Problem 14 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The reaction between \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\) was investigated at \(298 \mathrm{K}\) using the initial rate method \\[ \mathrm{H}_{2} \mathrm{PO}_{4}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{HPO}_{4}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{aq}) \\] The following results were obtained. (Section 9.5 ) $$\begin{array}{llll} & \begin{array}{l} \text { Initial rate } / 10^{-3} \\ \text {moldm }^{-3} \text {min }^{-1} \end{array} & \begin{array}{l} {[\mathrm{OH}]_{0} / 10^{-3}} \\ \text {moldm }^{-3} \end{array} & \begin{array}{l} {\left[\mathrm{H}_{2} \mathrm{PO}_{4}\right]_{0} / 10^{-3}} \\ \mathrm{mol} \mathrm{dm}^{-3} \end{array} \\ \hline \text { Experiment 1 } & 2.0 & 0.40 & 3.0 \\ \text { Experiment 2 } & 3.7 & 0.55 & 3.0 \\ \text { Experiment 3 } & 7.1 & 0.75 & 3.0 \end{array}$$ (a) Plot a log-log graph to determine the order of reaction with respect to \(\mathrm{OH}^{-}(\mathrm{aq})\) (b) What further experiments would you need to do to find the order with respect to \(\mathrm{H}_{2} \mathrm{PO}_{4}-?\)

Step-by-Step Solution

Answer

(a) The reaction is first order with respect to \( OH^{-}\). (b) To find the order with respect to \(\ H_2PO_4^-\), vary its concentration while keeping \(\ OH^{-}\) constant.

Step 1: Examine the Given Data

We are provided with initial rates, initial concentrations of OH^{-} , and H_2PO_4^- across three experiments. By analyzing these, we can derive the order of reaction with respect to OH^{-} .

Step 2: Determine Change in Reaction Rates

Calculate the change in reaction rates between experiments where only OH^{-} concentration varies. Here, only [ OH^{-}] changes while [ H_2PO_4^- ] is kept constant.

Step 3: Calculate Ratios of Change

From the data, observe the changes. \(\frac{3.7}{2.0} = 1.85\) for Experiment 2 over 1 and \(\frac{0.55}{0.40} = 1.375\). Similarly, for Experiments 3 over 2: reaction rate ratio is \(\frac{7.1}{3.7} = 1.92\) and concentration ratio is \(\frac{0.75}{0.55} = 1.36\).

Step 4: Logarithm of Ratios for Reaction Order

Use the rate law, rate = k [OH^{-}]^mto establish the order (m) with respect to OH^{-} by following the relationship: \( \log_{10}(\frac{u_2}{u_1}) = m \log_{10}(\frac{[OH^{-}]_2}{[OH^{-}]_1}) \) So, \(m = \frac{\log_{10}(1.85)}{\log_{10}(1.375)}\approx 1\), confirming a first-order reaction.

Step 5: Determine Additional Experiment Requirements

To find the order with respect to H_2PO_4^- , conduct additional experiments varying only the initial concentration of H_2PO_4^- while keeping OH^{-} concentration constant, and measure any changes in the initial rate.

Key Concepts

Reaction Order

The concept of reaction order is essential in understanding the rate of a chemical reaction. It refers to the sum of the powers to which all reactant concentrations appearing in the rate law are raised. For any given reaction, knowing the order helps us to relate how the rate depends on the concentration of reactants. In the provided reaction between \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\), determining the reaction order involves experimentally observing how changes in reactant concentrations affect the reaction rate. Reaction order can be first-order, second-order, or even zero-order, depending on how the concentration impacts the rate.
A first-order reaction means that the rate is directly proportional to the concentration of one reactant. That is what we find for \(\mathrm{OH}^{-}\) in this case. This is determined by observing how varying the concentration of \(\mathrm{OH}^{-}\) while keeping \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) constant impacts the initial rate. By plotting a log-log graph, the slope gives the order with respect to \(\mathrm{OH}^{-}\). Ultimately, understanding reaction order helps predict how changes in concentration could affect the progress of a reaction.

Rate Law

The rate law of a chemical reaction provides a mathematical relationship that links the reaction rate with the concentration of reactants. This relationship is crucial for predicting how fast a reaction will occur under specific conditions.
For the reaction between \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\), the general rate law expression can be written as: \[ \text{Rate} = k [\mathrm{H}_{2} \mathrm{PO}_{4}^{-}]^n [\mathrm{OH}^{-}]^m \]
Here, \(k\) is the rate constant, and \(n\) and \(m\) represent the reaction orders with respect to \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\), respectively. It is through experiments that we establish the values of \(n\) and \(m\) and subsequently determine the complete rate law. For our specific scenario with the given datasets, the order with respect to \(\mathrm{OH}^{-}\) is found to be first order already (i.e., \(m = 1\)). Additional experiments would be needed to determine \(n\).
Understanding rate laws allows chemists to explore reaction mechanisms further, as each different order can imply different steps or complexities within the chemical reaction itself.

Initial Rate Method

The initial rate method is a widely used technique to determine the order of a chemical reaction. This approach involves measuring the initial rate of reaction right at the start when the concentrations of reactants have not changed significantly.
In our example reaction, different sets of experiments are conducted with varying initial concentrations of \(\mathrm{OH}^{-}\) while maintaining \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) constant. By doing this, we focus only on how one reactant impacts the rate without interference from the other.
The procedure begins with measuring the initial reaction rates for each set of reactions. Data analysis includes calculating the ratio of initial rates and the concentration ratios of varying reactant. By comparing these ratios through logarithmic calculations, we deduce the order of the reaction with respect to a particular reactant.
It's a powerful technique, as it avoids complexities from complex reaction progression and provides more straightforward mathematical manipulation to find orders. The initial rate method can be applied efficiently to reactions in both inorganic and organic chemistry.

Inorganic Chemistry

Inorganic chemistry focuses on the properties and behavior of inorganic compounds, which encompass a wide range of substances except most organic (carbon-hydrogen) compounds. In the context of our given reaction, we are examining a classic acid-base reaction involving ions, which falls under the umbrella of inorganic chemistry.
The reaction between \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\) demonstrates several principles of inorganic chemistry. Here, we explore how ionic species interact in aqueous solution, which is a common scenario in many laboratory and industrial processes. Such reactions can tell us a lot about the properties of ions, how they may interact, and binding energies.
Studying inorganic reactions is crucial for understanding fundamental chemical processes and for advancing fields like materials science, catalysis, and environmental chemistry. The insights gained from inorganic reactions help not only in theoretical chemistry but also in practical applications that can influence technology and sustainability efforts.