2.3 Constructing Rotations 2.4 Constructing Translations 2.5 Identifying Series and Determining Congruence or Similarity 2.6:The Sum of Angles in a Triangle 2.7 Similar Triangles 2.8 Parallel Lines Cut by a Transversal Unit 3: Functions CC STANDARDS COVERED: 8.F.1, 8.F.3, 8.F.4, 8.F.5, 8.EE.6 3.1 Intro to Functions 3.2 Graphing Functions. Discover everything Scribd has to offer, including books and audiobooks from major publishers. (sec 2) 3/2 = (d 2 y/dx 2)/ (1+tan 2) 3/2 =. A =PL 2 /2EI slope. I hope these materials make this your most successful year of slope lessons yet! Enjoy the journey, and have a great year of Algebra!:) Feel free to share other ideas for exploring slope in the comments! I bet doing a life-size measuring layout with actual step-stools would be incredible if you have the time and can set it up!

This fun activity is designed to provide students with practice on how to discover points on a line using the slope definition of rise over run. Students work in pairs to race across a coordinate plane only using moves that support the directions for slope.Through this activity, students learn how. Slope of a Line Worksheet 3 RTF Slope of a Line Worksheet 3 PDF View Answers. Slope of a Line (Graphed Points) Worksheet 4 - Here is a 9 problem worksheet where you will asked to find the rise and run between two points on a line, then determine the slope of the line. This worksheet includes some points located at the origin of the coordinate.

Thermogravimetric analysis or thermal gravimetric analysis (TGA) is a method of thermal analysis in which the mass of a sample is measured over time as the temperature changes. This measurement provides information about physical phenomena, such as phase transitions, absorption, adsorption and desorption; as well as chemical phenomena including chemisorptions, thermal decomposition, and solid-gas reactions (e.g., oxidation or reduction).^[1]

Thermogravimetric analyzer[edit]

Thermogravimetric analysis (TGA) is conducted on an instrument referred to as a thermogravimetric analyzer. A thermogravimetric analyzer continuously measures mass while the temperature of a sample is changed over time. Mass, temperature, and time are considered base measurements in thermogravimetric analysis while many additional measures may be derived from these three base measurements.

A typical thermogravimetric analyzer consists of a precision balance with a sample pan located inside a furnace with a programmable control temperature. The temperature is generally increased at constant rate (or for some applications the temperature is controlled for a constant mass loss) to incur a thermal reaction. The thermal reaction may occur under a variety of atmospheres including: ambient air, vacuum, inert gas, oxidizing/reducing gases, corrosive gases, carburizing gases, vapors of liquids or 'self-generated atmosphere'; as well as a variety of pressures including: a high vacuum, high pressure, constant pressure, or a controlled pressure.

The thermogravimetric data collected from a thermal reaction is compiled into a plot of mass or percentage of initial mass on the y axis versus either temperature or time on the x-axis. This plot, which is often smoothed, is referred to as a TGA curve. The first derivative of the TGA curve (the DTG curve) may be plotted to determine inflection points useful for in-depth interpretations as well as differential thermal analysis.

A TGA can be used for materials characterization through analysis of characteristic decomposition patterns. It is an especially useful technique for the study of polymeric materials, including thermoplastics, thermosets, elastomers, composites, plastic films, fibers, coatings, paints, and fuels.

Types of TGA[edit]

There are three types of thermogravimetry:

• Isothermal or static thermogravimetry: In this technique, the sample weight is recorded as a function of time at constant temperature.
• Quasistatic thermogravimetry: In this technique, the sample temperature is raised in sequential steps separated by isothermal intervals, during which the sample mass reaches stability before the start of the next temperature ramp.
• Dynamic thermogravimetry: In this technique the sample is heated in an environment whose temperature is changed in a linear manner.

Applications[edit]

Thermal stability[edit]

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TGA can be used to evaluate the thermal stability of a material. In a desired temperature range, if a species is thermally stable, there will be no observed mass change. Negligible mass loss corresponds to little or no slope in the TGA trace. TGA also gives the upper use temperature of a material. Beyond this temperature the material will begin to degrade.

