Review of Separation Principles

Review of Separation Principles – p.1/23

An hour in the library could save you six weeks in the laboratory.

George Quadarer Mixer and Splitter Reactor Flash Distillation Absorption Multicomponent Systems Linear Balances

Before we develop linear models for flash and distillation units, we will review some fundamental principles of separations that were covered in previous courses.

Mixer and Splitter Reactor Flash Distillation Absorption Multicomponent Systems Linear Balances

Before we develop linear models for flash and distillation units, we will review some fundamental principles of separations that were covered in previous courses.

Mixer and Splitter Reactor Flash Distillation Absorption Multicomponent Systems Linear Balances

Before we develop linear models for flash and distillation units, we will review some fundamental principles of separations that were covered in previous courses.

Vapor Liquid Equilibrium

1. \underline{Pure Substance}

When a pure liquid is placed in a sealed container, part of the liquid evaporates.

At equilibrium, the pressure exerted by the vapor on the liquid is called the vapor pressure of the liquid.

The value of the vapor pressure is independent of the amount of liquid in the container.

Vapor pressure increases with temperature.

The boiling point of a liquid is the temperature at which the vapor pressure equals the total pressure. At this point, liquid and vapor may co-exist in any proportion indefinitely.

Review of Separation Principles – p.4/23

1. \underline{Pure Substance}

When a pure liquid is placed in a sealed container, part of the liquid evaporates.

At equilibrium, the pressure exerted by the vapor on the liquid is called the vapor pressure of the liquid.

The value of the vapor pressure is independent of the amount of liquid in the container.

Vapor pressure increases with temperature.

The boiling point of a liquid is the temperature at which the vapor pressure equals the total pressure. At this point, liquid and vapor may co-exist in any proportion indefinitely.

Review of Separation Principles – p.4/23

1. \underline{Pure Substance}

When a pure liquid is placed in a sealed container, part of the liquid evaporates.

At equilibrium, the pressure exerted by the vapor on the liquid is called the vapor pressure of the liquid.

The value of the vapor pressure is independent of the amount of liquid in the container.

Vapor pressure increases with temperature.

The boiling point of a liquid is the temperature at which the vapor pressure equals the total pressure. At this point, liquid and vapor may co-exist in any proportion indefinitely.

Review of Separation Principles – p.4/23

1. \underline{Pure Substance}

When a pure liquid is placed in a sealed container, part of the liquid evaporates.

At equilibrium, the pressure exerted by the vapor on the liquid is called the vapor pressure of the liquid.

The value of the vapor pressure is independent of the amount of liquid in the container.

Vapor pressure increases with temperature.

The boiling point of a liquid is the temperature at which the vapor pressure equals the total pressure. At this point, liquid and vapor may co-exist in any proportion indefinitely.

Review of Separation Principles – p.4/23

1. \underline{Pure Substance}

When a pure liquid is placed in a sealed container, part of the liquid evaporates.

At equilibrium, the pressure exerted by the vapor on the liquid is called the vapor pressure of the liquid.

The value of the vapor pressure is independent of the amount of liquid in the container.

Vapor pressure increases with temperature.

The boiling point of a liquid is the temperature at which the vapor pressure equals the total pressure. At this point, liquid and vapor may co-exist in any proportion indefinitely.

Review of Separation Principles – p.4/23

Liquid and vapor represented by points on the vapor pressure curve are called saturated liquid and saturated vapor respectively.

Liquid and vapor represented by points on the vapor pressure curve are called saturated liquid and saturated vapor respectively.

Vapor or gas at temperature above that corresponding to saturated is called superheated.

Vapor or gas at temperature above that corresponding to saturated is called superheated.

All three states co-exist at the triple point.

Review of Separation Principles – p.7/23

All three states co-exist at the triple point.

Review of Separation Principles – p.7/23

At the critical point, distinction between liquid and vapor disappears and all properties of the liquid, such as density, viscosity, refractive index etc. are identical to that of the vapor.

At the critical point, distinction between liquid and vapor disappears and all properties of the liquid, such as density, viscosity, refractive index etc. are identical to that of the vapor.

The substance at the temperature above the critical point is called a gas and it will not be liquified at this temperature regardless of how high the pressure might be imposed.

