The rate at which crickets chirp depends on the ambient...

The rate at which crickets chirp depends on the ambient temperature, because crickets are cold-blooded insects whose body temperature follows the temperature of their environment. It has been found that the Celsius temperature can be estimated by counting the number of chirps in 8 seconds and then adding $4 .$ In other words, $t_{c}=($ number of chirps in 8 seconds $)+4$ (a) Calculate the number of chirps in 8 seconds for temperatures of $20,25,30,$ and $35^{\circ} \mathrm{C}$. (b) The number of chirps per unit of time is directly proportional to the rate constant for a biochemical reaction involved in the cricket's chirp. (c) On the basis of this assumption, make a graph of $\ln ($ chirps in $8 \mathrm{~s}$ ) versus $(1 / T)$ (d) Calculate the activation energy for the biochemical reaction involved. (e) How many chirps would a cricket make in 8 seconds at a temperature of $40^{\circ} \mathrm{C}$ ?

The rate at which crickets chirp depends on the ambient temperature, because crickets are cold-blooded insects whose body temperature follows the temperature of their environment. It has been found that the Celsius temperature can be estimated by counting the number of chirps in 8 seconds and then adding $4 .$ In other words, $t_{c}=($ number of chirps in 8 seconds $)+4$
(a) Calculate the number of chirps in 8 seconds for temperatures of $20,25,30,$ and $35^{\circ} \mathrm{C}$.
(b) The number of chirps per unit of time is directly proportional to the rate constant for a biochemical reaction involved in the cricket's chirp.
(c) On the basis of this assumption, make a graph of $\ln ($ chirps in $8 \mathrm{~s}$ ) versus $(1 / T)$
(d) Calculate the activation energy for the biochemical reaction involved.
(e) How many chirps would a cricket make in 8 seconds at a temperature of $40^{\circ} \mathrm{C}$ ?

Key Concepts

Data Linearization in Kinetics

Data linearization is a technique used to transform exponential relationships, such as those described by the Arrhenius equation, into a linear form. By taking the natural logarithm of the rate constant and plotting it against the inverse of the temperature, one obtains a straight line. The slope of this line is related to the activation energy, allowing for its determination through a simple linear regression analysis. This approach simplifies the analysis of temperature-dependent reaction rates.

Temperature Dependence of Reaction Rates

This concept refers to the observation that the rates of chemical or biochemical reactions generally increase with temperature. In biochemical systems, higher temperatures provide molecules with more kinetic energy, making it easier to overcome energy barriers associated with molecular interactions and reactions. This principle underlies many experimental observations and predictions in chemical kinetics and thermodynamics.

Direct Proportionality

Direct proportionality describes a linear relationship where an increase in one quantity results in a proportional increase in another. In many experimental contexts, this relationship simplifies the analysis since the two variables change at a constant rate relative to each other. This approach is particularly useful when one measured quantity is used as a proxy for another variable that may be more difficult to measure directly.

Arrhenius Equation

The Arrhenius equation is a fundamental relation in chemical kinetics that models how the rate constant of a reaction depends on temperature. It is typically expressed as k = A exp(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. This equation explains the exponential increase in reaction rates with temperature and lays the groundwork for quantitative activation energy analysis.

Activation Energy

Activation energy is the minimum energy required to initiate a chemical or biochemical reaction. It represents the energy barrier that reactant molecules must overcome for the reaction to proceed. Determining the activation energy is crucial in understanding reaction kinetics because it provides insight into how sensitive a reaction is to temperature changes and the energy dynamics of the reaction pathway.

Transcript

00:01 So on this problem, we're modeling cricket chirping, and we're given this table of data where the temperatures here and the chirps per second of the crickets is listed.

00:16 And we're asked to use a graphing utility to draw a scattered diagram of this data and figure out what type of relationship appears.

00:25 And then to find either a linear or quadratic model that best models this data.

00:30 And then to use that to predict if the temperature's 80 degrees.

00:35 All right, so i went to desmus .com, brought down the graphing calculator here.

00:41 So the first thing i do is go into the plus and do table, and i start entering the data here, okay? so 88 .6, 93 .3, 80 .6, 99 .7, 69 .4, 79 .4, 79 .4.

01:16 Point six, 80 .6, 76 .3, and 75 .2.

01:38 Okay.

01:39 And then our chirps, 20, 18, 8, and 17 .1, 17 .1, and 14 .7 .7.

02:04 And 14 .7.

02:08 And 15 .4.

02:13 And 15 and 16 and 14 .4 and 15 .5.

02:35 So here's our data right here, right, like this.

02:53 So this data looks fairly flat.

02:58 Then it kind of problem i have is the scales i'm using right now.

03:01 So let's change our scales here for a minute.

03:04 So our xs go from roughly what 75 to almost 100 so let's go from 75 to 100 on the x and then the y scale here well let's see our chirps run anywhere from 14 something up to 20 so let's do 10 up to about 21...

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