The fundamental principle is Dalton’s law combined with the ideal gas law: a carrier gas becomes vapor-saturated when the vapor’s partial pressure equals the liquid’s equilibrium vapor pressure at the operating temperature. If the total pressure is (P) and the liquid vapor pressure is (p_v), the vapor volume fraction is (p_v/P), while its mass fraction additionally depends on the molecular weights of the vapor and carrier gas.

Under ideal-gas conditions, volumetric composition is governed by pressure ratios, whereas weight composition is governed by pressure ratios and molecular weights. Temperature affects composition primarily through the liquid’s vapor pressure.

How Saturated Gas Composition Is Established

Equilibrium at the Liquid Surface

When a dry carrier gas contacts a volatile liquid, liquid molecules enter the gas phase until equilibrium is reached. At saturation,

[ p_v = p_v^*(T) ]

where (p_v^*(T)) is the liquid’s saturation vapor pressure at the prevailing temperature.

The vapor cannot normally have a partial pressure greater than the equilibrium vapor pressure without condensation occurring, assuming the liquid phase remains present.

Dalton’s Law of Partial Pressures

For an ideal gas mixture, the total pressure equals the sum of the component partial pressures:

[ P = p_v + p_c ]

where (p_c) is the partial pressure of the dry carrier gas. Therefore,

[ p_c = P-p_v ]

This relationship is the basis for calculating both vapor content and the remaining carrier-gas content.

Ideal-Gas Equivalence of Volume and Mole Fraction

At a common temperature and pressure, each ideal-gas component occupies a volume proportional to its number of moles. Consequently,

[ \frac{V_v}{V}=\frac{n_v}{n_{\text{total}}} =\frac{p_v}{P} ]

The vapor volume fraction, or volume percentage, is therefore

[ \boxed{\text{vol% vapor}=100\frac{p_v^*(T)}{P}} ]

This is also the vapor mole fraction (y_v).

Determining Volumetric Composition

Pure-Component Vapor Volume

The hypothetical pure-component volume associated with the vapor component is

[ \boxed{V_v=V\frac{p_v^*(T)}{P}} ]

Here, (V_v) is the volume the vapor would occupy at the mixture temperature and total pressure if separated while retaining its component amount.

This expression does not mean that the vapor occupies a distinct physical region inside the mixture. Ideal-gas components are intermingled; the volume is a bookkeeping basis for composition.

Dependence on Temperature and Pressure

The saturated vapor volume fraction depends on the liquid vapor pressure and total system pressure:

[ y_v=\frac{p_v^*(T)}{P} ]

Because (p_v^*(T)) is a property of the liquid, increasing temperature generally increases the vapor fraction. Increasing total pressure at fixed temperature generally decreases the vapor fraction, provided the liquid vapor pressure remains unchanged.

Independence from Carrier-Gas Identity

For ideal gases at the same temperature and total pressure, the vapor volume percentage does not depend on whether the carrier is air, nitrogen, oxygen, hydrogen, methane, or another ideal carrier gas.

The carrier identity still matters for mass-based calculations, density, transport properties, heat capacity, and equipment behavior. It does not alter the ideal-gas volume fraction determined by (p_v^*/P).

Converting Vapor Content to Weight Composition

Mass Fraction of Vapor

Let (M_v) be the molecular weight of the vapor and (M_c) the molecular weight of the carrier gas. Since (n_i=p_iV/(RT)), the component masses are proportional to (p_iM_i).

The vapor mass fraction in a binary saturated mixture is therefore

[ \boxed{ w_v= \frac{p_v^*M_v} {p_v^M_v+(P-p_v^)M_c} } ]

The vapor weight percentage is

[ \boxed{\text{wt% vapor}=100w_v} ]

Unlike volume percentage, weight percentage changes when the carrier gas changes because the carrier molecular weight changes.

Vapor Mass per Mass of Dry Carrier Gas

Pilot-plant calculations often use a dry-carrier basis rather than total-mixture mass. The mass ratio is

[ \boxed{ \frac{m_v}{m_c}

\frac{p_v^M_v} {(P-p_v^)M_c} } ]

This basis is especially useful for drying towers, humidification columns, saturation vessels, and gas-liquid contacting experiments.

Vapor Mass per Unit Mixture Volume

Using the ideal gas law, the vapor concentration by mass is

[ \boxed{ \frac{m_v}{V}

\frac{p_v^*M_v}{RT} } ]

The pressure must be expressed in units consistent with (R), and molecular weight must be converted consistently with the desired mass units.

This quantity is useful when the pilot plant measures volumetric gas flow and the operator needs to determine the rate of liquid evaporated:

[ \dot m_v

\dot V\frac{p_v^*M_v}{RT} ]

Choosing the Calculation Basis

Volume or Mole Basis

Use volume percentage or mole fraction when analyzing ideal-gas flow rates, partial pressures, and gas-phase material balances:

[ y_v=\frac{p_v^*}{P} ]

The corresponding vapor-to-dry-carrier molar ratio is

[ \boxed{ \frac{n_v}{n_c}

\frac{p_v^}{P-p_v^} } ]

This ratio is often more practical than volume percentage for calculating the amount of dry gas required to absorb or carry a specified quantity of vapor.

