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Integration of Rational Functions Using Partial Fractions01:29

Integration of Rational Functions Using Partial Fractions

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Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
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Applications of Integration to Find Hydrostatic Pressure01:30

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Hydrostatic force is a fluid's total force at rest on a surface. For a horizontal surface submerged at a fixed depth, the pressure is constant and calculated as the product of fluid density, gravitational acceleration, and depth. In the case of a vertical dam wall submerged in water, this force is not evenly distributed due to the increasing pressure with depth. This variation arises from the cumulative weight of the water above each point. Integration is used to account for the continuous...
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Integration by Parts: Indefinite Integrals01:26

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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Integration by Parts: Definite Integrals01:23

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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Unless individual gases chemically react with each other, the individual gases in a mixture of gases do not affect each other’s pressure. Each gas in a mixture exerts the same pressure that it would exert if it were present alone in the container. The pressure exerted by each individual gas in a mixture is called its partial pressure.
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While the differential rate law relates the rate and concentrations of reactants, a second form of rate law called the integrated rate law relates concentrations of reactants and time. Integrated rate laws can be used to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law helps determine the length of time a radioactive material must be stored for its...
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Determining the Partial Pressure of Volatile Components via Substrate-Integrated Hollow Waveguide Infrared

Vjekoslav Kokoric1, Johannes Theisen2, Andreas Wilk1

  • 1Institute of Analytical and Bioanalytical Chemistry , Ulm University , Albert-Einstein-Allee 11 , 89081 Ulm , Germany.

Analytical Chemistry
|March 6, 2018
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Summary

A novel microfluidic-infrared sensor system (μFLUID-IR) measures chemical activity of volatile compounds in complex fluids. This system enables precise analysis without sample perturbation, crucial for advanced process control.

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Area of Science:

  • Analytical Chemistry
  • Chemical Engineering
  • Spectroscopy

Background:

  • Chemical activity is a critical parameter for understanding complex fluids, yet it is underutilized in process analysis.
  • Traditional methods like headspace gas chromatography involve sample perturbation, limiting real-time analysis.
  • A need exists for advanced analytical techniques that can measure chemical activity non-invasively.

Purpose of the Study:

  • To develop a microfluidic system integrated with hollow waveguide (iHWG) vapor phase infrared spectroscopy for chemical activity evaluation.
  • To demonstrate the system's capability in measuring partial pressures of volatile compounds in complex fluids.
  • To validate the system's performance against established methods and literature values.

Main Methods:

  • Integration of a microfluidic device with substrate-integrated hollow waveguide (iHWG) for vapor phase mid-infrared spectroscopy.
  • Controlled evaporation and permeation of volatile compounds from a sample solution into the iHWG.
  • Measurement of partial pressures at thermodynamic equilibrium for chemical activity deduction.

Main Results:

  • The microfluidic-iHWG mid-infrared sensor system (μFLUID-IR) successfully evaluated chemical activity in a water/ethanol mixture.
  • Data derived from partial pressure measurements at equilibrium conditions correlated with established literature values.
  • Achieved sensor response time <150 s (t90) and recovery time <300 s (t_recovery).

Conclusions:

  • The μFLUID-IR system provides a perturbation-free method for analyzing chemical activity in complex fluids.
  • The developed system offers rapid response and recovery times, suitable for process analysis and control.
  • This technology enables systematic studies on phase diagrams and aggregation processes without sampling artifacts.