Knowledge Revolution

Isotherm Program

This page is from the Isotherm Program on-line help.

Contents

Water Activity - an explanation

       Effect of Water Activity on Microorganisms

Saturated salt solutions

Isotherms - an explanation and example plot

Data Requirements - to create an isotherm

Models:- equations and explanations

     BET

     GAB

Program Features - how to use each feature

     Data Entry

     Recalculate

     Estimate

     Database Functions

     Printing

     Scale

     Setup

Water Activity

Water activity is a measure of the water in a food (or other substance) available to organisms.

If you place a sample of food in a sealed container, the moisture in the air-space will come to an equilibrium value. Water activity is defined as the equilibrium relative humidity (ERH) of this air divided by 100.

                a_w = ERH/100

Another form of the definition for water activity is:     a_w=p/p_o

where:

p =partial pressure of water in food
p_o =saturated vapour pressure for pure water at the same temp.

The water activity of a food is important because it may affect the stability to degradation by chemical reactions, moulds, yeasts, bacteria and textural changes. Water activity may therefore give an indication of shelf life.
 

Isotherm

Water activity is a function of moisture and temperature of the food. The graphical relationship between moisture and water activity at constant temperature is called an isotherm. Isotherms are generally drawn with water activity on the horizontal axis.

An isotherm allows changes in moisture level of the food to be used to predict changes in water activity. It also allows prediction of the equilibrium moisture level of a food stored in an environment of known RH and temperature.

Typical Isotherm:  

Data Requirements

This program produces isotherms based on your data.

Data requirements:

• At least 3 data points of a_w / moisture, measured at the same temperature.
• a_w may be entered as a value or a salt, or as a refractive index if using the CSIRO method.
• Moisture may be entered as a value (dry or wet basis), or as a set of pan/sample weights.

Models

The isotherms are based on the BET and GAB models.
The BET model is widely believed to give the best fit to the data at a_w's of up to 0.55.
The GAB model provides a high level of accuracy up to at least 0.9 a_w.

    BET Model a_w/[(1-a_w)*m] = 1/(mo*c) + [(c-1)/(mo*c)]*a_w

where:

m = dry basis moisture content

T = temperature in °K

a_w= water activity

mo = monolayer moisture content

c = surface heat constant

also c = exp(Qs/RT)

where:

Qs = excess heat capacity at the monolayer (cal/mole)

R = the gas constant , 1.986 cal/mol°K

A plot of a_w/[(1-a_w)*m] vs a_w allows the determination of the following parameters:

mo = 1/(intercept+slope)

c = 1/(intercept*mo)

GAB Model
        a_w/[(1-k*a_w)*m] = 1/(mo*c*k) + [(c-1)/(mo*c)]*a_w

    or m = mo*c*k*a_w/[(1-k*a_w)(1-k*a_w+c*k*a_w)]

This equation has a similar form to BET, but has an extra constant, k. BET is actually a special case of GAB, with k=1.

The GAB equation can be rearranged to a polynomial expression:

    a_w/m = k/mo*(1/c-1)*a_w² + *(c-2)/(mo*c)*a_w+ 1/(mo*c*k)

where:

k = constant, in the range 0.6-1

a_w= water activity

mo = GAB monolayer moisture

c = constant (not equal to the BET c)

The polynomial can be represented by: a_w/m = A*a_w² + B*a_w + C
or    A*a_w² + (B-1/m)*a_w + C = 0
Then a_w = [-B1-sqrt(B1² -4*B0*B2)]/(2*B2)

where:

B0 = 1/(mo*c*k) = C

B1 = (c-2)/(mo*c) -1/m= B-1/m

B2 = k/mo*(1/c-1)= A

Modified GAB

The modified GAB equation replaces c with c/T, where T is the temperature in °C. This enables isotherms to be estimated for any temperature, based on data measured at one temperature.

However, the accuracy of this is approximate only, as it assumes all materials are affected by temperature identically.
 

Features of the Isotherm Program

The program provides the following features, under the menu Program menu:

1. Enter new data, and calculate BET and GAB parameters. Draw isothermsIsotherms.

2. Recalculate the isotherm parameters.

3. Estimate a_w or moisture at any temperature.

4. Add the product to the database
    Delete a product from the database
    Work with a product from the database
    Add the GAB constants for a product to the database
    Overlay a second isotherm on a chart (dry or wet basis moisture)

5. Print

6. Change the isotherm scales

7. Change the set-up.

1. Enter a new data set of a_w / moisture (or add to or modify an existing data set), and the program will calculate the following:

BET model:

monolayer

Qs BET sorption heat

c

GAB model:

monolayer

c

k

r²

Linear region:

intercept

slope

The BET parameters are calculated from the log. plot as described above.

The GAB parameters are calculated using matrices. You may also determine them iteratively using Solver, which may slightly improve the fit.

The isotherm generally has an almost linear region at intermediate values of a_w. The program automatically calculates the extent of this region, and the slope and intercept of the straight line of best fit. The region is determined by calculating the widest a_w range, within which the average difference between the isotherm curve and the straight line is <0.005 g water/g solid.

