Consider the two reactions:
O + N₂ → NO + N E_a = 315 kJ/mol
Cl + H₂ → HCl + H E_a = 23 kJ/mol
b. The frequency factors for these two reactions are very close to each other in value. Assuming that they are the same, calculate the ratio of the reaction rate constants for these two reactions at 25 °C.

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Identify the Arrhenius equation: \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the frequency factor, \( E_a \) is the activation energy, \( R \) is the gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin.

Convert the temperature from Celsius to Kelvin: \( T = 25 + 273.15 = 298.15 \text{ K} \).

Since the frequency factors \( A \) are assumed to be the same for both reactions, the ratio of the rate constants \( \frac{k_1}{k_2} \) can be expressed as \( \frac{e^{-\frac{E_{a1}}{RT}}}{e^{-\frac{E_{a2}}{RT}}} = e^{-\frac{E_{a1} - E_{a2}}{RT}} \).

Substitute the given activation energies into the equation: \( E_{a1} = 315 \text{ kJ/mol} = 315,000 \text{ J/mol} \) and \( E_{a2} = 23 \text{ kJ/mol} = 23,000 \text{ J/mol} \).

Calculate the exponent: \( -\frac{(315,000 - 23,000)}{8.314 \times 298.15} \) and use this to find the ratio \( \frac{k_1}{k_2} = e^{\text{calculated exponent}} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arrhenius Equation

The Arrhenius equation describes how the rate constant of a chemical reaction depends on temperature and activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. This equation highlights the exponential relationship between temperature and reaction rates, indicating that higher temperatures generally lead to faster reactions.

Activation Energy (Ea)

Activation energy is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to transform into products. In the context of the given reactions, the activation energies of 315 kJ/mol and 23 kJ/mol indicate that the first reaction is significantly more energy-intensive than the second, which affects their respective reaction rates at a given temperature.

Reaction Rate Constant (k)

The reaction rate constant (k) is a proportionality factor that relates the rate of a reaction to the concentrations of the reactants. It is influenced by factors such as temperature and activation energy. In the context of the question, calculating the ratio of the rate constants for the two reactions involves using the Arrhenius equation, allowing for a comparison of how the different activation energies impact the overall reaction rates at 25 °C.

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This reaction has an activation energy of zero in the gas phase: CH₃ + CH₃ → C₂H₆ a. Would you expect the rate of this reaction to change very much with temperature?

Textbook Question

Consider the two reactions:

O + N₂ → NO + N E_a = 315 kJ/mol

Cl + H₂ → HCl + H E_a = 23 kJ/mol

a. Why is the activation barrier for the first reaction so much higher than that for the second?

Textbook Question

Anthropologists can estimate the age of a bone or other sample of organic matter by its carbon-14 content. The carbon-14 in a living organism is constant until the organism dies, after which carbon- 14 decays with first-order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains 19.5% of the C-14 found in living organisms. How old is the bone?

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Textbook Question

Consider the gas-phase reaction: H₂(g) + I₂(g) → 2 HI(g) The reaction was experimentally determined to be first order in H₂ and first order in I₂. Consider the proposed mechanisms. Proposed mechanism I: H₂(g) + I₂(g) → 2 HI(g) Single step Proposed mechanism II: I₂(g) Δk1k-12 I(g) Fast H₂( g) + 2 I( g) → k22 HI( g) Slow a. Show that both of the proposed mechanisms are valid.