The substance at the temperature above the critical point is called a gas and it will not be liquified at this temperature regardless of how high the pressure might be imposed.

2. \underline{Mixtures} Consider a binary mixture A − B where component A is more volatile.

Similar to pure substances, there will be a Pressure-Temperature relationship.

However, there is an additional variable, concentration (usually represented by mole fraction)

x:  mole fraction of A in liquid
y∗:  corresponding mole fraction of A in vapor
(1 − x): mole fraction of B in liquid
(1 − y∗): corresponding mole fraction of A in vapor

2. \underline{Mixtures} Consider a binary mixture A − B where component A is more volatile.

Similar to pure substances, there will be a Pressure-Temperature relationship.

However, there is an additional variable, concentration (usually represented by mole fraction)

x:  mole fraction of A in liquid
y∗:  corresponding mole fraction of A in vapor
(1 − x): mole fraction of B in liquid
(1 − y∗): corresponding mole fraction of A in vapor

2. \underline{Mixtures} Consider a binary mixture A − B where component A is more volatile.

Similar to pure substances, there will be a Pressure-Temperature relationship.

However, there is an additional variable, concentration (usually represented by mole fraction)

x:  mole fraction of A in liquid
y∗:  corresponding mole fraction of A in vapor
(1 − x): mole fraction of B in liquid
(1 − y∗): corresponding mole fraction of A in vapor

2. \underline{Mixtures} Consider a binary mixture A − B where component A is more volatile.

Similar to pure substances, there will be a Pressure-Temperature relationship.

However, there is an additional variable, concentration (usually represented by mole fraction)

x:  mole fraction of A in liquid
y∗:  corresponding mole fraction of A in vapor
(1 − x): mole fraction of B in liquid
(1 − y∗): corresponding mole fraction of A in vapor

At point H, the mixture is a saturated liquid.

At point J, the mixture is a saturated vapor.

At point E, the mixture is a two phase mixture with liquid phase composition of H and vapor phase composition of J such that the average composition of the entire mixture is E.

Review of Separation Principles – p.11/23

At point H, the mixture is a saturated liquid.

At point J, the mixture is a saturated vapor.

At point E, the mixture is a two phase mixture with liquid phase composition of H and vapor phase composition of J such that the average composition of the entire mixture is E.

Review of Separation Principles – p.11/23

At point H, the mixture is a saturated liquid.

At point J, the mixture is a saturated vapor.

At point E, the mixture is a two phase mixture with liquid phase composition of H and vapor phase composition of J such that the average composition of the entire mixture is E.

Review of Separation Principles – p.11/23

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

Review of Separation Principles – p.12/23

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

Review of Separation Principles – p.12/23

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

Review of Separation Principles – p.12/23

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

Review of Separation Principles – p.12/23

At point G, the mixture is a liquid.

As temperature is raised, at point H, the mixture is a saturated liquid.

At point Q, the mixture is a two phase mixture with liquid phase composition of L and vapor phase composition of K.

At point M, the mixture is a saturated vapor.

At point O, the vapor is superheated.

Note that “boiling” occurs between H and M, and not at a constant temperature as in pure substances.

Review of Separation Principles – p.12/23

Relative Volatility

If we plot y^{∗} vs. x at constant pressure, we get a plot that illustrates how easy it is to separate A and B. A numerical measure of ease of separation is called relative volatility

Review of Separation Principles – p.13/23

Relative Volatility

If we plot y^{∗} vs. x at constant pressure, we get a plot that illustrates how easy it is to separate A and B. A numerical measure of ease of separation is called relative volatility

Review of Separation Principles – p.13/23

Relative Volatility

If we plot y^{∗} vs. x at constant pressure, we get a plot that illustrates how easy it is to separate A and B. A numerical measure of ease of separation is called relative volatility

Review of Separation Principles – p.13/23

Relative Volatility

If we plot y^{∗} vs. x at constant pressure, we get a plot that illustrates how easy it is to separate A and B. A numerical measure of ease of separation is called relative volatility

Review of Separation Principles – p.13/23

Relative Volatility

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Relative Volatility, α, is the ratio of the concentration of A and B in one phase to that in the other.

