# Introduction to Modeling of Mass Transfer Processes

This chapter is from the book

## 1.4 Concentration Jump at Interface

Another important point to understand is that the concentration is not continuous at a phase boundary (e.g., gas–liquid interface), unlike temperature. Consider air–water system as an example. Is the oxygen concentration on the air side of the interface the same as the oxygen concentration on the water side of the gas–liquid interface? The answer is no. Because oxygen has a poor solubility in water, the oxygen concentration in the water phase in much lower than that in the air phase. This difference in concentration is called the concentration jump at the interface. The thermodynamic relations needed to calculate the jump are for various cases are discussed next.

### 1.4.1 Gas–Liquid Interface: Henry’s Law

The concentrations on the gas side and on the liquid side of the interface are often assumed to be linearly proportional—a concept summarized by Henry’s law. The constant of proportionality is known as the Henry’s law constant. Various definitions are used for this constant, depending on the units used to measure the concentrations. The common form is as follows:

where pA is the partial pressure of the species in the gas phase and xA is the mole fraction in the liquid. The constant HA has the units of atm or Pa or bars here. The values of the Henry’s law constant for some common gases are shown in Table 1.2.

#### Table 1.2 Henry’s Law Constant for Some Common Gaseous Species in Water at 298.15 K

 Gas H, atm Hydrogen 7.099 × 104 Oxygen 4.259 × 104 Nitrogen 8.65 × 104 Ozone 4570 Carbon dioxide 1630 Sulfur dioxide 440 Ammonia 30

The larger the value of the Henry’s law constant, HA, the less soluble the gas is in the liquid phase. The effect of total pressure in the system is to increase the solubility in accordance with Henry’s law. In general, increasing the temperature decreases the solubility. Hence the Henry’s law constant is usually an increasing function of temperature. An exception occurs with hydrogen, which shows a retrograde behavior. Here the solubility increases at first with an increase in temperature, reaches a maximum, and then decreases thereafter.

#### Other Definitions of Henry’s Law Constant

Other definitions of the Henry’s law constant are also used, depending on which unit is used for the concentrations in the two phases. Two common definitions are as follows:

where HA,pc is the Henry’s law constant (unit of Pa m3/mol) with partial pressure, Pa, as the unit for the gas phase and mol/m3 as the concentration unit for the liquid phase, respectively. Another form is

where HA,cp is the Henry’s law constant (unit of mol/Pa m3) and is the reciprocal of HA,pc.

Since the concentration in the gas phase can also be used instead of the partial pressure unit, we have yet more ways of writing the Henry’s law relationship! Example 1.1 shows the application of the Henry’s law and the concentration jump at a gas–liquid interface.

Note: Mass transport is usually performed under non-equilibrium conditions. Hence Henry’s law should be applied only at the interface, which is assumed to still be at equilibrium. Concentrations in phase 1 and phase 2, away from the interface, will not be the equilbrium values. Otherwise, no mass transfer will occur.

The schematic of the concentration variation for interfacial mass transfer is shown in Figure 1.3. Although the concentration varies in both phases, the interfacial concentration values are related by thermodynamic considerations. The subscript i in Figure 1.3 refers to the interface and yAi and xAi are related by Henry’s law.

### 1.4.2 Vapor–Liquid Interface: Raoult’s Law

For volatile liquid-phase species, Raoult’s law is often used to relate the interfacial concentrations. It states that at the vapor–liquid interface, a pure liquid exerts a partial pressure equal to the vapor pressure of the liquid at that temperature. For an ideal liquid mixture, the partial pressure at the interface is equal to the vapor pressure multiplied by the mole fraction in the liquid. Hence the interfacial relation is

pA,i = xA,i pvap,A

The vapor pressure pvap varies as a function of temperature and is often correlated by the Antoine equation:

Values of the constants A, B, and C are tabulated in many books (for example, Reid et al., 1987) and websites. The units are often in mm Hg for vapor pressure and temperature (T) is in degrees Celsius rather than in standard S.I. units. Hence caution must be exercised when extracting these values from the literature.

### 1.4.3 Liquid–Liquid Interface: Partition Constant

A simple partition constant (denoted as mA for species A) is often used to describe the interfacial equilibrium between two liquids:

yA,i = mAxA,i

where yA,i is the interfacial mole fraction in one of the liquid phases and xA,i is the interfacial mole fraction at the interface of the second liquid. The partition coefficient, in turn, is related to ratio of activity coefficients of A in the two phases:

The value of this coefficient is often predicted using thermodynamic models for activity coefficients.

### 1.4.4 Fluid–Solid Interface: Adsorption Isotherm

Thermodynamic jump at gas–solid or liquid–solid interfaces is defined in a similar manner, using an adsorption equilibrium constant that has a same status as the solubility or partition coefficient. Often a linear relation is used: qA = KACA where qA is the equilibrium concentration of A in the solid phase and CA is its concentration in the gas phase. KA is referred to as the (linear) adsorption equilibrium constant.

### 1.4.5 Nonlinear Equilibrium Models

A note is in order regarding the linear models described in the previous sections. These models are widely used, even when they do not hold exactly for simplification of the models. However, more complex models are needed to describe equilibrium in non-ideal liquid mixtures—and in systems where there is a strong adsorbed layer due to differences between the interfacial tensions of various species dissolved in the liquids. This layer often has completely different property from the two liquids in contact and is sometimes referred to as a microphase.

On a similar note for fluid–solid systems, linear adsorption equilibria are often used, although these are actually the limiting case of nonlinear relations (the classical Langmuir equation and other isotherms are described in Section 29.3). The linear relation is a good approximation for dilute systems. More generally, the Langmuir isotherm is used to represent equilibrium for gas–solid and liquid–solid systems for concentrated solutions. For gases that undergo a reaction in the liquid, the solubility has to be viewed as the first step for the dissolution equilibrium constant—for example, A(gas) to A(aq) equilibrium. Dissolved gas may further react; hence the quantity of gas absorbed will also depend on the equilibrium constant for these reactions, which are accounted for separately. This value can exceed that calculated using the solubility parameter alone. An example of such system is SO2 in water, where the dissolved SO2 undergoes reaction to form or ions depending on the pH of the solution.

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.