To plot the given data and calculate the activation energy, we need to follow these steps:

1. Understand the relationship: The data suggests that $T(c)$ corresponds to temperature in Celsius, and $k(s-1)$ is the rate constant (presumably in units of $s^{-1}$).

2. Use the Arrhenius equation:
   The Arrhenius equation expresses the temperature dependence of reaction rates as:

   $k = A \cdot e^{-\frac{E_a}{RT}}$

   Where:

   • $k$ is the rate constant,
   • $A$ is the pre-exponential factor,
   • $E_a$ is the activation energy,
   • $R$ is the gas constant (8.314 J/mol·K),
   • $T$ is the temperature in Kelvin.

   To find the activation energy $E_a$, we take the natural logarithm of both sides of the equation:

   $\ln(k) = \ln(A) - \frac{E_a}{R} \cdot \frac{1}{T}$

   This is a linear equation of the form $y = mx + b$, where:

   • $y = \ln(k)$,
   • $x = \frac{1}{T}$,
   • $m = -\frac{E_a}{R}$ (the slope).

3. Steps to calculate activation energy:

   • Convert the temperature from Celsius to Kelvin by adding 273.15.
   • Plot $\ln(k)$ vs. $\frac{1}{T}$.
   • Determine the slope of the line, which will give us $-\frac{E_a}{R}$.
   • Solve for $E_a$.

I’ll now plot the graph and calculate the activation energy.

Let me first plot the data. numpy as np import matplotlib.pyplot as plt

Given data

T_celsius = np.array([190, 199, 215, 230, 248]) # Temperature in Celsius k_s_1 = np.array([0.000025, 0.000053, 0.00025, 0.00063, 0.0032]) # Rate constant (s^-1)

Convert temperature to Kelvin

T_kelvin = T_celsius + 273.15

Calculate 1/T and ln(k)

inv_T = 1 / T_kelvin ln_k = np.log(k_s_1)

Plotting the graph of ln(k) vs. 1/T

plt.figure(figsize=(8, 6)) plt.plot(inv_T, ln_k, 'o-', label="Data points", color='b') plt.title("Arrhenius Plot: ln(k) vs. 1/T") plt.xlabel("1/T (1/K)") plt.ylabel("ln(k)") plt.grid(True) plt.legend() plt.show()The plot above shows the relationship between $\ln(k)$ and $\frac{1}{T}$. To calculate the activation energy, we need to determine the slope of the line, which corresponds to $-\frac{E_a}{R}$.

Let me now calculate the slope and determine the activation energy $E_a$. Perform linear regression to find the slope (m) and intercept (b) slope, intercept = np.polyfit(inv_T, ln_k, 1)

Calculate the activation energy (E_a) in Joules per mole

R = 8.314 # Gas constant in J/mol·K E_a_Joules_per_mol = -slope * R # E_a = -slope * R

Convert activation energy from Joules to kilojoules (1 kJ = 1000 J)

E_a_kJ_per_mol = E_a_Joules_per_mol / 1000

E_a_kJ_per_molThe activation energy $E_a$ is approximately 166 kJ/mol.

Would you like more details on the calculations, or have any further questions about this?

Here are some related questions you might find helpful:

1. How can I interpret the results of an Arrhenius plot in terms of reaction kinetics?
2. What does the pre-exponential factor $A$ represent in the Arrhenius equation?
3. How can temperature affect the rate constant $k$ in chemical reactions?
4. What are common units for activation energy and rate constants in chemical kinetics?
5. How would the activation energy change if the reaction took place at a different temperature range?

Tip: Always ensure temperatures are in Kelvin when using the Arrhenius equation, as it's essential for accurate calculations.