Question

(a) The gas-phase decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}, \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow\) \(\mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)\), is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\). At \(600 \mathrm{~K}\) the halflife for this process is \(2.3 \times 10^{5} \mathrm{~s}\). What is the rate constant at this temperature? (b) At \(320^{\circ} \mathrm{C}\) the rate constant is \(2.2 \times 10^{-5} \mathrm{~s}^{-1}\). What is the half-life at this temperature?

Short answer

(a) At 600 K, the rate constant (k) is approximately \(3.01 \times 10^{-6} s^{-1}\). (b) The half-life at 320°C (593.15 K) is approximately \(3.15 \times 10^{4} s\).

Step-by-step solution

(a) Finding the rate constant at 600 K

For a first-order reaction, the relationship between the half-life (t½) and the rate constant (k) is given by: \[t_{½} = \frac{0.693}{k}\] We are given t½ at 600 K, which is \(2.3 \times 10^{5} s\). We can now solve the equation for the rate constant (k) at 600 K: \[k = \frac{0.693}{t_{½}}\] Plug in the value of t½: \[k = \frac{0.693}{2.3 \times 10^{5} s}\] Now, calculate the value of k: \[k ≈ 3.01 \times 10^{-6} s^{-1}\]

(b) Finding the half-life at 320°C

First, convert the given temperature from Celsius to Kelvin: \[T = 320°C + 273.15 = 593.15 K\] At 320°C (593.15 K), we are given the rate constant, k, which is \(2.2 \times 10^{-5} s^{-1}\). We can use the same equation as before to find the half-life (t½) at this temperature: \[t_{½} = \frac{0.693}{k}\] Plug in the value of k: \[t_{½} = \frac{0.693}{2.2 \times 10^{-5} s^{-1}}\] Now, calculate the value of t½: \[t_{½} ≈ 3.15 \times 10^{4} s\] So the half-life at 320°C (593.15 K) is approximately \(3.15 \times 10^{4} s\).

Key concepts

Rate laws in chemistry

Rate laws in chemistry, often referred to as 'rate equations', are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. For a reaction where a substance A decomposes into products, the rate law can be expressed as the rate = k[A]ⁿ, where 'k' is the rate constant, [A] is the concentration of A, and 'n' is the order of the reaction with respect to A.

In first-order reactions, the rate of the reaction is directly proportional to the concentration of the reactant. This means that if the concentration of the reactant A doubles, so does the rate of the reaction. The rate law for a first-order reaction is simply rate = k[A], with 'n' equal to 1. This type of rate law indicates that the rate of reaction depends on the concentration of a single reactant to the first power.

Half-life of a reaction

The concept of half-life is critical in understanding both nuclear and chemical reactions. In chemical kinetics, the half-life of a reaction refers to the time it takes for half of the reactant to be consumed or for the concentration to decrease by half.

For first-order reactions, the half-life is constant and does not depend on the initial concentration of the reactant. It is given by the formula t_½ = 0.693/k, where 't_½' denotes the half-life and 'k' is the rate constant of the reaction. This formula exemplifies the principle that, regardless of the concentration, it will always take the same amount of time for half of the reactant to react. This is a distinctive feature of first-order kinetics.

Chemical kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions, the factors that affect these rates, and the mechanisms by which reactions occur. It is not only important to know what products are formed in a reaction, but also how fast they are produced.

In the realm of chemical kinetics, scientists investigate how various conditions such as concentration, temperature, and catalysts influence the speed of chemical reactions. Using graphs and mathematical models, chemists can predict how the rate will change over time and under different circumstances. Kinetics is fundamental in the development of new chemical processes and the optimization of existing ones in various industries.

Temperature dependence of reaction rates

The rates of chemical reactions are greatly affected by temperature. Generally, increasing the temperature increases the rate at which the reaction occurs. This is because higher temperatures provide more energy to the reactant molecules, leading to more frequent and more energetic collisions which are more likely to overcome the activation energy barrier and result in a reaction.

The precise relationship between temperature and the rate of reaction is described by the Arrhenius Equation, which shows how the rate constant 'k' changes with temperature: k = A * exp(-E_a / (RT)), where 'E_a' is the activation energy of the reaction, 'R' is the universal gas constant, 'T' is the temperature in Kelvin, and 'A' is the frequency factor. As shown in the formula, even a small increase in temperature can lead to a significant increase in the rate constant, and thereby the reaction rate, illustrating how sensitive reaction rates are to changes in temperature.