}}

Result

α_{ij} = \frac{y _{i} / x _{i}}{y _{j} / x _{j}}

Source: Smith, J.M., Van Ness, H.C., & Abbott, M.M. (2018). Introduction to Chemical Engineering Thermodynamics. McGraw-Hill Education.

Visual intuition

Graph

The graph displays an inverse curve where alpha decreases rapidly as the liquid mole fraction of component i increases. For an engineering student, this shape illustrates that as the concentration of component i becomes dominant in the liquid phase, the relative volatility between the two components diminishes, making separation more difficult. The most important feature of this curve is that it never reaches zero, meaning that even at high concentrations, a degree of relative volatility is always maintained between the components.

Graph type: inverse

Why it behaves this way

Intuition

Imagine two types of molecules, i and j, at a liquid-vapor interface; relative volatility quantifies how much more component i 'jumps' into the vapor phase compared to component j.

α_{ij}

A dimensionless measure of the difference in volatility between component i and component j

A higher value means component i is much more likely to be in the vapor phase than component j, making separation easier.

y_{i} / x_{i}

The K-value (vapor-liquid equilibrium ratio) for component i, indicating its tendency to partition into the vapor phase relative to the liquid phase

If this ratio is greater than 1, component i prefers the vapor phase; if less than 1, it prefers the liquid phase.

y_{j} / x_{j}

The K-value (vapor-liquid equilibrium ratio) for component j, indicating its tendency to partition into the vapor phase relative to the liquid phase

Similar to component i, this ratio shows component j's preference for the vapor or liquid phase.

Signs and relationships

\frac{y_i / x_i}{y_j / x_j}: The ratio of K-values directly compares the 'vapor-seeking' tendency of component i to that of component j. A value significantly different from 1 indicates a difference in volatility.

Free study cues

Insight

Canonical usage

Relative volatility is calculated as a ratio of dimensionless equilibrium ratios (K-values) or mole fractions, resulting in a dimensionless index used to quantify separation ease.

Common confusion

Attempting to calculate relative volatility using mass fractions instead of mole fractions, which is incorrect for distillation design unless the components have identical molecular weights.

Dimension note

Relative volatility is a ratio of ratios. Because mole fractions and K-values are dimensionless, the resulting alpha value is a pure number independent of the system of units (SI or USCS)

Unit systems

$y_{i}, y_{j}$ dimensionless - Mole fraction of components i and j in the vapor phase; must be between 0 and 1.

$x_{i}, x_{j}$ dimensionless - Mole fraction of components i and j in the liquid phase; must be between 0 and 1.

$K_{i}, K_{j}$ dimensionless - Vapor-liquid equilibrium ratios (K-values) defined as y/x for each component.

$α_{ij}$ dimensionless - The resulting relative volatility; a value of 1.0 indicates no separation is possible by distillation.

One free problem

Practice Problem

In a vapor-liquid equilibrium system, component A has a vapor mole fraction ( $y_{A}$ ) of 0.7 and a liquid mole fraction ( $x_{A}$ ) of 0.3. Component B has a vapor mole fraction ( $y_{B}$ ) of 0.2 and a liquid mole fraction ( $x_{B}$ ) of 0.5. Calculate the relative volatility of A with respect to B (a_AB).

Vapor Mole Fraction (Component i)0.7 dimensionless

Liquid Mole Fraction (Component i)0.3 dimensionless

Vapor Mole Fraction (Component j)0.2 dimensionless

Liquid Mole Fraction (Component j)0.5 dimensionless

Solve for: $a l p h a_{i} j$

Hint: First calculate the K-values for each component.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When designing a distillation column to separate ethanol from water in a brewery, Relative Volatility is used to calculate the \alpha_{ij} value from Vapor Mole Fraction (Component i), Liquid Mole Fraction (Component i), and Vapor Mole Fraction (Component j). The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

A relative volatility (a_ij) of 1 means components i and j have the same volatility and cannot be separated by simple distillation.
Always ensure that y and x values correspond to the same component (i or j) and are mole fractions.
The K-value (K = y/x) is a measure of a component's tendency to partition into the vapor phase.
Relative volatility is often assumed constant over a small temperature range, but it can vary significantly with temperature and pressure.

Avoid these traps

Common Mistakes

Mixing up components i and j in the numerator and denominator, leading to a_ji instead of a_ij.
Using mass fractions instead of mole fractions for y and x.
Assuming relative volatility is constant across wide temperature or pressure ranges without verification.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relative volatility is defined as the ratio of the K-values (vapor-liquid equilibrium ratios) of two components in a mixture.

This equation is essential in chemical engineering for designing and analyzing separation processes, particularly distillation. It's applied when determining the feasibility and efficiency of separating a mixture into its components based on their boiling points. A relative volatility significantly greater than 1 indicates a practical separation, while a value close to 1 suggests a difficult or impossible separation by simple distillation.

Relative volatility is a cornerstone of distillation column design, which is critical in industries like petroleum refining, chemical manufacturing, and pharmaceuticals. It directly impacts the number of stages required in a distillation column, energy consumption, and overall process economics. Understanding and manipulating relative volatility allows engineers to optimize separation processes and produce high-purity products efficiently.

Mixing up components i and j in the numerator and denominator, leading to a_ji instead of a_ij. Using mass fractions instead of mole fractions for y and x. Assuming relative volatility is constant across wide temperature or pressure ranges without verification.

When designing a distillation column to separate ethanol from water in a brewery, Relative Volatility is used to calculate the \alpha_{ij} value from Vapor Mole Fraction (Component i), Liquid Mole Fraction (Component i), and Vapor Mole Fraction (Component j). The result matters because it helps size components, compare operating conditions, or check a design margin.

A relative volatility (a_ij) of 1 means components i and j have the same volatility and cannot be separated by simple distillation. Always ensure that y and x values correspond to the same component (i or j) and are mole fractions. The K-value (K = y/x) is a measure of a component's tendency to partition into the vapor phase. Relative volatility is often assumed constant over a small temperature range, but it can vary significantly with temperature and pressure.

References

Sources

J. M. Smith, H. C. Van Ness, M. M. Abbott, 'Introduction to Chemical Engineering Thermodynamics'
R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, 'Transport Phenomena'
Wikipedia: Relative volatility
Seader, Henley, and Roper, Separation Process Principles
Perry's Chemical Engineers' Handbook
IUPAC Gold Book
Seader, J. D., Henley, E. J., & Roper, D. K. Separation Process Principles with Applications Using Process Simulators. 4th ed.
Bird, R. B., Stewart, W. E., & Lightfoot, E. N. Transport Phenomena. 2nd ed. John Wiley & Sons, 2007.

Overview

Variables

Derivation

Define K-value for a Component:

Define Relative Volatility:

Substitute K-value Definitions:

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

K-Value (Vapor-Liquid Equilibrium Ratio)

Raoult's Law

Frequently Asked Questions

Sources