TGA is used in the analysis of polymers. Polymers usually melt before they decompose, thus TGA is mainly used to investigate the thermal stability of polymers. Most polymers melt or degrade before 200 °C. However, there is a class of thermally stable polymers that are able to withstand temperatures of at least 300 °C in air and 500 °C in inert gases without structural changes or strength loss, which can be analyzed by TGA.^[2][3][4]

Oxidation and combustion[edit]

The simplest materials characterization is the residue remaining after a reaction. For example, a combustion reaction could be tested by loading a sample into a thermogravimetric analyzer at normal conditions. The thermogravimetric analyzer would cause ion combustion in the sample by heating it beyond its ignition temperature. The resultant TGA curve plotted with the y axis as percentage of initial mass would show the residue at the final point of the curve.

Oxidative mass losses are the most common observable losses in TGA.^[5]

Studying the resistance to oxidation in copper alloys is very important. For example, NASA (National Aeronautics and Space Administration) is conducting research on advanced copper alloys for their possible use in combustion engines. However, oxidative degradation can occur in these alloys as copper oxides form in atmospheres that are rich in oxygen. Resistance to oxidation is very important because NASA wants to be able to reuse shuttle materials. TGA can be used to study the static oxidation of materials such as these for practical use.

Combustion during TG analysis is identifiable by distinct traces made in the TGA thermograms produced. One interesting example occurs with samples of as-produced unpurified carbon nanotubes that have a large amount of metal catalyst present. Due to combustion, a TGA trace can deviate from the normal form of a well-behaved function. This phenomenon arises from a rapid temperature change. When the weight and temperature are plotted versus time, a dramatic slope change in the first derivative plot is concurrent with the mass loss of the sample and the sudden increase in temperature seen by the thermocouple. The mass loss could be the result of particles of smoke released from burning caused by inconsistencies in the material itself, beyond the oxidation of carbon due to poorly controlled weight loss.

Different weight losses on the same sample at different points can also be used as a diagnosis of the sample's anisotropy. For instance, sampling the top side and the bottom side of a sample with dispersed particles inside can be useful to detect sedimentation, as thermograms will not overlap but will show a gap between them if the particle distribution is different from side to side.^[6][7]

Thermogravimetric kinetics[edit]

Thermogravimetric kinetics may be explored for insight into the reaction mechanisms of thermal (catalytic or non-catalytic) decomposition involved in the pyrolysis and combustion processes of different materials.^{[8][9][10][11][12][13][14]}

Activation energies of the decomposition process can be calculated using Kissinger method.^[15]

Though a constant heating rate is more common, a constant mass loss rate can illuminate specific reaction kinetics. For example, the kinetic parameters of the carbonization of polyvinyl butyral were found using a constant mass loss rate of 0.2 wt %/min.^[16]

Operation in combination with other instruments[edit]

Thermogravimetric analysis is often combined with other processes or used in conjunction with other analytical methods.

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For example, the TGA instrument continuously weighs a sample as it is heated to temperatures of up to 2000 °C for coupling with Fourier-transform infrared spectroscopy (FTIR) and mass spectrometry gas analysis. As the temperature increases, various components of the sample are decomposed and the weight percentage of each resulting mass change can be measured.

References[edit]

