Question

Experiments during a recent summer on a number of fireflies (small beetles, Lampyridaes photinus) showed that the average interval between flashes of individual insects was 16.3 \(\mathrm{s}\) at \(21.0^{\circ} \mathrm{C}\) and 13.0 \(\mathrm{s}\) at \(27.8^{\circ} \mathrm{C}\) a. What is the apparent activation energy of the reaction that controls the flashing? b. What would be the average interval between flashes of an individual firefly at \(30.0^{\circ} \mathrm{C} ?\) c. Compare the observed intervals and the one you calculated in part b to the rule of thumb that the Celsius temperature is 54 minus twice the interval between flashes.

Short answer

The apparent activation energy of the reaction that controls the flashing is approximately \(4.59 \times 10^4 J/mol\). The average interval between flashes of an individual firefly at \(30.0^{\circ}C\) would be approximately 11.6 seconds. The rule of thumb (\(T_C = 54 - 2(\Delta t)\)) produces reasonably accurate predictions for the temperature, but caution should be taken when using this rule for specific situations.

Step-by-step solution

Convert temperatures to Kelvin

Since the Arrhenius equation requires temperature to be measured in Kelvin, we must convert the given temperatures from Celsius to Kelvin: \(T_K = T_C + 273.15\) At \(T_C = 21.0^{\circ}C\): \(T_1 = 21.0 + 273.15 = 294.15 K\) At \(T_C = 27.8^{\circ}C\): \(T_2 = 27.8 + 273.15 = 300.95 K\)

Determine the activation energy

To determine the activation energy, we will use the Arrhenius equation. Let's rewrite the equation to find the ratio of reaction rates: \(\frac{k_1}{k_2} = \frac{Ae^{-\frac{E_a}{R T_1}}}{Ae^{-\frac{E_a}{R T_2}}}\) Simplify the equation by canceling out the pre-exponential factor A: \(\frac{k_1}{k_2} = e^{\frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)}\) We are given the intervals between firefly flashes at each respective temperature. We can assume that the flash intervals are inversely proportional to the reaction rates, meaning that we can write the ratio of reaction rates as the inverse ratio of the flash intervals: \(\frac{k_1}{k_2} = \frac{t_2}{t_1}\) Now, using the given intervals \(t_1 = 16.3s\) and \(t_2 = 13.0s\), and the temperature values we calculated, we can solve for \(E_a\): \(E_a = \frac{R \ln{\frac{t_2}{t_1}}}{\frac{1}{T_2} - \frac{1}{T_1}}= \frac{8.314 \ln{\frac{13.0}{16.3}}}{\frac{1}{300.95} - \frac{1}{294.15}} \approx 4.59 \times 10^4 J/mol\) So, the apparent activation energy is approximately \(4.59 \times 10^4 J/mol\).

Calculate average interval at 30.0°C

Now, we will use the activation energy we found to calculate the average interval between flashes at \(T_C = 30.0^{\circ}C\). First, convert the temperature to Kelvin: \(T_3 = 30.0 + 273.15 = 303.15 K\) Next, use the Arrhenius equation, replacing one of the reaction rate ratios with the inverse ratio of the flash intervals: \(\frac{t_3}{t_1} = e^{\frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_3}\right)}\) Solve for \(t_3\): \(t_3 = t_1e^{\frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_3}\right)} = 16.3 \times e^{\frac{4.59 \times 10^4}{8.314}\left(\frac{1}{294.15} - \frac{1}{303.15}\right)} \approx 11.6 s\) So, the average interval between flashes at \(30.0^{\circ}C\) would be approximately 11.6 seconds.

