Question

Estimate the rate of the reaction, $$ \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q) \longrightarrow \mathrm{H}_{2} \mathrm{O} $$ given the rate law for the reaction is$$\text { rate }=\left(1.3 \times 10^{11} \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}\right)\left[\mathrm{OH}^{-}\right]\left[\mathrm{H}^{+}\right]$$ for neutral water, where \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-7} M\) and \(\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-7} M\)

Short answer

The rate of the reaction in neutral water is 1.3 x 10^-3 M/s.

Step-by-step solution

- Understanding the Rate Law

The rate law for the given reaction is expressed as rate = (1.3 x 10^11 L mol^-1 s^-1)[OH^-][H^+]. This equation tells us that the rate of the reaction is proportional to the product of the concentrations of the hydroxide ions [OH^-] and the hydronium ions [H^+].

- Insert Given Concentrations

To find the rate of the reaction for neutral water, we need to insert the given concentrations into the rate law. For neutral water, we have [OH^-] = 1.0 x 10^-7 M and [H^+] = 1.0 x 10^-7 M.

- Calculate the Reaction Rate

Now, we calculate the rate of the reaction by multiplying the rate constant with the concentrations of the reactants: rate = (1.3 x 10^11 L mol^-1 s^-1)(1.0 x 10^-7 M)(1.0 x 10^-7 M).

Key concepts

Rate Law

Understanding the rate law is essential when calculating the pace at which chemical reactions proceed. The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It is usually determined experimentally and can take the form of a simple product of reactant concentrations, each raised to a power. These powers are known as 'reaction orders' and they indicate how sensitive the reaction rate is to changes in each reactant's concentration.

The general form of a rate law is: \( rate = k[A]^m[B]^n \) ... where \( k \) is the rate constant, \( [A] \) and \( [B] \) are the concentrations of reactants, and \( m \) and \( n \) are the orders of the reaction with respect to each reactant. It's important to note that the rate law is specific to each reaction and must be determined through experiments; it cannot be deduced from the balanced chemical equation alone.

When you plug in the concentrations of the reactants into the rate law, you can calculate the reaction rate. This is a fundamental step for chemists when predicting how fast a reaction will occur under various conditions.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the way in which they change over time. It investigates the different factors that influence the speed of chemical reactions, such as temperature, pressure, reactant concentrations, and the presence of catalysts. By understanding kinetics, chemists can design processes that optimize reaction conditions for industrial applications, control reaction rates for safety, and explore reaction mechanisms.

One of the central equations in chemical kinetics is the Arrhenius equation, which relates the rate constant \( k \) to the temperature and activation energy of a reaction: \( k = A e^{\frac{-E_a}{RT}} \) ...where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, \( T \) is the temperature (in Kelvin), and \( e \) is the base of the natural logarithm. This equation demonstrates how increasing the temperature can increase the reaction rate by providing the necessary energy to overcome the activation energy barrier.

Reaction Concentration

The concept of reaction concentration plays a pivotal role in understanding chemical reactions. Concentrations are usually denoted in molarity (M), which is the number of moles of a solute per liter of solution. In the context of reaction rate calculations, the concentrations of reactants determine the number of collisions between particles and, consequently, the likelihood of reactions occurring.

In a rate law, increasing the concentration of reactants generally leads to an increase in the reaction rate, since more reactant molecules are available to interact. For instance, doubling the concentration of a reactant will, all else being equal, double the rate of reaction if the reactant's order is one. However, this is not always a linear relationship; the actual effect on rate depends on the reaction order with respect to each reactant.

The rate of a reaction is not only dependent on the initial concentrations but can change over time as reactants are consumed and products are formed. Continuous monitoring of reactant concentrations is key in reaction kinetics studies to accurately analyze the rate at which reactants are transformed into products.

Neutral Water Chemistry

Neutral water chemistry refers to a situation where the concentrations of hydronium ions \( [H^+] \) and hydroxide ions \( [OH^-] \) in an aqueous solution are equal, typically at \( 1.0 \times 10^{-7} M \) for each at 25°C. This leads to a neutral pH of 7 in pure water. The concept becomes particularly important when discussing acid-base reactions and the self-ionization of water, which can be represented by the equation: \( 2H_2O(l) \longleftrightarrow H^+(aq) + OH^-(aq) \) ...

In a neutral water solution, adding an acid (which increases \( [H^+] \)) or a base (which increases \( [OH^-] \)), will disrupt the equilibrium and change the pH. For chemical reactions occurring in water, understanding the neutral pH balance is important, as the state of the water can significantly affect the rate and outcome of the reaction. In the case of the reaction between hydronium \( [H^+] \) and hydroxide \( [OH^-] \) ions to form water, the neutral concentrations provide a base point for calculating the reaction rate under these specific conditions.