Two isotherm charts are shown, one with moisture on a dry basis and one on a wet basis. The charts show the data points, together with isotherms calculated by the GAB equation (and BET if selected under set-up). The straight line calculated for the linear region is also shown, if selected under set-up.

Data entry method:

Select the menu item Program menu - Data. A dialog box will appear, where you enter the product name and the temperature at which the data was measured.

You may choose to:

clear the existing data points and enter new data

add to or modify the existing data

The data entry dialog box will appear.

• Data is entered as a set of a_w/ moisture points.
• Water activity may be entered either directly, or by entering the salt solution used for equilibration with the sample.
• Moisture may be entered directly (on a wet or dry basis), or as raw data (in either of two formats, as described below).
• The appearance of the dialog box will change to suit if you click on the option buttons in the a_w data or Moisture data boxes.
• If you are entering a_w as a value, enter the value (0-1) in the a_w box.
• If you are entering a_w as a salt a drop-down list of salts will appear. Click on a salt, and its a_w at the data temperature will be automatically entered in the a_w box.
• If you are entering the moisture as a value, you may choose to do so on a dry or wet basis, and a g/g or g/100g basis.
• If you are entering the a_w using the CSIRO method, enter the refractive index.
• If you are entering the moisture as raw data you must enter pan/sample weights:
  raw data 1:

  1. Empty pan weight

  2. Pan + initial moist sample weight (The sample that has been with the salt)

  3. Pan + final dry sample weight (The sample after drying in an oven)
  or
  raw data 2:

  1. Init. moisture (Moisture dry basis of the original sample)

  2. Empty pan weight

  3. Pan + initial sample weight (The sample prior to putting with the salt)

  4. Pan + equilib. sample weight (The sample in equilibrium with the salt)

When you click on Enter sample data (or hit ENTER), the a_w, weights if entered, and moisture for the sample will be added to the list box on the right.

If you click on "Delete sample data", then the sample with the number entered in the sample# box will be deleted, and the remaining samples re-numbered.

When you have entered all the sample data, click on Continue to enter the data into the isotherm program. The isotherm BET and GAB constants, and the linear region will be calculated and graphed.

The GAB parameters are calculated on a polynomial basis, using a least squares regression. This regression has no constraints, and may calculate values outside the acceptable range.

You will be presented with a summary of the data and isotherm. You may choose to use Solver to try to improve on the fit if you wish. At least a marginal improvement usually occurs. Solver is constrained to keep 0.7<=k<=1 and c>=1.

2. You can recalculate the parameters at any time. by selecting the menu item Program menu - Recalculate. This is necessary if you have changed the Solver options under Setup.

3. Estimate the a_w for a known moisture, or the moisture for a known a_w, at any temperature.
The estimate will appear between the two isotherm charts, and a GAB isotherm at the new temperature will be added to these charts.

4. Database functions.

The program contains a database of products, with raw data points and calculated parameters.
You may add a new poduct to this database, or work with a product from the database.
If you work with a product from the database, you may make changes to the data, recalculate the parameters, and overwrite the old database record with the new one.

You may also display an isotherm from the database overlaid on the isotherm you are working with, on a wet or dry moisture basis.

Database Method:

Select the menu item "Program menu - Database".
A dialog box with the following choices will appear:

• Add the current product to the database
• Work with a product from the database
• Add GAB constants for a product from an external source
• Add an overlay to the isotherm plot (dry basis moisture)
• Add an overlay to the isotherm plot (wet basis moisture)

If you choose to add to the database,and the product is already there, you will be asked whether you want to overwrite the existing record.

If you choose to work with an existing product, you must choose a product/temperature from the list. The data for the chosen product will be entered into the model, and the linear region recalculated.

If you to add the GAB constants for a product, you will be prompted for the following details: Product name and temperature GAB constants
This feature allows you to add isotherms to the database if you know the GAB constants, but have no data.

If you choose to add an overlay to your isotherm plot, you will be asked to pick any product/temperature from the database. The isotherm for this product will then be overlaid on your current isotherm.

5. Print

2 pages may be printed.

The first shows both isotherms with the BET and GAB parameters.

The second shows a plot of residuals, the log plot used to calculate the BET parameters, and the raw data.

Method: Select menu item Print 1 or Print 1-2 from the Program menu .

6. Scale

This allows you to alter the x and y axis scales on the two isothermsIsotherms.

7. Setup

You may also choose whether to display:

• The linear region isotherm line
• The BET isotherm line

Effect of Water Activity on Microorganisms

It is water activity, rather than moisture levels, that determine whether a microorganism can live on or in a food or other substance.  A low water activity indicates that the moisture is bound to the substance, therefore unavailable to the microorganism.

Saturated Salt Solutions

The water activity of many saturated salt solutions is accurately known, and published in the literature. The water activity / temperature relationship of a saturated salt solution generally follows the following model:

    ln(a_w) = k₁/T - k₂

where:

k₁ and k₂ are constants, different for each salt

T is temperature in Kelvin

When entering data sets in the program, the a_w of the selected salt will automatically be calculated at the sample temperature.

Click here for values of k₁ and k₂ for various salts

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Last modified: 10-Aug-2008