1. ^Coats, A. W.; Redfern, J. P. (1963). 'Thermogravimetric Analysis: A Review'. Analyst. 88 (1053): 906–924. Bibcode:1963Ana....88..906C. doi:10.1039/AN9638800906.
2. ^Liu, X.; Yu, W. (2006). 'Evaluating the Thermal Stability of High Performance Fibers by TGA'. Journal of Applied Polymer Science. 99 (3): 937–944. doi:10.1002/app.22305.
3. ^Marvel, C. S. (1972). 'Synthesis of Thermally Stable Polymers'. Ft. Belvoir: Defense Technical Information Center.
4. ^Tao, Z.; Jin, J.; Yang, S.; Hu, D.; Li, G.; Jiang, J. (2009). 'Synthesis and Characterization of Fluorinated PBO with High Thermal Stability and Low Dielectric Constant'. Journal of Macromolecular Science, Part B. 48 (6): 1114–1124. Bibcode:2009JMSB...48.1114Z. doi:10.1080/00222340903041244. S2CID98016727.
5. ^Voitovich, V. B.; Lavrenko, V. A.; Voitovich, R. F.; Golovko, E. I. (1994). 'The Effect of Purity on High-Temperature Oxidation of Zirconium'. Oxidation of Metals. 42 (3–4): 223–237. doi:10.1007/BF01052024. S2CID98272654.
6. ^Lopresti, Mattia; Alberto, Gabriele; Cantamessa, Simone; Cantino, Giorgio; Conterosito, Eleonora; Palin, Luca; Milanesio, Marco (January 28, 2020). 'Light Weight, Easy Formable and Non-Toxic Polymer-Based Composites for Hard X-ray Shielding: A Theoretical and Experimental Study'. International Journal of Molecular Sciences. 21 (3): 833. doi:10.3390/ijms21030833. PMC7037949. PMID32012889.
7. ^Lopresti, Mattia; Palin, Luca; Alberto, Gabriele; Cantamessa, Simone; Milanesio, Marco (20 November 2020). 'Epoxy resins composites for X-ray shielding materials additivated by coated barium sulfate with improved dispersibility'. Materials Today Communications: 101888. doi:10.1016/j.mtcomm.2020.101888.
8. ^Reyes-Labarta, J.A.; Marcilla, A. (2012). 'Thermal Treatment and Degradation of Crosslinked Ethylene Vinyl Acetate-Polyethylene-Azodicarbonamide-ZnO Foams. Complete Kinetic Modelling and Analysis'. Industrial & Engineering Chemistry Research. 51 (28): 9515–9530. doi:10.1021/ie3006935.
9. ^Reyes-Labarta, J.A.; Marcilla, A. (2008). 'Kinetic Study of the Decompositions Involved in the Thermal Degradation of Commercial Azodicarbonamide'(PDF). Journal of Applied Polymer Science. 107 (1): 339–346. doi:10.1002/app.26922. hdl:10045/24682.
10. ^Marcilla, A.; Gómez, A.; Reyes, J.A. (2001). 'MCM-41 Catalytic Pyrolysis of Ethylene-Vinyl Acetate Copolymers. Kinetic Model'. Polymer. 42 (19): 8103–8111. doi:10.1016/S0032-3861(01)00277-4.
11. ^Marcilla, A.; Gómez, A.; Reyes-Labarta, J.A.; Giner, A. (2003). 'Catalytic pyrolysis of polypropylene using MCM-41. Kinetic model'. Polymer Degradation and Stability. 80 (2): 233–240. doi:10.1016/S0141-3910(02)00403-2.
12. ^Marcilla, A.; Gómez, A.; Reyes-Labarta, J.A.; Giner, A.; Hernández, F. (2003). 'Kinetic study of polypropylene pyrolysis using ZSM-5 and an equilibrium fluid catalytic cracking catalyst'. Journal of Analytical and Applied Pyrolysis. 68–63: 467–480. doi:10.1016/S0165-2370(03)00036-6.
13. ^Conesa, J.A.; Caballero, J.A.; Reyes-Labarta, J.A. (2004). 'Artificial Neural Network for Modelling Thermal Decompositions'. Journal of Analytical and Applied Pyrolysis. 71: 343–352. doi:10.1016/S0165-2370(03)00093-7.
14. ^Reyes, J.A.; Conesa, J.A.; Marcilla, A. (2001). 'Pyrolysis and combustion of polycoated cartons recycling. kinetic model and ms analysis'. Journal of Analytical and Applied Pyrolysis. 58–59: 747–763. doi:10.1016/S0165-2370(00)00123-6.
15. ^Janeta, Mateusz; Szafert, Sławomir (2017-10-01). 'Synthesis, characterization and thermal properties of T8 type amido-POSS with p-halophenyl end-group'. Journal of Organometallic Chemistry. Organometallic Chemistry: from Stereochemistry to Catalysis to Nanochemistry honoring Professor John Gladysz's 65 birthday. 847 (Supplement C): 173–183. doi:10.1016/j.jorganchem.2017.05.044.
16. ^Tikhonov, N. A.; Arkhangelsky, I. V.; Belyaev, S. S.; Matveev, A. T. (2009). 'Carbonization of polymeric nonwoven materials'. Thermochimica Acta. 486 (1–2): 66–70. doi:10.1016/j.tca.2008.12.020.