13.8
Arrhenius Plots
The Arrhenius equation relates the activation energy and the rate constant, k, for chemical reactions. In the Arrhenius equation, k = Ae−Ea/RT, R is t…
The rate of a chemical reaction is highly sensitive to changes in temperature. This temperature dependence is mathematically explained using the Arrhenius equation; which expresses the relationship between the rate constant, the absolute temperature, the frequency factor, and the activation energy.
The activation energy and frequency factor can also be determined graphically by converting the Arrhenius equation into a non-exponential form.
Using the natural logarithms on both sides, an equation for a linear function is generated. The slope value corresponds to the negative value of activation energy over gas constant, and the y-intercept corresponds to the natural log of the frequency factor.
This equation can be used to generate a graph called the Arrhenius plot, in which the natural log of the rate constant is denoted as a function of the inverse of temperature in kelvin.
Kinetic data of experiments and reactions can be illustrated and analyzed using this Arrhenius plot. In this example, the graph yields a straight line. The slope value given in kelvin is set equal to the negative value of activation energy over R. After assigning the value for the gas constant and solving for the activation energy, a value of 93.1 kJ/mol is obtained.
Besides, the y-intercept of 26.8 is equal to the natural log of the frequency factor. Thus, solving for A gives the value of 4.36 × 1011 with the unit one-over-molarity-seconds — the same unit as the rate constant.
In cases of limited kinetic data or difficulties with graphical representation, a two-point form of the Arrhenius equation can be utilized to calculate the activation energy in a non-graphical manner.
In such cases, the non-exponential form of the Arrhenius equation is modified to include rate constants at two different temperatures.
Subsequent subtraction and rearrangement of the expression yield the two-point form of the Arrhenius equation, which is used to calculate the activation energy from experimentally-generated rate constants at two different temperatures. By substituting the values, the activation energy for this reaction is calculated to be 145 kJ/mol.
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Q1: What does the Arrhenius equation show about reaction rates and temperature?
The Arrhenius equation expresses the relationship between the rate constant, absolute temperature, frequency factor, and activation energy. It demonstrates that reaction rates are highly sensitive to temperature changes. The equation k = Ae−Ea/RT allows chemists to quantify how temperature affects the speed of chemical reactions.
Q2: How do you convert the Arrhenius equation into a linear form for graphing?
Taking natural logarithms of both sides of the Arrhenius equation generates a linear function: ln k = ln A − (Ea/R)(1/T). This linear form allows you to plot ln k versus 1/T to create an Arrhenius plot. The slope equals −Ea/R and the y-intercept equals ln A, making it easy to extract kinetic parameters graphically.
Q3: What information can you extract from the slope and y-intercept of an Arrhenius plot?
The slope of an Arrhenius plot equals the negative value of activation energy divided by the gas constant R. Solving for activation energy yields its numerical value in kJ/mol. The y-intercept equals the natural log of the frequency factor A, which represents collision frequency and molecular orientation. Both parameters are essential for understanding reaction kinetics.
Q4: What is the two-point form of the Arrhenius equation used for?
The two-point form of the Arrhenius equation calculates activation energy from rate constants measured at two different temperatures without graphing. This non-graphical method is useful when kinetic data is limited or graphical representation is difficult. It involves rearranging and subtracting the Arrhenius equation at two temperatures to solve directly for Ea.
Q5: What does the frequency factor A represent in the Arrhenius equation?
The frequency factor A is a constant related to the frequency of molecular collisions and the proper orientation of reacting molecules. It has the same units as the rate constant and can be determined from the y-intercept of an Arrhenius plot by calculating the antilog of ln A. The frequency factor reflects how often molecules collide with favorable geometry.
Q6: Why is the graphical approach typically more reliable than the two-point method for experimental data?
The graphical approach uses multiple data points to establish a best-fit line, which averages out experimental errors and uncertainties. The two-point method relies on only two data pairs, making it more sensitive to measurement errors in those specific points. Using more experimental data points through graphical analysis generally yields more accurate activation energy values.
Q7: How do you calculate activation energy from an Arrhenius plot with experimental kinetic data?
Plot ln k versus 1/T using experimental rate constants at different temperatures to generate a linear Arrhenius plot. Calculate the slope using any two data points on the line. Set the slope equal to −Ea/R, substitute the gas constant value (8.314 J/mol·K), and solve algebraically for activation energy in kJ/mol.