Problem 102 - General Chemistry: Principles and Modern Applications

Question

A first-order reaction, \(\mathrm{A} \longrightarrow\) products, has a halflife of \(75 \mathrm{s},\) from which we can draw two conclusions. Which of the following are those two (a) the reaction goes to completion in 150 s; (b) the quantity of \(A\) remaining after 150 s is half of what remains after 75 s; (c) the same quantity of A is consumed for every 75 s of the reaction; (d) one- quarter of the original quantity of A is consumed in the first 37.5 s of the reaction; (e) twice as much A is consumed in 75 s when the initial amount of \(\mathrm{A}\) is doubled; (f) the amount of \(\mathrm{A}\) consumed in 150 s is twice as much as is consumed in 75 s.

Step-by-Step Solution

Answer

Based on the analysis of the statements, the two correct conclusions that we can draw from the information on the reaction's half-life are: (b) the quantity of \(A\) remaining after 150 seconds is half of what remains after 75 seconds and (e) twice as much \(A\) is consumed in 75 seconds when the initial amount of \(A\) is doubled.

Step 1: Analyze Statement (a)

Statement (a) suggests the reaction completes in 150 seconds, which is twice the half-life. This isn't correct for a first-order reaction. After one half-life (75 seconds), half of the original quantity of A remains. After another half-life (another 75 seconds), only half of that remaining amount is consumed, not the entire remaining amount. Therefore, statement (a) is incorrect.

Step 2: Analyze Statement (b)

Statement (b) suggests that the quantity of A remaining after 150 seconds is half of what remains after 75 seconds. This is correct. After 75 seconds (one half-life), half of the original amount of A remains. After another half-life (another 75 seconds), only half of the remaining amount is left. Therefore, statement (b) is correct.

Step 3: Analyze Statement (c)

Statement (c) suggests that the same quantity of A is consumed for every 75 seconds of the reaction. This is not correct. While it is true that the reaction rate is proportional to the amount of reactant, in a first-order reaction, the actual quantity of A consumed decreases with each successive half-life. Therefore, statement (c) is incorrect.

Step 4: Analyze Statement (d)

Statement (d) is claiming that one quarter of A is consumed in the first 37.5 seconds. This isn’t true for a first-order reaction. The half-life is defined as the time it takes for half of the reactant to be consumed, not a quarter. Therefore, statement (d) is incorrect.

Step 5: Analyze Statement (e)

Statement (e) is asserting that if the initial amount of A is doubled, twice as much A is consumed in 75 seconds. This is true for a first-order reaction. Since the reaction rate is proportional to the concentration of the reactant, if we start with twice as much reactant, we'll consume twice as much in the same amount of time. Hence, statement (e) is correct.

Step 6: Analyze Statement (f)

Statement (f) suggests that the amount of A consumed in 150 seconds is twice as much as is consumed in 75 seconds. It's the case with the first-order reaction that half the reactant remains after one half-life (75 seconds in this case), and half of the remaining reactant is consumed after the next half-life. Actually, the amount consumed in the first 75 seconds is more than that consumed in the next 75 seconds, so statement (f) is incorrect.

Key Concepts

Reaction Kinetics

Reaction kinetics is the branch of chemistry focused on understanding the rates of chemical reactions and how different conditions affect them. It provides insights into how quickly a reaction proceeds and what factors influence this speed.

For a better understanding, consider a reaction where substance A converts to products. This is a perfect example of a first-order reaction. Here, the rate of reaction depends directly on the concentration of A. As the concentration of A changes, so does the speed of the reaction.

• In first-order reactions, the rate is proportional to the concentration of a single reactant.
• The rate constant ( k ext{ extbackslash}) can be determined experimentally and helps in quantifying how quickly a reaction approaches completion.
• Every reaction has a unique rate constant depending on conditions like temperature and pressure.

When analyzing reaction kinetics, it’s crucial to pay attention to how the concentration changes over time. This helps chemists control reactions efficiently in industrial processes.
By understanding the kinetics, scientists can predict how long a reaction will take and optimize conditions to make reactions faster, safer, and more efficient.

Half-Life

Half-life is an essential concept in both chemistry and physics. It defines the time required for half of a reactant to be consumed in a reaction. For first-order reactions, the half-life is a constant value, unaffected by the initial concentration.

Take the reaction where A is being converted to products. If we know the half-life ( 75 ext{ s} extbackslash) of this first-order reaction, we can deduce the amount of A present at any given time.

• After one half-life, half of A is left, meaning if you start with a certain amount of A, only 50% remains at the end of the half-life.
• After two half-lives, only 25% of the original A remains, as A keeps being halved in equal time intervals.

Understanding half-life is key to predicting how quickly a reaction approaches equilibrium or completes.
In practical applications, this knowledge helps in fields such as pharmacology and radioactive decay, where knowing the duration of a substance's effectiveness or safety is critical.

Chemical Reactions

Chemical reactions are processes in which reactants are transformed into products. This transformation involves the breaking and forming of bonds and is governed by reaction kinetics and thermodynamics.

Each reaction has its unique characteristics, such as rate, energy changes, and mechanisms, which are studied intensively in chemistry. First-order reactions are particularly interesting because they have predictable rates and behaviors based on the concentration of the reactants.

• One key feature of a first-order reaction is that the rate decreases over time as the concentration of the reactant decreases.
• This means that in a reaction where substance A turns into a product, the amount of A diminishing follows an exponential decay curve.
• The simplicity of first-order reactions allows for easy mathematical modeling using logarithmic functions.

To sum up, chemical reactions are fundamental processes that scientists must understand to innovate and improve the numerous applications of chemistry, from medicine to industry. By mastering the principles of reactions like those of the first-order, chemists can predict and control the outcomes of complex processes effectively.