Home > Articles

This chapter is from the book

1.2 Preliminaries: Continuum and Concentration

The modeling of mass transfer process primarily involves the computation of the concentration profile and the rate at which species A, B, C, and so on are transported across any surface in the media. The intrinsic rate of transport (the transport rate per unit area) is quantified in terms of the flux vector, and this section introduces the preliminaries. The notion of the flux vector is then discussed in Section 1.3.

1.2.1 The Continuum Assumption

The key concept in transport modeling is the continuum assumption, which states that matter is distributed continuously in space. Thus we address the bulk properties rather than the molecular-level interactions. The continuum concept permits us to assign local values to variables at each and every point in an Euclidean space. We can then speak of a temperature field, concentration field, velocity field, and so on.

1.2.2 Concentration: Mole Units

In the context of the continuum approximation, the local concentration, CA may be defined as


where V is the volume element and MA is the moles of species A present in that volume. In this book, the S.I. system of units is used, with kg, m, and s being the basic units. However, the basic unit of concentration, is mol/m3, where mol refers to g mol, rather than kg mol. The molarity, denoted as M, is also often used for concentration; it is equivalent to mol/L. Hence 1 M = 1000 mole/m3—a useful conversion factor. Note that units of kmol/m3 can also be used to represent concentration if convenient. In the final calculations, this will need to be converted to gmol/m3 to maintain consistency of units.

The total molar concentration of the mixture C (at any position and at any time) is defined as the summation of all the species concentrations:


where ns is the total number of species present in the mixture.

The total concentration for a gas mixture CG and the total pressure for an ideal gas mixture are related as


Here Rg is the gas constant, which has a value of 8.314(Pa m3/mol K in the S.I. system of units. Note that the units of Rg can also be expressed as J/mol K.

For example, for a gas at 1 bar pressure and 300 K, the total concentration is calculated as 1 × 105 Pa/[8.314(Pa m3/mol K) × 300 K] and has a value of 37 mol/m3.

Total concentration in liquid can be calculated if the partial molar volume of the components in the mixture solution is known. For dilute solutions, it can be based on the density of the solvent.

For example, for a dilute aqueous solution we can use the density of water as the representative density of the mixture. Hence ρ = 1000 kg/m3. The molecular weight of water is expressed as 18 × 10–3 kg/mol rather than as 18 g/mol. Hence the total concentration in dilute solution can be approximated as 1000 kg/m3/18 × 10–3 kg/mol = 55556 mol/m3.

Very often it is necessary to refer to two or more phases—for example, a gas and a liquid. In such cases we use CAG to represent the concentration of A in the gas phase and CAL to represent the concentration in the liquid. Thus we use the first subscript to represent the species by either by numbers i = 1, 2, . . . or simply by names as A, B, . . . , especially for binary mixtures. The second subscript is used to represent the phase (or the location) where it is present. Similarly, CG rather than C is used for the total concentration in the gas phase and CL for the total concentration in the liquid phase. The ratio of CA to C is called the mole fraction, denoted by yA for gas mixtures or xA for liquid mixtures:


The average molecular weight of the mixture 01equin01.jpg is calculated as


where MA is the molecular weight of species A. Also note that


where ρ is the density of the mixture.

1.2.3 Concentration: Mass Units

Another way of expressing the species concentration is to use the mass units. The mass concentration (also called the partial density) is designated by ρA and is defined as the mass of species A per unit volume of the mixture locally at a point in the fluid continuum. This quantity is related to CA simply by the species molecular weight:


Note that ρA is the density of A in the mixture (same as mass of A per unit volume of the mixture) and should not be confused with the pure component density of A. The latter will be denoted in this book with a superscript 01equin02.jpg if we have to refer to both quantities in the same context.

The summation of this for all the species is the mixture density, ρ:


The ratio of ρA to ρ is the mass fraction of species A, denoted often by ωA:


The average molecular weight of the mixture is calculated from the mass fraction values by the reciprocal weighting rule:


1.2.4 Concentration: Partial Pressure Units

In gaseous systems, a measure of concentration is the partial pressure of a species. This follows from the ideal gas law:




Partial pressure is therefore a direct measure of concentration in an ideal gas mixture and used commonly for such cases. The partial pressure difference can be used as a driving force for mass transfer instead of concentration differences for (ideal) gas-phase systems. The two sets of units can be interconverted easily, if needed, by the previous equations.

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.


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.


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.

Cookies and Related Technologies

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.


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


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


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.


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.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020