Name: finding rate law from initial rates
Uploaded: 2020-09-24T15:06:50.000Z
Duration: 2 min 53 s
Channel: JoVE

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Determining Order of Reaction

For a chemical reaction, the rate law expresses the relationship between the reaction rate and reactant concentration. The exponents of the reactant concentrations influence the reaction rate individually and are called reaction orders.
The reaction order is determined experimentally by employing the method of initial rates, where a chemical reaction is repeated multiple times with varying reactant concentrations to measure the initial reaction rates.
An increase in the reactant concentration, producing a linearly-proportional increase in the reaction rate, characterizes a first-order reaction. If doubling the reactant concentration quadruples the initial rate, then a second-order reaction is observed.
However, if the change of reactant concentrations does not affect the initial rate values, then a zero-order reaction is observed.
When the obtained data of initial rates present an indiscernible relationship between the change in initial reactant concentration and the corresponding rate, a ratio of rate laws is calculated.
Here, any two concentration values of the reactant and their corresponding reaction rates are used to determine the reaction order.
But what about reactions with multiple reactants? First, using the method of initial rates, the reaction order of each reactant is determined individually.
Next, the individual reaction orders are expressed as exponents to their respective reactant concentrations to formulate the rate law. Lastly, the summation of individual exponents from the rate law determines the overall reaction order.
While the reaction order depicts the reaction rate&#8217;s dependence on the reactant concentration, a direct measure of the relative reaction speed is indicated by the rate constant.
The rate constant k, is the proportionality coefficient relating the reaction rate to the product of reactant concentrations.
The unit of a rate constant depends on the overall reaction order and can be determined by rearranging the rate law to solve for the rate constant. For a zero-order reaction, k has the unit molarity per seconds. The unit for a first-order reaction is 1/s, and for a second-order reaction it is 1/M&#183;s.
A large rate constant indicates a fast reaction. Conversely, a smaller rate constant indicates a slow reaction. A value of zero signifies the absence of any chemical reaction.

