Problem 67 - Chemistry: Principles and Reactions

Question

For the reaction $$2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)$$ the rate constant is \(0.066 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(565^{\circ} \mathrm{C}\) and \(22.8 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(728^{\circ} \mathrm{C} .\) (a) What is the activation energy of the reaction? (b) What is \(k\) at \(485^{\circ} \mathrm{C}\) ? (c) At what temperature is \(k\), the rate constant, equal to \(11.6 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min} ?\)

Step-by-Step Solution

Answer

At what temperature will the rate constant be 11.6 L/mol•min? Answer: To estimate the activation energy, we use the Arrhenius equation and the given rate constants at two different temperatures. In part (a), we found the activation energy (Ea) by solving the Arrhenius equation with the given data. In part (b), we used the found activation energy to calculate the rate constant (k) at 485°C. In part (c), we found the temperature at which the rate constant becomes 11.6 L/mol•min using the activation energy obtained in part (a).

Step 1: Write the Arrhenius equation

The Arrhenius equation is $$k=Ae^{\frac{-E_{a}}{RT}}$$ where \(k\) is the rate constant, \(E_{a}\) is the activation energy, \(R\) is the gas constant (8.314 \(\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\)), \(T\) is the temperature in Kelvin, and \(A\) is the pre-exponential factor.

Step 2: Convert temperatures to Kelvin

To use the Arrhenius equation, we need to convert the given temperatures from Celsius to Kelvin: \(565^{\circ} \mathrm{C} + 273.15 = 838.15 \mathrm{K}\) \(728^{\circ} \mathrm{C} + 273.15 = 1001.15 \mathrm{K}\)

Step 3: Apply the Arrhenius equation to given data

We have two sets of given data - one for each given temperature. At \(565^{\circ} \mathrm{C}\) or \(838.15 \mathrm{K}\) : \(k_{1} = 0.066 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) At \(728^{\circ} \mathrm{C}\) or \(1001.15 \mathrm{K}\) : \(k_{2} = 22.8 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) We can form two equations from the Arrhenius equation using this data: $$k_{1}=Ae^{\frac{-E_{a}}{R \times 838.15}}$$ $$k_{2}=Ae^{\frac{-E_{a}}{R \times 1001.15}}$$

Step 4: Divide the two equations and solve for activation energy (\(E_{a}\))

Dividing the two equations, the \(A\) pre-exponential factor cancels out: $$\frac{k_{2}}{k_{1}} = \frac{e^{\frac{-E_{a}}{R \times 1001.15}}}{e^{\frac{-E_{a}}{R \times 838.15}}}$$ Taking the natural logarithm, we can solve for \(E_{a}\): $$E_{a} = R \times \frac{838.15 \times 1001.15 \times \ln{\frac{k_{2}}{k_{1}}}}{1001.15-838.15}$$ Plugging in the given values for \(k_{1}\) and \(k_{2}\), and using \(R=8.314\,\mathrm{J/mol\cdot K}\), the activation energy \(E_{a}\) can be calculated. #b) Calculating k at 485°C #

Step 1: Convert the temperature to Kelvin

\(485^{\circ} \mathrm{C} + 273.15 = 758.15 \mathrm{K}\)

Step 2: Use the Arrhenius equation with the activation energy (\(E_{a}\)) found in part (a)

Using either of the equations from step 3 in part (a), we can find the rate constant \(k\) at 485°C: $$k=0.066e^{\frac{-E_{a}}{8.314 \times (838.15-758.15)}}$$ Plug the value of \(E_{a}\) obtained in part (a) and solve for \(k\). #c) Rate constant value at k=11.6 L/mol•min#

Step 1: Use the Arrhenius equation to find the temperature

With the activation energy found in part (a), we can rewrite the Arrhenius equation as: $$11.6 = 0.066\,e^{\frac{-E_{a}}{8.314 \times (T - 838.15)}}$$

Step 2: Solve for the temperature T in Kelvin

To find the temperature T, we need to rearrange the equation and take the natural logarithm: $$T = 838.15 + \frac{-E_{a}}{8.314\,(\ln{\frac{11.6}{0.066}})}$$ Plug the value of \(E_{a}\) obtained in part (a) and solve for \(T\).

Step 3: Convert the temperature from Kelvin to Celsius

Lastly, convert the temperature T from Kelvin to Celsius: \(T^{\circ} \mathrm{C} = T \mathrm{K} - 273.15\) Now we have the temperature where the rate constant is equal to \(11.6 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\).

Key Concepts

Activation Energy

Activation energy, denoted as \(E_a\), refers to the minimum amount of energy that reacting particles must have for a chemical reaction to occur. It's a crucial concept within chemical kinetics, acting as an energy barrier that must be overcome for reactants to transform into products. In the context of the Arrhenius equation, \(E_a\) can be calculated using temperature and rate constants associated with the reaction.

Understanding activation energy helps in explaining why certain reactions happen quickly while others do not. For example, when you strike a match, the activation energy required for the chemicals in the match head to react is rapidly provided by the frictional heat, initiating combustion. On the molecular level, the activation energy corresponds to the peak in potential energy that reactants must reach before descending to a lower energy, stable product state.

Rate Constant

The rate constant, represented as \(k\), is a proportionality factor that shows up in the rate laws of chemical reactions. It's a measure of the rate at which reactants are transformed into products. In the Arrhenius equation, the rate constant directly correlates with the temperature and activation energy of a reaction.

Factors that influence the rate constant include the physical state of the reactants, the concentration of the reactants, pressure (for gases), and the presence of a catalyst. A larger rate constant indicates a faster reaction. The dependency of the rate constant on temperature is particularly significant because even a small increase in temperature can lead to a substantial increase in the rate constant, thereby speeding up the reaction.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the factors that affect these rates. It helps us understand the speed (or slowness) of reactions and how various conditions might affect this speed. Key principles within kinetics include the reactions' rate laws, the role of catalysts, and the determination of reaction mechanisms.

The study of kinetics involves not only measuring how fast a reaction occurs but also deducing the sequence of steps (mechanism) that leads to the product formation. This understanding can be applied to control reactions in industrial processes, predict the shelf life of pharmaceuticals, and even aid in environmental assessments concerning pollutant degradation.

Temperature Conversion

Temperature conversion in chemistry is essential because most formulas, including the Arrhenius equation, require temperatures to be in Kelvin (K). The Kelvin scale is an absolute temperature scale, with 0 K being absolute zero, where theoretically, particles have no kinetic energy.

To convert Celsius to Kelvin, which is a common task in chemical kinetics, you add 273.15 to the Celsius temperature. This adjustment is necessary because the Arrhenius equation uses the gas constant \(R\), which is based on the Kelvin scale. Abiding by these conversion rules is vital for calculating quantities accurately, such as the activation energy or rate constant of a reaction.