α = \frac{ratio of A and B in vapor phase}{ratio of A and B in liquid phase}
= \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)}
= \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

If y^{∗} = x (except at x = 0 or x = 1), α = 1.0 and no separation is possible. The larger the value of α above unity, the easier it is to separate A and B.

Review of Separation Principles – p.14/23

Ideal Gas

An ideal gas is one that follows the relationship:

P V = nRT

At a given pressure and temperature, one mole of any gas that is ideal will have exactly the same volume. One does not need to do experiments to determine this volume.

An ideal gas is one that follows the relationship:

P V = nRT

At a given pressure and temperature, one mole of any gas that is ideal will have exactly the same volume. One does not need to do experiments to determine this volume.

An ideal gas is one that follows the relationship:

P V = nRT

At a given pressure and temperature, one mole of any gas that is ideal will have exactly the same volume. One does not need to do experiments to determine this volume.

Ideal Solution

In a binary vapor-liquid mixture, if the liquid phase is an ideal solution, we can compute the equilibrium partial pressure of the gas from the solution, without having to do experiments. Characteristics of Ideal Solutions:

1. The average intermolecular forces of attraction and repulsion in the solution are unchanged on mixing the constituents.

2. The volume of the solution varies linearly with composition.

3. The is neither absorption nor evolution of heat in mixing the constituents.

4. The total vapor pressure of the solution varies linearly

with composition expressed as mole fraction.

Review of Separation Principles – p.16/23

In a binary vapor-liquid mixture, if the liquid phase is an ideal solution, we can compute the equilibrium partial pressure of the gas from the solution, without having to do experiments. Characteristics of Ideal Solutions:

1. The average intermolecular forces of attraction and repulsion in the solution are unchanged on mixing the constituents.

2. The volume of the solution varies linearly with composition.

3. The is neither absorption nor evolution of heat in mixing the constituents.

4. The total vapor pressure of the solution varies linearly

with composition expressed as mole fraction.

Review of Separation Principles – p.16/23

In a binary vapor-liquid mixture, if the liquid phase is an ideal solution, we can compute the equilibrium partial pressure of the gas from the solution, without having to do experiments. Characteristics of Ideal Solutions:

1. The average intermolecular forces of attraction and repulsion in the solution are unchanged on mixing the constituents.

2. The volume of the solution varies linearly with composition.

3. The is neither absorption nor evolution of heat in mixing the constituents.

4. The total vapor pressure of the solution varies linearly

with composition expressed as mole fraction.

Review of Separation Principles – p.16/23

In a binary vapor-liquid mixture, if the liquid phase is an ideal solution, we can compute the equilibrium partial pressure of the gas from the solution, without having to do experiments. Characteristics of Ideal Solutions:

1. The average intermolecular forces of attraction and repulsion in the solution are unchanged on mixing the constituents.

2. The volume of the solution varies linearly with composition.

3. The is neither absorption nor evolution of heat in mixing the constituents.

4. The total vapor pressure of the solution varies linearly

with composition expressed as mole fraction.

Review of Separation Principles – p.16/23

Raoult’s Law

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be

replaced by fugacity.

Review of Separation Principles – p.17/23

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be

replaced by fugacity.

Review of Separation Principles – p.17/23

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be

replaced by fugacity.

Review of Separation Principles – p.17/23

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be

replaced by fugacity.

Review of Separation Principles – p.17/23

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be replaced by fugacity.

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be replaced by fugacity.

When a gas mixture in equilibrium with an ideal liquid solution follows the ideal gas law, the partial pressure p^{∗} of the solute gas A equals the product of its vapor pressure, p^{0}, at the same temperature and its mole fraction, x, in the liquid phase.

p^{∗} = p^{0}x

1. The nature of the solvent liquid does not enter into consideration.

2. Solubility of a particular gas in any solvent is the same.

3. If the gas behavior is non-ideal, the pressure can be replaced by fugacity.

Non-Ideal Liquid Solutions

Henry’s Law is used for non-ideal solutions.

y^{∗} = \frac{p^{∗}}{p_{t}} = Kx

where K is a “constant”. K is a function of P, T, and x.

Review of Separation Principles – p.18/23

y^{∗} = \frac{p^{∗}}{p_{t}} = Kx

where K is a “constant”. K is a function of P, T, and x.