Mass Basis

Use weight fraction when the experiment involves liquid consumption, feed and product weighing, or total mass balances:

[ w_v= \frac{y_vM_v} {y_vM_v+(1-y_v)M_c} ]

This form emphasizes the general rule: convert mole fractions to mass fractions by weighting each component by its molecular weight.

Concentration Basis

For reactor and mass-transfer calculations, the vapor molar concentration is

[ \boxed{ C_v=\frac{p_v^*}{RT} } ]

More generally, for any gas component (A),

[ C_A=\frac{p_A}{RT}=\frac{y_AP}{RT} ]

This relation connects equilibrium composition to gas-phase concentration and is central to absorption, evaporation, stripping, and humidification studies.

Understanding the Trade-offs

Ideal-Gas Assumption

The preceding equations assume that gas molecules behave independently, occupy negligible molecular volume, and experience negligible intermolecular attraction.

This approximation is generally appropriate for educational pilot plants operating at low to moderate pressure and at ambient or elevated temperature.

Conditions Requiring Caution

At high pressure or very low temperature, gas density increases and real-gas effects become significant. In those conditions, fugacity, compressibility factors, or a real-gas equation of state may be required.

The ideal-gas law can then produce inaccurate volumes, densities, and flow rates, even if the partial-pressure framework remains a useful first approximation.

Carrier-Gas Simplification

The simple binary equations assume one vapor and one dry carrier gas. If the carrier contains additional vapors or condensable species, each component must be included in the total pressure and material balance.

For multicomponent systems,

[ P=\sum_i p_i ]

and each component’s mole fraction is (y_i=p_i/P), subject to the appropriate phase-equilibrium relationships.

Vapor-Pressure Limitations

The vapor pressure used in the calculation must correspond to the actual liquid composition and temperature. For a pure liquid, the saturation vapor-pressure relation is direct; for a liquid mixture, activity coefficients or other vapor-liquid equilibrium models may be necessary.

Temperature measurement is therefore critical because even modest temperature changes can substantially alter the vapor pressure of volatile liquids.

Applying the Principles in a Pilot Plant

A practical calculation begins by specifying the absolute temperature (T), absolute total pressure (P), liquid vapor pressure (p_v^*(T)), vapor molecular weight (M_v), and carrier molecular weight (M_c).

The workflow is:

Determine (p_v^*(T)) from reliable vapor-pressure data.
Calculate the vapor mole or volume fraction using (y_v=p_v^*/P).
Calculate the vapor-to-dry-carrier molar ratio using (p_v^/(P-p_v^)).
Convert to mass fraction or mass ratio using the relevant molecular weights.
Use (C_v=p_v^*/RT) or (\dot m_v=\dot Vp_v^*M_v/(RT)) for concentration and flow calculations.
Check whether pressure and temperature conditions justify the ideal-gas assumption.

Making the Right Choice for Your Goal

Select the composition basis that matches the measurement and material balance required by the experiment.

If your primary focus is gas volume or mole balance: Use (y_v=p_v^*(T)/P), because ideal-gas volume fraction equals mole fraction.
If your primary focus is liquid evaporation or mass balance: Use the vapor-to-dry-carrier mass ratio, (\frac{p_v^M_v}{(P-p_v^)M_c}).
If your primary focus is carrier-gas requirement: Use the molar ratio (\frac{n_v}{n_c}=\frac{p_v^}{P-p_v^}) to calculate the dry-gas flow needed for saturation.
If your primary focus is mass-transfer or reactor analysis: Use the vapor concentration (C_v=p_v^*/RT) and verify that the gas remains near equilibrium.
If your primary focus is high-pressure or low-temperature operation: Evaluate real-gas behavior before relying on ideal-gas volumes or densities.

These principles let pilot-plant operators move consistently between pressure, volume, mole, and mass descriptions while preserving the underlying vapor-liquid equilibrium relationship.

Summary Table:

Composition basis	Key equation	Best used for
Vapor volume or mole fraction	`y_v = p_v^*(T) / P`	Gas flow and mole balances
Vapor-to-carrier molar ratio	`n_v/n_c = p_v^/(P-p_v^)`	Calculating dry carrier-gas requirements
Vapor mass fraction	`w_v = p_v^M_v / [p_v^M_v + (P-p_v^*)M_c]`	Total mixture mass balances
Vapor-to-dry-carrier mass ratio	`m_v/m_c = p_v^M_v / [(P-p_v^)M_c]`	Evaporation and liquid consumption
Vapor mass concentration	`m_v/V = p_v^*M_v/(RT)`	Volumetric flow and evaporation rates
Vapor molar concentration	`C_v = p_v^*/(RT)`	Absorption, stripping, and mass-transfer studies

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