Question

Experiments have shown the average frequency of chirping of individual snowy tree crickets (Oecanthus fultoni) to be $178{\min }^{-1}$ at ${25.0}^{\circ }\mathrm{C},126{\min }^{-1}$ at ${20.3}^{\circ }\mathrm{C}$, and $100.{\min }^{-1}$ at ${17.3}^{\circ }\mathrm{C}$. a. What is the apparent activation energy of the reaction that controls the chirping? b. What chirping rate would be expected at ${15.0}^{\circ }\mathrm{C}$ ? c. Compare the observed rates and your calculated rate from part b to the rule of thumb that the Fahrenheit temperature is 42 plus $0.80$ times the number of chirps in $15\mathrm{s}$.

Short answer

The apparent activation energy of the reaction that controls the chirping is approximately $54.26\mathrm{kJ}/\mathrm{mol}$. The expected chirping rate at ${15.0}^{\circ }\mathrm{C}$ would be around $84.63\stackrel{-1}{min}$. The observed rates and calculated rate are in good agreement with the rule of thumb, as both ${25}^{\circ }\mathrm{C}$ and ${15}^{\circ }\mathrm{C}$ cases yield values close to the Fahrenheit temperature calculated from the rule of thumb.

Step-by-step solution

Understand the Arrhenius equation

The Arrhenius equation is a formula for the temperature dependence of reaction rates. It can be written as: $$k=A{e}^{\frac{-E_a}{RT}}$$ where: - $k$ is the reaction rate constant - $A$ is the pre-exponential factor - $E_a$ is the activation energy - $R$ is the universal gas constant ($8.314\mathrm{J}/(\mathrm{mol}\cdot \mathrm{K})$) - $T$ is the absolute temperature in Kelvin (K)

Calculate the activation energy

We are given the reaction rate constants and temperatures, so we can set up the following equations using the Arrhenius equation: $$\frac{k_2}{k_1}=e^{\frac{-E_a}{R}(\frac{1}{T_2}-\frac{1}{T_1})}$$ $$\frac{k_3}{k_2}=e^{\frac{-E_a}{R}(\frac{1}{T_3}-\frac{1}{T_2})}$$ Here, the given temperature data must first be converted to Kelvin: $T_1=25.0+273.15=298.15\mathrm{K}$, $T_2=20.3+273.15=293.45\mathrm{K}$, $T_3=17.3+273.15=290.45\mathrm{K}$. Now plug in the known values: $$\frac{126}{178}=e^{\frac{-E_a}{8.314}(\frac{1}{293.45}-\frac{1}{298.15})}$$ $$\frac{100}{126}=e^{\frac{-E_a}{8.314}(\frac{1}{290.45}-\frac{1}{293.45})}$$ Solve these equations for the activation energy $E_a$, and we get $E_a\approx 54.26\mathrm{kJ}/\mathrm{mol}$.

Calculate the chirping rate at 15°C

Now we want to find the chirping rate ($k_{15}$) at ${15}^{\circ }\mathrm{C}$ (or $288.15\mathrm{K}$). We can use the Arrhenius equation for this with the calculated activation energy: $$k_{15}=178e^{\frac{-54.26\times {10}^3}{8.314}(\frac{1}{288.15}-\frac{1}{298.15})}$$ Solving for $k_{15}$, we get a chirping rate of approximately $84.63\stackrel{-1}{min}$.

Compare calculated rate to the rule of thumb

According to the rule of thumb, the Fahrenheit temperature is 42 plus 0.80 times the number of chirps in 15 seconds. Let's use this rule to calculate the temperature and compare it to the given data: At ${25}^{\circ }\mathrm{C}$ (or ${77}^{\circ }\mathrm{F}$), the number of chirps in 15 seconds is $178\stackrel{-1}{min}\times \frac{15}{60}=44.5$, so the rule of thumb gives: $$77\approx 42+0.80\times 44.5$$ At ${15}^{\circ }\mathrm{C}$ (or ${59}^{\circ }\mathrm{F}$), the calculated chirping rate is $84.63\stackrel{-1}{min}$. Thus, the number of chirps in 15 seconds is $84.63\stackrel{-1}{min}\times \frac{15}{60}=21.16$, so the rule of thumb gives: $$59\approx 42+0.80\times 21.16$$ In both cases, the calculated rates are in good agreement with the rule of thumb, suggesting that our activation energy and chirping rate calculations are reasonable.

Key concepts

Activation Energy

Activation energy is the minimum amount of energy required for a chemical reaction to occur. It's like a barrier that reactants must overcome to transform into products. In the context of the Arrhenius equation, the activation energy determines how sensitive a reaction rate is to temperature changes. A higher activation energy means that temperature has a more pronounced effect on how fast the reaction goes. Imagine it as a gatekeeper allowing the reaction to proceed when enough energy is provided.
In our exercise, we calculate the activation energy using crickets' chirp rates at different temperatures, following the Arrhenius equation. By comparing chirp rates at various temperatures, we can solve for the activation energy, giving insight into the biological processes influencing these sounds.

Reaction Rate Constant

The reaction rate constant, often denoted as $k$, is a crucial component in the Arrhenius equation. It provides a measure of how fast a reaction occurs. The value of $k$ changes with temperature, and for every reaction, there's a unique rate constant.
In simpler terms, the reaction rate constant tells us about the speed of crickets' chirping under specific conditions. The higher the value of $k$, the faster the chirping happens. To determine $k$ under different conditions, we use observed data to calculate its value at a specific temperature using the Arrhenius formula. This calculation helps us predict chirp rates at other temperatures.

Temperature Dependence of Reaction Rates

Temperature significantly impacts reaction rates, with most chemical reactions happening faster at higher temperatures. The Arrhenius equation quantifies this dependency, showing how $k$ changes with temperature. As temperature increases, molecules have more energy to overcome activation energy barriers.
In our example, snowy tree crickets chirp more frequently at higher temperatures. By applying the Arrhenius equation, we confirm this pattern and calculate how much chirp rates alter with temperature shifts. This scientific approach allows precise predictions about how changing conditions influence biological and chemical reactions.

Universal Gas Constant

The universal gas constant $R$ is a constant feature in thermodynamic equations, and it's vital for the Arrhenius equation. $R$ connects the amount of energy per mole per degree, making it possible to relate energy changes in a reaction to temperature changes. It has a value of $8.314\mathrm{J}/(\mathrm{mol}\cdot \mathrm{K})$, and it's what allows us to include temperatures measured in Kelvin directly in our calculations.
In the exercise, $R$ relates activation energy with temperature shifts influencing chirping rates. Without $R$, it would be challenging to unify energy, temperature, and rates for predictive models in chemistry. This constant bridges the reaction mechanics with the measurable temperature and energy changes.