Review of Separation Principles – p.18/23

y^{∗} = \frac{p^{∗}}{p_{t}} = Kx

where K is a “constant”. K is a function of P, T, and x.

Review of Separation Principles – p.18/23

y^{∗} = \frac{p^{∗}}{p_{t}} = Kx

where K is a “constant”. K is a function of P, T, and x.

Review of Separation Principles – p.18/23

y^{∗} = \frac{p^{∗}}{p_{t}} = Kx

where K is a “constant”. K is a function of P, T, and x.

Review of Separation Principles – p.18/23

Application of Raoult’s Law

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Consider a mixture A-B. The equilibrium partial pressure is given by

p_{A}^{∗} = p_{A}^{0}x

where the superscript 0 refers to the vapor pressure of the pure component and x is the mole fraction in the liquid. Then:

p_{B}^{∗} = p_{B}^{0}(1 − x)

If the vapor follows ideal gas behavior:

\begin{aligned}p_{t} & = & p_{A}^{∗} + p_{B}^{∗} \\ & = & p_{A}^{0}x + p_{B}^{0}(1 − x) \end{aligned}

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Also,

y^{∗} = \frac{p_{A}^{∗}}{p_{t}} = \frac{p_{A}^{0}x}{p_{t}}

Similarly:

1 − y^{∗} = \frac{p_{B}^{∗}}{p_{t}} = \frac{p_{B}^{0}(1 − x)}{p_{t}}

Thus, relative volatility is given by:

\begin{aligned}α & = & \frac{y^{∗}/(1 − y^{∗})}{x/(1 − x)} & = & \frac{p_{A}^{0}x/(p_{B}^{0}(1 − x)}{x/(1 − x)} \\ & & & = & \frac{p_{A}^{0}}{p_{B}^{0}} \end{aligned}

Thus, while vapor pressures of A and B may vary considerably at different temperatures, their ratio will be constant

provided the solution is ideal. Review of Separation Principles – p.20/23

Example

Compute the vapor-liquid equilibria of n-heptane (A) and n-octane (B) at 760 mm Hg given the following vapor pressure data:

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)
98.4  760  333
105.0  940  417
110.0  1050  484
115.0  1200  561
120.0  1350  650
125.6  1540  760

Verify that α is nearly constant in the above temperature range.

Solution Strategy

Using p_{t} = 760 mm Hg, compute x from:

p_{t} = p_{A}^{0}x + p_{B}^{0}(1 − x)

For each x compute the corresponding y^{∗} from:

y^{∗} = \frac{p_{A}^{0}x}{p_{t}}

Compute α from:

α = \frac{p_{A}^{0}}{p_{B}^{0}}

Review of Separation Principles – p.22/23

Solution Strategy

Using p_{t} = 760 mm Hg, compute x from:

p_{t} = p_{A}^{0}x + p_{B}^{0}(1 − x)

For each x compute the corresponding y^{∗} from:

y^{∗} = \frac{p_{A}^{0}x}{p_{t}}

Compute α from:

α = \frac{p_{A}^{0}}{p_{B}^{0}}

Solution Strategy

Using p_{t} = 760 mm Hg, compute x from:

p_{t} = p_{A}^{0}x + p_{B}^{0}(1 − x)

For each x compute the corresponding y^{∗} from:

y^{∗} = \frac{p_{A}^{0}x}{p_{t}}

Compute α from:

α = \frac{p_{A}^{0}}{p_{B}^{0}}

Solution

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23

T (^{o}C) p_{A}^{0} (mm Hg) p_{B}^{0} (mm Hg)  x  y^{∗}  α
98.4  760  333 1.000 1.000 2.28
105.0  940  417 0.655 0.810 2.25
110.0  1050  484 0.487 0.674 2.17
115.0  1200  561 0.312 0.492 2.14
120.0  1350  650 0.157 0.279 2.08
125.6  1540  760 0.000 0.000 2.03

Although p_{A}^{0} and p_{B}^{0} vary considerably, α is nearly constant. An average value of α is 2.16.

The equation α = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})} can be rearranged to give:

y^{∗} = \frac{2.16x}{1 + 1.16x}

Review of Separation Principles – p.23/23