determining order of reaction

Rate laws describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In a rate law, the rate constant k and the reaction orders are determined experimentally by observing how the rate of reaction changes as the concentrations of the reactants are changed. A common experimental approach to the determination of rate laws is the method of initial rates. This method involves measuring reaction rates for multiple experimental trials carried out using different initial reactant concentrations. Comparing the measured rates for these trials permits determination of the reaction orders and, subsequently, the rate constant, which together are used to formulate a rate law.
The rate of a reaction, for example, involving nitric oxide with ozone [NO (g) + O3 (g) &#10230; NO2 (g) + O2 (g)] can be determined from the experimental data of method of initial rates, in the laboratory.
<table style="margin-left: auto; margin-right: auto;">
	<tbody>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;"> Trial </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> [NO] (mol/L)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">[O3] (mol/L)</td>
			<td style="color: black; border: 1px solid black; text-align: center;"> &#916;[NO2]/&#916;t (mol/L&#183;s) </td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1</td>
			<td style="color: black; border: 1px solid black; text-align: center;"> 1.00 &#215; 10&#8722;6 </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> 3.00 &#215; 10&#8722;6 </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> 6.60 &#215; 10&#8722;5 </td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">2</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">6.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.32 &#215; 10&#8722;4</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">9.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.98 &#215; 10&#8722;4</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">4</td>
			<td style="color: black; border: 1px solid black; text-align: center;">2.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">9.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">3.96 &#215; 10&#8722;4</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">5</td>
			<td style="color: black; border: 1px solid black; text-align: center;">3.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">9.00 &#215; 10&#8722;6</td>
			<td style="color: black; border: 1px solid black; text-align: center;">5.94 &#215; 10&#8722;4</td>
		</tr>
	</tbody>
</table>
From the rate data, a generic rate law; rate = k[NO]m[O3]n is formulated. The values of the reaction orders m and n, and rate constant k are determined from the experimental data using a three-part process:
In step 1, the value of m is determined from the data in which [NO] varies, and [O3] is constant. In trials 3, 4 and 5, [NO] varies while [O3] remains constant. When [NO] doubles from trial 3 to 4, the rate doubles, and when [NO] triples from trial 3 to 5, the rate also triples. Thus, the rate is also directly proportional to [NO], and m in the rate law is equal to 1.
In step 2, the value of n is determined from data in which [O3] varies, and [NO] is constant. In trials 1,2 and 3, [NO] is constant and [O3] varies. The reaction rate changes in direct proportion to the change in [O3]. When [O3] doubles from trial 1 to 2, the rate doubles; when [O3] triples from trial 1 to 3, the rate increases also triples. Thus, the rate is directly proportional to [O3], and n is equal to 1. The rate law is thus: rate = k [NO]1 [O3]1 = k [NO][O3]
In step 3, the value of k is determined from one set of concentrations (for instance, the data from trial 1) and its corresponding rate.
<img alt="Chromatography diagram with benzylpenicillin formula; C16H18N2O4S structure for education." src="/files/ftp_upload/11376/11376_Equation%201.png" style="text-align: center; height: 145px;" data-altupdatedat="1754904545000" data-alt="Bacterial transformation diagram showing plasmid insertion via heat shock, process illustration." />
In reactions where the data from the method of initial rates does not directly imply the relation between initial concentrations and initial rates, a calculation involving the ratio of rate laws can be employed to calculate the reaction order and rate constant.
For example, the general rate law for the reaction 2 NO (g) + Cl2 (g) &#10230; 2 NOCl (g) is expressed as: rate = k [NO]m [Cl2]n.
The data from the method of initial rates are:
<table style="margin-left: auto; margin-right: auto;">
	<tbody>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;"> Trial </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> [NO] (mol/L) </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> [Cl2] (mol/L) </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> Initial Rate (mol/L&#183;s) </td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.10</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.10</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.00300</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">2</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.10</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.00450</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.10</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.00675</td>
		</tr>
	</tbody>
</table>
The values of m and n can be determined from the experimental data using an algebraic approach, following which the value of k is determined.
In step 1, the value of m is determined from the data in which [NO] varies and [Cl2] is constant. A ratio of rate laws is expressed by substituting data from two different trials (for instance trial 3 and trial 1).
<img alt="DNA recombination diagram showing insertion of foreign gene into plasmid for genetic engineering." src="/files/ftp_upload/11376/11376_Equation%202.png" style="text-align: center; height: 260px;" data-altupdatedat="1754904545000" data-alt="Chemical diagram of phenylalanine conversion showing structural formulae and reaction arrows." />
In step 2, the value of n is determined from data in which [Cl2] varies, and [NO] is constant.
<img alt="Static equilibrium diagram; pulley system; forces &#931;Fx=0, &#931;Fy=0; educational physics concept." src="/files/ftp_upload/11376/11376_Equation%204.png" style="text-align: center; height: 120px;" data-altupdatedat="1754904545000" data-alt="Free body diagram, static equilibrium, &#931;Fx=0, &#931;Fy=0, showing forces, tensions, weight balance." />
Using the computed values of m and n the rate law is expressed as rate = k [NO]2 [Cl2].
In step 3, the numerical value of the rate constant k is determined with appropriate units. The units for the rate of a reaction are mol/L&#183;s. The units for k is concluded by substituting the units of all other parameters in the rate law. In this example, the concentration units are mol3/L3. The units for k should be L2/mol2&#183;s so that the rate is in terms of mol/L&#183;s. The value of k is determined once the rate law expression has been solved, by simply substituting the values from any of the experimental trials (for instance trial 1).
<img alt="Static equilibrium diagram; &#931;Fx=0, &#931;Fy=0; free body diagram with forces and angles labeled." src="/files/ftp_upload/11376/11376_Equation%205.png" style="text-align: center; height: 70px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium &#931;Fx=0, &#931;Fy=0 equations; mathematical diagram; mechanics principles." />
Reaction Order and Rate Constant Units
In some reactions, the reaction orders in the rate law happen to be the same as the coefficients in the chemical equation for the reaction. This is merely a coincidence and very often not the case. Rate laws may exhibit fractional orders for some reactants, and negative reaction orders are sometimes observed when an increase in the concentration of one reactant causes a decrease in reaction rate. Rate laws are determined by experiment only and are not reliably predicted by reaction stoichiometry.
The units for a rate constant will vary as appropriate to accommodate the overall order of the reaction. The unit of the rate constant for a zero-order reaction is mol/L&#183;s (or M/s) and that for a first-order reaction is 1/s. The unit of the rate constant for a second-order reaction is L/mol&#183;s (or 1/M&#183;s) and that for a third-order reaction is L2/mol2&#183;s. Although the specific units for concentration and time are indicated as (mol/L) and (s), any other valid units can be used to represent the properties of concentration and time.
This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/12-3-rate-laws#57033">Openstax, Chemistry 2e, Section 12.3: Rate Laws.</a>

