As always, it should be noted that the order of this reaction, like the order for all chemical reactions, cannot be deduced from the chemical equation, but must be determined experimentally. The Haber process : The Haber process produces ammonia from hydrogen and nitrogen gas. The reverse of this process the decomposition of ammonia to form nitrogen and hydrogen is a zero-order reaction. The integrated rate laws derive from calculus, and they relate the concentrations of reactants with time.
Graph integrated rate laws for zero-, first-, and second-order reactions in order to obtain information about the rate constant and concentrations of reactants. The rate law is a differential equation, meaning that it describes the change in concentration of reactant s per change in time. Using calculus, the rate law can be integrated to obtain an integrated rate equation that links concentrations of reactants or products with time directly.
We can rearrange this equation to combine our variables, and integrate both sides to get our integrated rate law:. This is the final form of the integrated rate law for a first-order reaction.
Here, [A] t represents the concentration of the chemical of interest at a particular time t , and [A] 0 represents the initial concentration of A. 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.
For a reaction that is second-order overall, and first-order in two reactants, A and B, our rate law is given by:. There are two possible scenarios here. The first is that the initial concentrations of A and B are equal, which simplifies things greatly. This is the standard form for second-order rate law, and the integrated rate law will be the same as above.
Note here that a plot of [A] versus t will yield a straight line with the slope -k. The y-intercept of this plot will be the initial concentration of A, [A] 0.
The important thing is not necessarily to be able to derive each integrated rate law from calculus, but to know the forms, and which plots will yield straight lines for each reaction order. A summary of the various integrated rate laws, including the different plots that will yield straight lines, can be used as a resource. Summary of integrated rate laws for zero-, first-, second-, and n th-order reactions : A summary of reactions with the differential and integrated equations.
The half-life of a reaction is the amount of time it takes for the concentration of a reactant to decrease to one-half of its initial value. The half-life is the time required for a quantity to fall to half its initial value, as measured at the beginning of the time period.
If we know the integrated rate laws, we can determine the half-lives for first-, second-, and zero-order reactions. For this discussion, we will focus on reactions with a single reactant. Half-life : The half-life of a reaction is the amount of time it takes for it to become half its quantity. If we plug this in for [A] in our integrated rate law, we have:.
By rearranging this equation and using the properties of logarithms, we can find that, for a first order reaction:. What is interesting about this equation is that it tells us that the half-life of a first-order reaction does not depend on how much material we have at the start.
It takes exactly the same amount of time for the reaction to proceed from all of the starting material to half of the starting material as it does to proceed from half of the starting material to one-fourth of the starting material.
In each case, we halve the remaining material in a time equal to the constant half-life. Keep in mind that these conclusions are only valid for first-order reactions. Consider, for example, a first-order reaction that has a rate constant of 5. To find the half-life of the reaction, we would simply plug 5.
Thus the half-life of a second-order reaction, unlike the half-life for a first-order reaction, does depend upon the initial concentration of A. Consider, for example, a second-order reaction with a rate constant of 3 M -1 s -1 in which the initial concentration of A is 0. Therefore, for a zero-order reaction, half-life and initial concentration are directly proportional. As initial concentration increases, the half-life for the reaction gets longer and longer. Privacy Policy.
Skip to main content. Chemical Kinetics. Search for:. The Rate Law: Concentration and Time The Rate Law The rate law for a chemical reaction relates the reaction rate with the concentrations or partial pressures of the reactants. Learning Objectives Produce rate equations for elementary reactions. For elementary reactions, the rate equation can be derived from first principles using collision theory.
Philips, Marcy H. Journal of Chemical Education , 96 9 , Development of the Sci-math Sensemaking Framework: categorizing sensemaking of mathematical equations in science. Parobek , Patrick M. Chaffin , Marcy H. Location-thinking, value-thinking, and graphical forms: combining analytical frameworks to analyze inferences made by students when interpreting the points and trends on a reaction coordinate diagram.
Chemistry Education Research and Practice , 22 3 , Rodriguez , Kinsey Bain , Marcy H. Rodriguez , Avery R. Stricker , Nicole M. Chemistry Education Research and Practice , 21 2 , An interactive virtual laboratory addressing student difficulty in differentiating between chemical reaction kinetics and equilibrium.
Computer Applications in Engineering Education , 28 1 , Rodriguez , Marcy H. Analysis of student reasoning about Michaelis—Menten enzyme kinetics: mixed conceptions of enzyme inhibition. Chemistry Education Research and Practice , 20 2 , Pair your accounts.
Your Mendeley pairing has expired. Please reconnect. The letters a and b represent the order of the reaction with respect to A and the order of the reaction with respect to b.
Their values are determined experimentally. Together, they give the order of the reaction, n:. For example, if doubling the concentration of A doubles the reaction rate or quadrupling the concentration of A quadruples the reaction rate, then the reaction is first order with respect to A. The rate constant is:.
If you double the concentration of A and the reaction rate increases four times, the rate of the reaction is proportional to the square of the concentration of A. The reaction is second order with respect to A. The rate constant may also be expressed using the Arrhenius equation :. Here, A is a constant for the frequency of particle collisions, Ea is the activation energy of the reaction, R is the universal gas constant, and T is the absolute temperature. From the Arrhenius equation, it is apparent that temperature is the main factor that affects the rate of a chemical reaction.
Ideally, the rate constant accounts for all of the variables impacting reaction rate. The units of the rate constant depend on the order of reaction.
For higher order reactions or for dynamic chemical reactions, chemists apply a variety of molecular dynamics simulations using computer software. Despite its name, the rate constant isn't actually a constant. It only holds true at a constant temperature. It's affected by adding or changing a catalyst, changing the pressure, or even by stirring the chemicals. It doesn't apply if anything changes in a reaction besides the concentration of the reactants.
Also, it doesn't work very well if a reaction contains large molecules at a high concentration because the Arrhenius equation assumes reactants are perfect spheres that perform ideal collisions. Actively scan device characteristics for identification. Use precise geolocation data. Select personalised content. Create a personalised content profile.
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