Compare to the rule of thumb

Finally, we will compare our calculated interval to the given rule of thumb: \(T_C = 54 - 2(\Delta t)\). Using the intervals at each temperature, calculate the predicted temperatures using the rule of thumb: For \(t_1 = 16.3s\): \(T_{C1} = 54 - 2(16.3) = 21.4^{\circ}C\) For \(t_2 = 13.0s\): \(T_{C2} = 54 - 2(13.0) = 28^{\circ}C\) For \(t_3 = 11.6s\): \(T_{C3} = 54 - 2(11.6) = 30.8^{\circ}C\) Comparing these values to the original temperatures (\(21.0^{\circ}C\), \(27.8^{\circ}C\), and \(30.0^{\circ}C\)), we observe that the rule of thumb produces reasonably accurate predictions for the temperature, particularly for the second and third intervals. However, it should be noted that the rule of thumb may only be accurate within a certain range of temperatures and flash intervals, and caution should be taken when using this rule for specific situations.

Key concepts

Activation Energy

Activation energy is a fundamental concept in chemistry, referring to the minimum energy required to initiate a chemical reaction. Think of it as the energy barrier that reactants must overcome to transform into products.
In the context of fireflies, activation energy governs the biochemical reaction leading to their luminous flashing. This energy is not constant and can be influenced by factors like temperature. To determine the activation energy (\(\Delta E_a\)), we use the Arrhenius equation, which shows the exponential relation between the rate of a reaction and its temperature:

\[ k = Ae^{-\frac{E_a}{RT}} \] where:

• \( k \) is the reaction rate constant.
• \( A \) is the pre-exponential factor (depends on reaction specifics).
• \( E_a \) is the activation energy (in Joules/mole).
• \( R \) is the universal gas constant (8.314 J/mol·K).
• \( T \) is the absolute temperature in Kelvin.

Understanding activation energy helps predict how changes in conditions will affect reaction speeds, such as how increasing the temperature results in faster flashing intervals in fireflies.

Reaction Rates

Reaction rates are all about how quickly or slowly a reaction proceeds. In the case of fireflies, it refers to how frequently they emit flashes.
The rate is affected by various factors, including temperature. As temperatures rise, reaction rates typically increase. This is due to molecules moving more energetically, colliding more often with enough energy to surpass the activation energy.

For example, in the firefly scenario, the rate at which the insects flash is faster at higher temperatures because the biochemical reactions driving the light emissions are accelerated. Mathematically, we can express reaction rates using the relation:

\[ k_2 = k_1 \cdot \left(\frac{t_1}{t_2}\right) \] Here, \( t_1 \) and \( t_2 \) represent the time intervals at different temperatures, indirectly indicating the reaction rates. Studies of reaction rates are crucial, as we can control and predict the reactions by understanding how factors affect these rates.

Temperature Conversion

In reaction kinetics, especially when dealing with the Arrhenius equation, converting temperature to Kelvin is crucial. This uniform unit avoids errors that arise due to the negative scale of Celsius.
The formula you use to convert Celsius to Kelvin is simple:

\[ T_K = T_C + 273.15 \]This ensures consistent calculations within the Arrhenius equation, avoiding problems that come with using Celsius directly.
For example:

• If the temperature is 21°C, in Kelvin, that's \(21 + 273.15 = 294.15 K\).
• At 27.8°C, it becomes \(27.8 + 273.15 = 300.95 K\).

Using Kelvin allows us to apply mathematical models correctly to assess the effect of temperature changes on reaction rates and activation energies.

Reaction Kinetics

Reaction kinetics is the branch of chemistry concerned with the rates and mechanisms of chemical reactions. In simpler terms, it studies how quickly reactions happen and what steps occur during these processes.
It is pivotal in understanding phenomena like the bioluminescence in fireflies. By applying kinetics principles, we predict the impact of temperature changes on flashing intervals.

Furthermore, kinetics provides insights into the efficiency and control of chemical processes. It utilizes the Arrhenius equation extensively to link reaction rates with activation energy and temperature. Two main components of reaction kinetics include:

• **Reaction Mechanism:** The step-by-step sequence that defines how reactants transform into products.
• **Rate Laws:** Equations that express the rate of a chemical process based on reactant concentrations and rate constants.

For students, grasping these elements is crucial as reaction kinetics plays a role in fields ranging from industrial synthesis processes to understanding and optimizing biological reactions like the hypnotic allure of a firefly's flash.