Rate laws describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In a rate law, the rate constant k and ...

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KEY TERMS AND CONCEPTS

finding rate law from initial rates

Rate laws connect the speed of a chemical reaction to the concentration of its reactants. The rate constant, k, and the reaction orders are not taken from the chemical equation. They are found experimentally by watching how the rate changes when the reactant concentrations change.

One common way to do this is the method of initial rates. In this method, several trials are run with different starting concentrations. The measured rates are then compared to determine the reaction orders and, after that, the rate constant. These values are used to write the rate law.

For the reaction of nitric oxide with ozone, NO(g) + O3(g) → NO2(g) + O2(g), the data show how this works in the lab. When [NO] is held constant and [O3] changes, the rate changes in the same way. When [O3] doubles, the rate doubles. When [O3] triples, the rate triples. This shows that the reaction is first order in O3. A similar comparison using trials where [O3] stays constant shows that the rate is also directly proportional to [NO], so the reaction is first order in NO as well. The rate law is therefore rate = k[NO][O3].

In other reactions, the orders are found with an algebraic ratio of rate laws. For the reaction 2 NO(g) + Cl2(g) → 2 NOCl(g), the rate law is written as rate = k[NO]m[Cl2]n. Using the trial data, one step gives m = 2 and another gives n = 1. The rate law becomes rate = k[NO]2[Cl2]. The value of k is then found by using one trial and solving for the constant.

The units of the rate constant depend on the overall order of the reaction. Rate is measured in mol/L·s, and the units for k must match the rate law. For this example, the concentration terms give mol3/L3, so k must have units of L2/mol2·s. In general, a zero-order reaction has k in mol/L·s, a first-order reaction has k in 1/s, a second-order reaction has k in L/mol·s, and a third-order reaction has k in L2/mol2·s.

Reaction order does not always match the coefficients in the balanced equation. That match can happen, but it is only a coincidence. Rate laws can also include fractional orders, and some reactants can have negative orders if a higher concentration lowers the reaction rate. Because of this, rate laws are determined by experiment, not by stoichiometry. Although the examples here use mol/L and s, other valid concentration and time units can also be used.

Finding Rate Law from Initial Rates

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Use initial-rate data to find reaction order and the rate constant. Explore examples with NO, ozone, and NOCl, plus how rate constant units change.

Finding Rate Law from Initial Rates | JoVE K12

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IPC.7.D

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a ...

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that ...

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of ...

Quantum Numbers

It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one energy ...

Balmer Series

The Bohr Model
Niels Bohr proposed a model for the hydrogen atom in 1913 that described discrete energy states are associated with a fixed electron orbit ...

Beer's Law

Absorbance and Fluorescence
When light hits a substance, it is either absorbed, transmitted, or reflected. Typically, a substance interacts with a range ...

The Wave Nature of Light

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms ...

Explain how atomic energy levels and emission spectra present evidence for the wave particle duality.

IPC.7.E

Reaction Rate

The rate of reaction is the change in the amount of a reactant or product per unit time. Reaction rates are therefore determined by measuring the time ...

Polarimetry finds application in chemical kinetics to measure the concentration and reaction kinetics of optically active substances during a chemical ...

Concentration and Rate Law

The rate of a reaction is affected by the concentrations of reactants. Rate laws (differential rate laws) or rate equations are mathematical expressions ...

Plan and conduct an investigation to provide evidence that the rate of reaction or dissolving is affected by multiple factors such as particle size, stirring, temperature, and concentration.

Science concepts. The student knows that relationships exist between the structure and properties of matter. The student is expected to:

Finding Rate Law from Initial Rates

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