Problem 60 - Chemistry The Molecular Nature of Matter

Question

The decomposition of phosphine, a very toxic gas, forms phosphorus and hydrogen in the following reaction: $$4 \mathrm{PH}_{3}(g) \longrightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g)$$ (a) Express the rate with respect to each of the reactants and products. (b) If the instantaneous rate with respect to \(\mathrm{PH}_{3}\) is \(0.34 M \mathrm{~s}^{-1}\), what is the instantaneous rate of the reaction?

Step-by-Step Solution

Answer

Rate of disappearance of PH3 = -0.34 M s^-1, rate of formation of P4 = 1/4 * 0.34 M s^-1, and rate of formation of H2 = 1/6 * 0.34 M s^-1. The instantaneous rate of the reaction is -0.34 M s^-1.

Step 1: Understand rate of reaction

The rate of a chemical reaction is defined as the change in concentration of a reactant or a product per unit time.

Step 2: Express rate with respect to reactants and products

For the given balanced chemical equation, the rate of formation of products (P4 and H2) is directly related to the rate of decomposition of the reactant (PH3). Using stoichiometry, the rate of disappearance of PH3 is -1/4 times the rate of formation of P4 and -1/6 times the rate of formation of H2.

Step 3: Calculate the rates for each species

The rate of disappearance of PH3 is given by the negative change in its concentration divided by the change in time, -\(\frac{\Delta [PH_3]}{\Delta t}\). The rates of appearance for P4 and H2 can be given as: Rate of P4 formation = 1/4 * (-Rate of PH3 disappearance), Rate of H2 formation = 1/6 * (-Rate of PH3 disappearance).

Step 4: Relate the instantaneous rate with respect to PH3

Substitute the given instantaneous rate of PH3 (-0.34 M s^-1) into the equations for P4 and H2 to find their rates.

Step 5: Calculate the instantaneous rates of P4 and H2

Instantaneous rate of P4 formation = 1/4 * (0.34 M s^-1), Instantaneous rate of H2 formation = 1/6 * (0.34 M s^-1).

Step 6: Express the final rate of the reaction

The rate of the reaction can be expressed as one of the rates calculated before since they are all directly related. Thus, using the instantaneous rate of PH3 disappearance (-0.34 M s^-1) as the rate of the reaction would be accurate.

Key Concepts

Reaction Stoichiometry

Reaction stoichiometry is central to the field of chemistry as it describes the quantitative relationships between reactants and products in a chemical reaction. Understanding this concept allows chemists to predict the amounts of substances consumed and produced in a reaction.

For example, consider the decomposition of phosphine (PH₃), in the reaction:
\[4 \mathrm{PH}_{3}(g) \longrightarrow \mathrm{P}_{4}(g) + 6 \mathrm{H}_{2}(g)\].
Here, stoichiometry indicates that four moles of phosphine produce one mole of phosphorus and six moles of hydrogen gas. The coefficients in the balanced reaction represent the mole ratio of the reactants and products and are essential in determining the rate of the reaction with respect to each substance.

In practice, stoichiometry can be used to scale up reactions from the laboratory to industrial scales and is critical for efficient resource management and to predict yields of products. Hence, mastering stoichiometry is not only foundational for understanding chemical processes but also for practical applications in various scientific and engineering fields.

Instantaneous Rate of Reaction

The instantaneous rate of reaction is a measure of how quickly reactants are converted into products at any given moment during a chemical reaction. It is the rate at which a reaction proceeds at a specific time and can be determined by the slope of the concentration vs. time graph at that point.

To illustrate, if we are given the instantaneous rate of decomposition of phosphine (PH₃) as \(0.34 \, M \cdot s^{-1}\), this indicates the speed at which PH₃ concentrations decrease per second at that particular instant. It is calculated using the negative change in concentration of the reactant over a very small, almost zero, time interval: \(-\frac{\Delta [\mathrm{PH}_3]}{\Delta t}\).

The concept of instantaneous rate is crucial when determining reaction mechanisms and conditions, as it can inform chemists about the kinetics of a reaction—how the reaction rate alters as the reaction progresses. Moreover, understanding instantaneous rates can lead to optimizing reaction conditions for maximum efficiency in chemical manufacturing.

Rate of Decomposition

The rate of decomposition is a measure of how quickly a reactant breaks down into simpler products. It is directly related to the reaction rate but specifically concerns the breakdown of compounds.

In our given exercise, the decomposition of phosphine is the focus: \(4 \mathrm{PH}_{3}(g) \longrightarrow \mathrm{P}_{4}(g) + 6 \mathrm{H}_{2}(g)\). The rate at which phosphine (PH₃) decomposes can be tied to the rates of formation of the products phosphorus (P₄) and hydrogen (H₂) via the stoichiometry of the reaction.

It is crucial to note that the rates of formation for the products (expressed as positive values) are dependent on the rate of disappearance of the reactant (expressed as a negative value) to respect the law of conservation of mass. Therefore, by knowing the rate of decomposition of a reactant, one can derive the rates of product formation. This concept is immensely useful in environmental chemistry, where decomposition rates determine the lifespan and impact of chemicals in the environment.