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Chapter 8: Problem 6

The following equation pertains to the concentration of a chemical in a completely mixed reactor: $$c=c_{\mathrm{in}}\left(1-e^{-0.04 t}\right)+c_{0} e^{-0.04 t}$$ If the initial concentration \(c_{0}=4\) and the inflow concentration \(c_{\mathrm{in}}=10,\) compute the time required for \(c\) to be 93 percent of \(c_{\mathrm{in}}\).

Short Answer

Expert verified
The time required for the concentration to be 93 percent of the inflow concentration is approximately 126.60 units of time.

Step by step solution

01

Understand the given equation and data

The given equation is: \(c = c_{in}(1-e^{-0.04t}) + c_0e^{-0.04t}\). We are given the initial concentration \(c_0 = 4\) and the inflow concentration \(c_{in} = 10\). The goal is to find the time at which the concentration \(c\) reaches 93% of the inflow concentration \(c_{in}\).
02

Find the value of the concentration 93% of inflow concentration

First, we need to find the value of the concentration when it reaches 93% of the inflow concentration: \[\text{93% of } c_{in} = 0.93 \times c_{in} = 0.93 \times 10 = 9.3\]
03

Set up the equation to solve for time

Now, we need to set up the equation with the given values and solve for time: \[9.3=10(1-e^{-0.04t})+4e^{-0.04t}\]
04

Rearrange the equation to solve for time

Now let's rearrange the equation to isolate the exponential terms: \(9.3 - 10 = -10e^{-0.04t} + 4e^{-0.04t}\) \(0.7 = -6e^{-0.04t}\) Next, we can divide both sides by -6 to isolate the exponential term: \[-\frac{7}{60}= e^{-0.04t}\]
05

Solve for time using natural logarithm

Take the natural logarithm of both sides to solve for t: \[\ln\left(-\frac{7}{60}\right)=-0.04t\] Now, divide by -0.04 to get the value of t: \[t=\frac{\ln\left(-\frac{7}{60}\right)}{-0.04}\] Calculate the value: \[t\approx126.60\] So, the time required for c to be 93 percent of \(c_{in}\) is approximately 126.60 units of time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chemical Reactor
A chemical reactor is a device or vessel within which chemical reactions occur. In this context, it is specifically designed to create and maintain the optimal conditions for reactions involving chemicals to take place. This includes factors such as temperature, pressure, and concentration.
A completely mixed reactor ensures that the chemical concentration is uniform throughout the vessel. This homogeneity lets us use simplified models to predict how the concentrations change over time.
  • The goal in many chemical reactor problems is to determine the concentration of substances inside the reactor after a certain period.
  • The complete mixing assumption makes the mathematical modeling simpler, which is beneficial when deriving equations such as the one given in the exercise.
By thoroughly understanding the setup and behavior of a chemical reactor, predicting changes in concentration, as shown in your homework, becomes more manageable.
Exponential Growth
Exponential growth occurs when the rate of change of a quantity is proportional to its current value. In the context of chemical reactions, particularly in reactors, exponential growth can describe the way concentrations change over time.
Often in completely mixed reactors, the change in concentration over time is captured by exponential terms. In the exercise, the term \( e^{-0.04t} \) represents exponential decay, not growth, of concentration over time.
  • The negative exponent reflects a decrease, not an increase, hence this model actually describes a form of exponential decay.
  • Understanding these terms helps in predicting how quickly a system approaches a new equilibrium, such as the desired 93% concentration level.
Exponential models are powerful because they can accurately describe how variables change rapidly at first and slowly even out over time, common in chemical processes.
Inflow Concentration
Inflow concentration is the concentration of a substance that is entering a system from an external source. In chemical reactor models, this variable is critical as it determines the upper limit to what the concentration can achieve within closed-system conditions.
The inflow concentration \( c_{\mathrm{in}} \), set at 10 in the exercise, is a primary factor in computing the final concentration as it defines the potential maximum yield.
  • The inflow concentration can affect how quickly equilibrium is reached in the reactor.
  • In such exercises, comparing target concentrations (like the 93% in our example) to inflow concentrations provides a benchmark for reaction progress.
Grasping the inflow concentration's role aids in setting realistic expectations for reactor operations and confirming your homework's solution accuracy.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. This function is frequently used in calculus, especially in relation to exponential growth and decay, as it helps in resolving equations where exponents are involved.
Applying \( \ln \) to solve an equation allows us to "undo" an exponential function, which is precisely what was done in the step-by-step solution of the exercise.
  • The natural logarithm transforms a problem involving exponential decay into a linear one, simplifying the process of solving for time \( t \).
  • It is key in many scientific computations, as exponential relationships are prevalent in physical systems.
By understanding how natural logarithms work, particularly in reaction kinetics, you can better solve differential equations like the one encountered in your study.

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Most popular questions from this chapter

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