Question

The slope obtained by drawing a tangent at time " \(\mathrm{t}\) " on the curve for the concentration of reactants vs time is equal to instantaneous rate.

Short answer

To find the instantaneous rate of a reaction at a specific time "t" on the curve of concentration of reactants vs. time, one must follow these steps: 1. Identify the function that describes the relationship between the concentration of reactants and time. 2. Take the derivative of the concentration function with respect to time. 3. Plug in the specific time "t" for which we want to find the instantaneous rate into the derivative function obtained in step 2. The result will give us the instantaneous rate of the reaction at time "t".

Step-by-step solution

Definition of the instantaneous rate

The instantaneous rate of a reaction is the rate at which the reaction proceeds at an exact moment in time. It can be found by taking the derivative of the concentration of reactants with respect to time. Mathematically, the instantaneous rate at time "t" can be denoted as: \(\displaystyle -\frac{d[A]}{dt}\), where [A] represents the concentration of reactants at time "t".

Explain why the slope of the tangent line represents the instantaneous rate

On the curve for the concentration of reactants vs time, the slope of the tangent line at a specific point tells us how fast the concentration is changing at that point. Since the change in concentration with respect to time gives us the rate of the reaction, the slope of the tangent line at a point on the graph illustrates the instantaneous rate of the reaction at that specific time, "t".

Finding the slope of a tangent line at a given point on the curve

To find the slope of the tangent line at a given point on the curve, we need to perform the following steps: a. Identify the function that describes the relationship between the concentration of reactants and time. This function is often given, or you can deduce it from a provided data or graph. b. Take the derivative of the concentration function with respect to time. The derivative tells us how the concentration is changing for each instant "t". c. Plug in the specific time "t" for which we want to find the instantaneous rate (the slope of the tangent line) into the derivative function we obtained in step (b). The result will give us the instantaneous rate of the reaction at time "t".

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is fundamental to grasping the dynamics of how substances interact and transform. The rate of reaction refers to how quickly reactants are converted into products over time. Chemists often measure this rate by evaluating the change in concentration of a reactant or product per unit of time. This rate can be expressed as an average over a period of time or as an instantaneous rate at a specific moment.

The rate is often affected by various factors such as temperature, pressure, concentration of reactants, and the presence of a catalyst. By comprehending the factors impacting the rate, scientists can optimize reactions for industrial processes, control the reaction's speed, and predict the outcomes more effectively.

Concentration vs Time Graph

A concentration vs time graph is a powerful visual tool used in chemistry to depict the change in concentration of a reactant or product over the course of the reaction. The graph typically shows time on the x-axis and concentration on the y-axis. As the reaction progresses, you can expect to see the concentration of the reactants decreasing, while the concentration of the products increases.

The shape of the curve can thell us much about the reaction's behavior. For instance, a steep decline in reactant concentration indicates a high reaction rate, whereas a more gradual slope suggests a slower reaction. These graphs not only help in understanding the kinetics of reactions but are also essential in calculating the instantaneous rate by finding the slope of the tangent line at any given time point.

Derivative of Concentration

The derivative of concentration is a mathematical concept that is crucial for determining the instantaneous rate of a reaction. In calculus, the derivative measures how a function changes as its input changes — in this context, it evaluates how the concentration of reactants changes over time. To calculate the derivative, we often use the notation \( \frac{d[A]}{dt} \) where \( [A] \) is the concentration of the reactant A, and \( t \) is time.

The process of taking the derivative can sometimes be straightforward, or it can involve more complex calculus depending on the form of the concentration function. Regardless, the derivative is a snapshot of the rate at a particular moment, providing us with the instantaneous rate when we substitute a specific time into the derived function.

Slope of Tangent Line

The slope of the tangent line on a concentration vs time graph is directly tied to the instantaneous rate of a reaction. A tangent line is a straight line that touches the graph at exactly one point, representing the slope, or rate of change, of the graph at that point. When we speak about the slope of the tangent line in chemistry, we're referring to the rise over run — the change in concentration over the change in time — at that very specific point.

Calculating Instantaneous Rate

To calculate the slope of the tangent, and thereby the instantaneous rate, we first need the concentration function. We then differentiate this function, and finally, we plug in the time of interest to this derivative. The slope at this single point reveals the instantaneous rate of reaction, a critical piece of information for scientists studying reaction kinetics. Together, the concepts of concentration-time graphs, derivatives, and tangent lines form a solid foundation for understanding the changing rates of chemical reactions.