T Plot Method Bet
- In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy or Kelly bet), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run (i.e. Approaching the limit as the number of bets goes to infinity).
- Method is an esports organisation dedicated to creating world-class entertainment that celebrates the MMO and RPG communities.
- Please solve and explain the easiest method to plot these wave signals for the figure. This is from a math and signals class learning signal operations. I know the question asks for figure a but I am asking it for the figure that is shown, figure b.
The BET method for measurement of surface area. The BET method uses a measurement of the physisorption of a gas to derive a value of “surface area” for a sample. The gas molecules can pass between particles and into all pores, cracks, and surface roughness, so that the measurement probes the full microscopic surface area of the sample.
T-plot Method Bet
Introduction to BET (Brunauer, Emmett and Teller)
By BET (Brunauer, Emmett and Teller) the specific surface area of a sample is measured – including the pore size distribution. This information is used to predict the dissolution rate, as this rate is proportional to the specific surface area. Thus, the surface area can be used to predict bioavailability. Further it is useful in evaluation of product performance and manufacturing consistency.
T Plot Method Bet
For a quote on BET: Use the contact form
The specific surface determined by BET relates to the total surface area (reactive surface) as all porous structures adsorb the small gas molecules. The surface area determined by BET is thus normally larger than the surface area determined by air permeability. The method used complies with Ph. Eu.2.9.26 Method II.
Instrument and measuring principle, BET
T Plot Method Between Two
| Technical info | |
| Instrument | Micromeritics Gemini 2375 and Gemini V |
| Sample requirement | Samples dried with Micromeritics Flowprep 060 |
| Measuring range | Micropores (1.8 nm – 115 nm) |
| Result | Specific surface area in m²/g or m²/cm³. |
| Sample amount | 1 – 2 g of dry substance is typically required for analysis. |
BET theory
The specific surface area of a powder is determined by physical adsorption of a gas on the surface of the solid and by calculating the amount of adsorbate gas corresponding to a monomolecular layer on the surface. Physical adsorption results from relatively weak forces (van der Waals forces) between the adsorbate gas molecules and the adsorbent surface area of the test powder. The determination is usually carried out at the temperature of liquid nitrogen. The amount of gas adsorbed can be measured by a volumetric or continuous flow procedure.
Multi-point measurements
The data are treated according to the Brunauer, Emmett and Teller (BET) adsorption isotherm equation:
| P | = | partial vapour pressure of adsorbate gas in equilibrium with the surface at 77.4 K (b.p. of liquid nitrogen), in pascals, |
| Po | = | saturated pressure of adsorbate gas, in pascals, |
| Va | = | volume of gas adsorbed at standard temperature and pressure (STP) [273.15 K and atmospheric pressure (1.013 × 105 Pa)], in millilitres, |
| Vm | = | volume of gas adsorbed at STP to produce an apparent monolayer on the sample surface, in millilitres, |
| C | = | dimensionless constant that is related to the enthalpy of adsorption of the adsorbate gas on the powder sample. |
A value of Va is measured at each of not less than 3 values of P/Po. Then the BET value:
is plotted against P/Po according to equation (1). This plot should yield a straight line usually in the approximate relative pressure range 0.05 to 0.3. The data are considered acceptable if the correlation coefficient, r, of the linear regression is not less than 0.9975; that is, r2 is not less than 0.995. From the resulting linear plot, the slope, which is equal to (C − 1)/VmC, and the intercept, which is equal to 1/VmC, are evaluated by linear regression analysis. From these values, Vm is calculated as 1/(slope + intercept), while C is calculated as (slope/intercept) + 1. From the value of Vm so determined, the specific surface area, S, in m2·g–1, is calculated by the equation:
| N | = | Avogadro constant (6.022 × 1023 mol−1), |
| a | = | effective cross-sectional area of one adsorbate molecule, in square metres (0.162 nm2 for nitrogen and 0.195 nm2 for krypton), |
| m | = | mass of test powder, in grams, |
| 22400 | = | volume occupied by 1 mole of the adsorbate gas at STP allowing for minor departures from the ideal, in millilitres. |
A minimum of 3 data points is required. Additional measurements may be carried out, especially when non-linearity is obtained at a P/Po value close to 0.3. Because non-linearity is often obtained at a P/Po value below 0.05, values in this region are not recommended. The test for linearity, the treatment of the data, and the calculation of the specific surface area of the sample are described above.
Single point measurement
Normally, at least 3 measurements of Va each at different values of P/Po are required for the determination of specific surface area by the dynamic flow gas adsorption technique (Method I) or by volumetric gas adsorption (Method II). However, under certain circumstances described below, it may be acceptable to determine the specific surface area of a powder from a single value of Va measured at a single value of P/Po such as 0.300 (corresponding to 0.300 mole of nitrogen or 0.001038 mole fraction of krypton), using the following equation for calculating Vm:
The specific surface area is then calculated from the value of Vm by equation (2) given above.
The single-point method may be employed directly for a series of powder samples of a given material for which the material constant C is much greater than unity. These circumstances may be verified by comparing values of specific surface area determined by the single-point method with that determined by the multiple-point method for the series of powder samples. Close similarity between the single-point values and multiple-point values suggests that 1/C approaches zero.
The single-point method may be employed indirectly for a series of very similar powder samples of a given material for which the material constant C is not infinite but may be assumed to be invariant. Under these circumstances, the error associated with the single-point method can be reduced or eliminated by using the multi-point method to evaluate C for one of the samples of the series from the BET plot, from which C is calculated as (1 + slope/intercept). Then Vm is calculated from the single value of Va measured at a single value of P/Po by the equation:
The specific surface area is calculated from Vm by equation (2) given above.
