Examples of differential equation in the following topics:
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- The reaction rate involves differential equations, but in non-mathematical terms it is simply the rate of change in the concentrations.
- Instead, the reaction rate can be accurately modeled by a rate equation.
- This is an example of a rate equation that might model the above reaction:
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- The rate law is a differential equation, meaning that it describes the change in concentration of reactant(s) per change in time.
- We can rearrange this equation to combine our variables, and integrate both sides to get our integrated rate law:
- Note that this equation can also be written in the following form:
- However, the integrated first-order rate law is usually written in the form of the exponential decay equation.
- Note that this equation is also of the form $y=mx+b$.
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- The solution to this first-order differential equation is the function:
- Apply the equation Nt=N0e−λt in the calculation of decay rates and decay constants
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- When all the forces are balanced, the curvature of the surface is a good measure of the surface tension, which is described by the Young-Laplace equation:
- where $\Delta P$ is the pressure differential across the interface, $\gamma$ is the measured surface tension, and $R_1, R_2$ are the principal radii of curvature, which indicate the degree of curvature.
- This equation describes the shape and curvature of water bubbles and puddles, the "footprints" of water-walking insects, and the phenomenon of a needle floating on the surface of water.
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- A chemical equation is the symbolic representation of a chemical reaction.
- This equation indicates that oxygen and CH4 react to form H2O and CO2.
- The equation also identifies that all the compounds are in the gaseous state.
- Symbols are used to differentiate among different types of reactions.
- Identify the symbols used to represent the states of matter in a chemical equation.
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- Stable carbon isotopes in carbon dioxide are utilized differentially by plants during photosynthesis.
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- The solution to the particle in a box can be found by solving the Schrödinger equation:
- Separating the variables reduces the problem to one of simply solving the spatial part of the equation:
- The above equation establishes a direct relationship between the second derivative of the the wave function and the kinetic energy of the system.
- The best way to visualize the time-independent Schrödinger equation is as a stationary snapshot of a wave at particular moment in time.
- Differential calculus then reveals that the energy of the particle is given by:
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- Thermochemical equations are chemical equations which include the enthalpy change of the reaction, $\Delta H_{rxn}$ .
- A thermochemical equation is a balanced stoichiometric chemical equation which includes the enthalpy change.
- The equations take the form: $A+B\rightarrow C,\: \Delta H =(\pm n)$
- The equation takes the form:
- The equation takes the form:
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- The equation can be derived from the formula of pKa for a weak acid or buffer.
- The balanced equation for an acid dissociation is:
- After taking the log of the entire equation and rearranging it, the result is:
- The equation for the reaction is:
- Calculate the pH of a buffer system using the Henderson-Hasselbalch equation.
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- Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass, which is the mass not considering the mass defect, must balance for each side of the equation.
- The complete equation therefore reads:
- Therefore, the equation should read:
- The visual representation of the equation we used as an example.
- Describes how to write the nuclear equations for alpha and beta decay.