Problem 3 - Basic Engineering Mathematics

Question

The solubility \(s\) of potassium chlorate is shown by the following table: \begin{tabular}{|l|cccccccc|} \hline\(t^{\circ} \mathrm{C}\) & 10 & 20 & 30 & 40 & 50 & 60 & 80 & 100 \\ \(s\) & \(4.9\) & \(7.6\) & \(11.1\) & \(15.4\) & \(20.4\) & \(26.4\) & \(40.6\) & \(58.0\) \\ \hline \end{tabular} The relationship between \(s\) and \(t\) is thought to be of the form \(s=3+a t+b t^{2}\). Plot a graph to test the supposition and use the graph to find approximate values of \(a\) and \(b\). Hence calculate the solubility of potassium chlorate at \(70^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

Plot data, fit a quadratic curve, estimate \(a\) and \(b\), then calculate solubility at 70°C using the equation.

Step 1: Understanding the Problem

We need to test if sodium chlorate's solubility follows the quadratic model \( s = 3 + a t + b t^2 \). We will first plot a graph using the solubility data against the temperature and fit a curve to find suitable values for \(a\) and \(b\).

Step 2: Plotting the Data Points

Plot the given solubility data on a graph with temperature (\(t\)) on the x-axis and solubility (\(s\)) on the y-axis:- At \(10^\circ C\), \(s = 4.9\)- At \(20^\circ C\), \(s = 7.6\)- Continue plotting for all given temperatures.

Step 3: Fitting a Quadratic Curve

Since we believe the relationship is quadratic, we use the equation \( s = 3 + a t + b t^2 \). Use a graphing calculator or software to fit a quadratic curve to the plotted points.

Step 4: Estimating Coefficients

The quadratic fit will provide an equation that estimates the best fit for the data points. Assume the equation is of the form \( s = 3 + a t + b t^2 \), where \(a\) and \(b\) are constants determined by the fit.

Step 5: Calculating Solubility at 70°C

Use the determined values of \(a\) and \(b\) from the curve to calculate \(s\) at \(t = 70\): \[ s = 3 + a(70) + b(70)^2 \] Calculate this using the values obtained.

Key Concepts

Solubility of Potassium Chlorate

The solubility of potassium chlorate refers to how much of the substance can be dissolved in a solvent (usually water) at a certain temperature. Generally, solubility increases with temperature. For potassium chlorate, we observe this in the data, where solubility at different temperatures (from 10°C to 100°C) shows a consistent increase. Understanding solubility is crucial in chemistry as it influences various chemical processes such as reactions and separations. It tells us how much of a substance will dissolve, which is important in applications like preparing solutions for experiments or industrial processes. Recognizing solubility patterns with temperature can also help in predicting outcomes under different conditions.

Graphical Data Analysis

Graphical data analysis is a method used to visualize data, identify trends, and interpret results. In this exercise, plotting the solubility of potassium chlorate against temperature reveals the pattern of change visually. This graphical representation not only helps in examining the underlying relationship but also aids in validating assumptions made about mathematical models. The graph helps us visually compare data points and observe the increasing trend clearly. By inspecting plotted points, we can infer if the data follows a linear, quadratic, or another trend. A graph acts as an intuitive bridge between raw data (the table) and the mathematical function (our quadratic equation). It also allows quick estimates and verification of theoretical calculations.

Curve Fitting Techniques

Curve fitting is an essential technique in data analysis where we adjust a curve so that it best represents the set of data points. Here, we hypothesize that the relationship between solubility and temperature follows a quadratic form: \[ s = 3 + a t + b t^2 \].Fitting a curve provides insights into complex relationships by simplifying them into equations that can be easily analyzed.To achieve an accurate fit, use graphing software or calculators that support least squares fitting or regression analysis. This tool computes the best fit curve by minimizing the differences between the data points and the values predicted by the curve. The coefficients \(a\) and \(b\) in our quadratic equation are specifically calculated to provide the best representation of the data.

Quadratic Functions in Chemistry

Quadratic functions play a significant role in chemistry for modeling relationships that exhibit a parabolic pattern. In the context of solubility, a quadratic equation can represent how solubility changes with temperature and help predict unknown values.For potassium chlorate, the quadratic function takes the form: \[ s = 3 + a t + b t^2 \].This equation calculates solubility at any temperature by using specific values of \(a\) and \(b\). The quadratic function allows chemists to interpolate or extrapolate data points, providing predictions beyond the measured range.Quadratics are not limited to solubility studies—they also apply in reaction rates, concentration changes, and radioactive decay models. Understanding how to work with these functions empowers chemists to glean insights from experimental data and make informed predictions about chemical behaviors.