The following section describes the methods to be used for the sample preparation, the dynamic flow gas adsorption technique (Method I) and the volumetric gas adsorption technique (Method II).
Sample preparation: Outgassing: Before the specific surface area of the sample can be determined, it is necessary to remove gases and vapours that may have become physically adsorbed onto the surface after manufacture and during treatment, handling and storage. If outgassing is not achieved, the specific surface area may be reduced or may be variable because an intermediate area of the surface is covered with molecules of the previously adsorbed gases or vapours. The outgassing conditions are critical for obtaining the required precision and accuracy of specific surface area measurements on pharmaceuticals because of the sensitivity of the surface of the materials.
Conditions: The outgassing conditions must be demonstrated to yield reproducible BET plots, a constant weight of test powder, and no detectable physical or chemical changes in the test powder. The outgassing conditions defined by the temperature, pressure and time should be chosen so that the original surface of the solid is reproduced as closely as possible. Outgassing of many substances is often achieved by applying a vacuum, by purging the sample in a flowing stream of a non-reactive, dry gas, or by applying a desorption-adsorption cycling method. In either case, elevated temperatures are sometimes applied to increase the rate at which the contaminants leave the surface. Caution should be exercised when outgassing powder samples using elevated temperatures to avoid affecting the nature of the surface and the integrity of the sample.
If heating is employed, the recommended temperature and time of outgassing are as low as possible to achieve reproducible measurement of specific surface area in an acceptable time. For outgassing sensitive samples, other outgassing methods such as the desorption-adsorption cycling method may be employed.
The volumetric method (Ph. Eu.2.9.26 Method II)
Principle: In the volumetric method (see Figure 2.9.26.-2), the recommended adsorbate gas is nitrogen which is admitted into the evacuated space above the previously outgassed powder sample to give a defined equilibrium pressure, P, of the gas. The use of a diluent gas, such as helium, is therefore unnecessary, although helium may be employed for other purposes, such as to measure the dead volume.
Since only pure adsorbate gas, instead of a gas mixture, is employed, interfering effects of thermal diffusion are avoided in this method.
Procedure: Admit a small amount of dry nitrogen into the sample tube to prevent contamination of the clean surface, remove the sample tube, insert the stopper, and weigh it. Calculate the weight of the sample. Attach the sample tube to the volumetric apparatus. Cautiously evacuate the sample down to the specified pressure (e.g. between 2 Pa and 10 Pa). Alternatively, some instruments operate by evacuating to a defined rate of pressure change (e.g. less than 13 Pa/30 s) and holding for a defined period of time before commencing the next step.
If the principle of operation of the instrument requires the determination of the dead volume in the sample tube, for example, by the admission of a non-adsorbed gas, such as helium, this procedure is carried out at this point, followed by evacuation of the sample. The determination of dead volume may be avoided using difference measurements, that is, by means of reference and sample tubes connected by a differential transducer. The adsorption of nitrogen gas is then measured as described below.
Raise a Dewar vessel containing liquid nitrogen at 77.4 K up to a defined point on the sample cell. Admit a sufficient volume of adsorbate gas to give the lowest desired relative pressure. Measure the volume adsorbed, Va. For multi-point measurements, repeat the measurement of Va at successively higher P/Po values. When nitrogen is used as the adsorbate gas, P/Po values of 0.10, 0.20, and 0.30 are often suitable.
Reference materials: Periodically verify the functioning of the apparatus using appropriate reference materials of known surface area, such as α-alumina, which should have a specific surface area similar to that of the sample to be examined.
Figure 2.9.26.-1. — Schematic diagram of the dynamic flow method apparatus dynamic flow method apparatus
You are here: Home » Testing & Characterization » Testing & Analysis Techniques » BET Surface Area Analysis & Pore Size and Volume Analysis
BET analysis provides precise specific surface area evaluation of materials by nitrogen multilayer adsorption measured as a function of relative pressure using a fully automated analyzer. The technique encompasses external area and pore area evaluations to determine the total specific surface area in m2/g, yielding important information in studying the effects of surface porosity and particle size in many applications.
BJH analysis can also be employed to determine pore area and specific pore volume using adsorption and desorption techniques. This technique characterizes pore size distribution independent of external area due to particle size of the sample.
- Rapid single point and multipoint specific BET surface area determinations
- Full BET surface area characterization of disperse, nonporous or macroporous materials pore diameter >50nm (type II isotherms) and mesoporous materials with pore diameter between 2 nm and 50 nm (type IV isotherms)
- BET surface area characterization of microporous materials (<2 nm, type I isotherms)
- Pore volume and pore area distributions in the mesopore and macropore ranges using BJH analysis with a full complement of adsorbate thickness models
- BJH adsorption and desorption average pore diameter (4V/A) determinations.
Typical Applications
Specific surface area determinations measuring the external surface area and open pores of macroporous and mesoporous materials, along with pore volume and area distributions that characterize porosity below the effective range of mercury intrusion porosimetry are techniques that can be especially useful for studies of artificial bone materials, catalysts, particle and gas filtration, fuel cell technology, absorbents, sintering studies, zeolites and reactivity studies of materials among a variety of other applications.
Typical Industries using BET and BJH Analysis
- Medical devices
- Materials
- Healthcare
- Toothpaste manufacture
- Environmental and filtration
- Ceramics
- Dyes, paints and pigments
- Pharmaceuticals
- Cosmetics
- Nanotechnology
- Textiles
- Fuel cell technology.
» Testing & Characterization home page