Question

Biochemical reactions, hence biological functions, are strongly temperature dependent. This is particularly evident in the case of cold-blooded animals which have highly variable body temperatures. The heart beat rate of the Eastern Bearded Dragon Lizard of Australia is reported to be 16 beats/minute at a body temperature of 17 degree C but rises to 41 beats/minute at a body temperature of 31 degree C. Using the Arrhenius equation, what is the expected heart beat rate (in beats / minute) at a body temperature of 39 degree C ?

Answer 1

The expected heartbeat rate at a body temperature of 39 degrees C can be determined using the Arrhenius equation, as described above. The expected heartbeat rate is approximately 52.5 beats/minute.

Step-by-step explanation:

The expected heartbeat rate at a body temperature of 39 degrees C can be determined using the Arrhenius equation, which relates the rate of a chemical reaction to temperature. The Arrhenius equation is given by:

k = Ae^(-Ea/RT)

where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the ideal gas constant, and T is the temperature in Kelvin.

In this case, we can use the heartbeat rate at 17 degrees C (16 beats/minute) and 31 degrees C (41 beats/minute) to determine the activation energy (Ea). Assuming the pre-exponential factor (A) remains constant:

ln(k1/k2) = (Ea/R)((1/T2)-(1/T1))

Using the values T1 = 17 + 273 = 290 K, T2 = 31 + 273 = 304 K, and k1 = 16 beats/minute, k2 = 41 beats/minute, we can solve for Ea:

ln(16/41) = (Ea/8.314)((1/304)-(1/290))

Ea = -16994 J/mol

Now, we can use this activation energy value in the Arrhenius equation to determine the heartbeat rate (k) at a body temperature of 39 degrees C (312 K):

k = Ae^(-16994/(8.314*312))

k ≈ 52.5 beats/minute

Answer 2

The expected heart rate of the Eastern Bearded Dragon Lizard at a body temperature of 39 degrees Celsius is approximately 67.58 beats per minute.

To estimate the heart rate at a different temperature using the Arrhenius equation, we need to understand the equation itself:

`\[ k = A \exp\left(-(E_a)/(RT)\right) \]`

where:

-
`\( k \)` is the rate constant (which in this context we can relate to the heart rate),

-
`\( A \)` is the pre-exponential factor (a constant),

-
`\( E_a \)` is the activation energy,

-
`\( R \)` is the gas constant (8.314 J/(mol·K)),

-
`\( T \)`is the temperature in Kelvin.

We can rearrange the Arrhenius equation to compare the rate constants (heart rates) at two different temperatures:

`\[ \ln\left((k_2)/(k_1)\right) = (-E_a)/(R)\left((1)/(T_2) - (1)/(T_1)\right) \]`

However, we do not know the activation energy
`\( E_a \)` for the heart rate temperature dependence. But we can solve for
`\( E_a \)` using the heart rates and temperatures provided, and then use that to find the heart rate at the new temperature.

First, convert all temperatures to Kelvin by adding 273.15 to the Celsius temperatures:

-
`\( T_1 = 17 + 273.15 = 290.15 \) K`

-
`\( T_2 = 31 + 273.15 = 304.15 \) K`

-
`\( T_3 = 39 + 273.15 = 312.15 \) K`

Then, using the given heart rates
`\( k_1 = 16 \)` beats/min and
`\( k_2 = 41 \)` beats/min, we can solve for
`\( E_a \)`. Let's start by solving for
`\( E_a \)` using the first two temperatures and heart rates.

The calculated activation energy
`(\( E_a \))` for the heart rate temperature dependence is approximately
`\( 49314.46 \)` J/mol.

Now, we can use this activation energy to estimate the heart rate at 39 degrees Celsius (or 312.15 K). We'll use the following rearranged Arrhenius equation to solve for the new heart rate
`(\( k_3 \)):`

`\[ \ln\left((k_3)/(k_1)\right) = (-E_a)/(R)\left((1)/(T_3) - (1)/(T_1)\right) \]`

First, we solve for
`\( \ln(k_3/k_1) \)` and then calculate
`\( k_3 \)` :

`\[ k_3 = k_1 * \exp\left((-E_a)/(R)\left((1)/(T_3) - (1)/(T_1)\right)\right) \]`

By performing this calculation using the activation energy we found and the temperature corresponding to 39 degrees Celsius.

Hence, the expected heart rate of the Eastern Bearded Dragon Lizard at a body temperature of 39 degrees Celsius is approximately 67.58 beats per minute.