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Chapter 2: Problem 39

State what you would plot to get a straight line if experimental ( \(x, y\) ) data are to be correlated by the following relations, and what the slopes and intercepts would be in terms of the relation parameters. If you could equally well use two different kinds of plots (e.g., rectangular or semilog), state what you would plot in each case. [The solution to Part (a) is given as an example.] (a) \(y^{2}=a e^{-b / x}\). (b) \(y^{2}=m x^{3}-n\) (c) \(1 / \ln (y-3)=(1+a \sqrt{x}) / b\) (d) \((y+1)^{2}=\left[a(x-3)^{3}\right]^{-1}\) (e) \(y=\exp (a \sqrt{x}+b)\) (f) \(x y=10^{\left[a\left(x^{2}+y^{2}\right)+b\right]}\) (g) \(y=[a x+b / x]^{-1}\)

Short Answer

Expert verified
The line equations derived from the equations will have the following slopes and intercepts: (a) The slope will be -b and intercept \(\ln(a)\), (b) the slope will be m with intercept 0, (c) the slope will be b with intercept 0, (d) the slope will be -1 with intercept 0, (e) the slope will be a with intercept b, (f) the slope will be a with intercept b, and (g) the form is not standard linear because the 'intercept' \(b/x\) is not constant. The dependent and independent variables correspond to what you would plot on the y-axis and x-axis respectively.

Step by step solution

01

Express equation (a) in the form of a straight line

From the equation \(y^{2}=a e^{-b / x}\), taking the natural log of both sides gives: \(2 \ln(y)=\ln(a)-\frac{b}{x}\). This equation is in the form of a straight line: \(y = mx + c\), where \(2 \ln(y)\) is the dependent variable \(y'\), \(1/x\) is the independent variable \(x'\), \(m = -b\) is the slope, and \(c = \ln(a)\) is the intercept.
02

Evaluate equation (b) for linear form

Rearrange the equation \(y^{2}=m x^{3}-n\) as \( y^{2} + n = m x^{3}\). This implies that the dependent variable \(y'\) is \(y^{2} + n\), the independent variable \(x'\) is \(x^{3}\), the slope \(m\) is \(m\), and the intercept \(c\) is 0.
03

Redefine equation (c) for a linear form

From the equation \(1 / \ln(y-3)=(1+a \sqrt{x}) / b\), rearrange it so: \( b / (1+a \sqrt{x}) = \ln(y - 3)\). This implies that the dependent variable \(y'\) is \(\ln(y - 3)\), the independent variable \(x'\) is \(1/(1+a \sqrt{x})\), the slope \(m\) is \(b\), and the intercept \(c\) is 0.
04

Transform equation (d) to a linear form

From the equation \((y+1)^{2}=\left[a(x-3)^{3}\right]^{-1}\), taking the natural log of both sides, we can write: \(2 \ln(y+1) = -\ln[a(x-3)^{3}]\). This indicates that the dependent variable \(y'\) is \(2 \ln(y+1)\), the independent variable \(x'\) is \(\ln[a(x-3)^{3}]\), the slope \(m\) is -1, and the intercept \(c\) is 0.
05

Derive a linear form from equation (e)

Given \(y=\exp (a \sqrt{x}+b)\), the equation can be simplified to \(\ln(y) = a \sqrt{x} + b\). This implies that the dependent variable \(y'\) is \(\ln(y)\), the independent variable \(x'\) is \(\sqrt{x}\), the slope \(m\) is \(a\), and the intercept \(c\) is \(b\).
06

Break down equation (f) into a linear form

Given \(x y=10^{\left[a\left(x^{2}+y^{2}\right)+b\right]}\), taking the logarithm base 10 of both sides yields: \(\log_{10}(xy) = a (x^{2} + y^{2}) + b\). This implies that the dependent variable \(y'\) is \(\log_{10}(xy)\), the independent variable \(x'\) is \(x^{2} + y^{2}\), the slope \(m\) is \(a\), and the intercept \(c\) is \(b\).
07

Convert equation (g) into a linear form

Given \(y=[a x+b / x]^{-1}\), it can be rewritten to \(1/y = a x + b/x\). Therefore, it suggests that the dependent variable \(y'\) is \(1/y\), the independent variable \(x'\) is \(x\), the slope \(m\) is \(a\), and the intercept \(c\) is \(b/x\). Note that this is not a standard linear form since the intercept is not constant.

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Key Concepts

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

Experimental Data Correlation
When analyzing experimental data, one of the key objectives is to determine the relationship or correlation between two variables. This is often done by attempting to fit the data to an underlying model, which in many cases is assumed to be linear due to its simple mathematical nature and ease of interpretation. Correlation allows us to establish a direct quantitative link, often expressed with parameters such as the correlation coefficient.

For a clear and effective analysis, the method of plotting is crucial. For instance, if we're given a set of experimental data points \( (x, y) \) and we need to fit these to a model like \( y^2 = ae^{-b/x} \) as in our initial exercise, we would look for a way to linearize the equation. By plotting the transformed variables that straighten the original curve, we make pattern recognition and parameter estimation more manageable. The slope and intercept derived from this plot directly relate to the parameters of our model, providing us with valuable information regarding the system being studied.

Depending on the form of the data and the model, different types of plots such as rectangular or semilogarithmic might be employed to achieve linearization. The choice between these plots depends on the model's equation and on which transformation will yield a straight line. For example, a semilogarithmic plot may be best for an equation involving an exponential term, as it can linearize the exponential relationship into a form where the slope and intercept can be determined.
Linearization of Equations
Linearization is the process of approximating a nonlinear relationship by a linear one, which is essential in handling complex equations in experimental data analysis. Nonlinear equations can be challenging to work with, especially when it comes to data fitting and analysis. Linear equations, in contrast, are much simpler to handle because the relationship between variables is direct and proportionate. By linearizing an equation, we can make use of linear regression techniques to extract the parameters that govern the relationship.

Each of the steps in the provided solution demonstrates a different approach to turning a complicated equation into a linear form. This process often involves identifying the appropriate mathematical operations that, when applied to the variables, yield a linear relationship. For example, taking the logarithm is a common transformation for equations involving exponential functions. Once the equation is linearized, the 'y' becomes the dependent variable \( y' \) with respect to some transformed independent variable \( x' \) expressed in a standard linear form \( y' = mx' + c \), where \( m \) and \( c \) represent the slope and intercept respectively.

Transformations might include raising to powers, taking roots, or employing trigonometric functions, among others. The ultimate goal is to achieve a linear correlation which significantly simplifies statistical analysis, making it more tractable for finding relationships within the data.
Logarithmic Transformation
Logarithmic transformation is a powerful tool in the arsenal of data analysis techniques, particularly when dealing with multiplicative relationships and exponential growth patterns. This form of transformation can linearize curves, enabling the use of linear regression for slope-intercept form estimation and ultimately simplifying the analysis process.

In many cases, taking the natural logarithm (ln) can turn a product into a sum or an exponential function into a linear one, as it does in steps 1, 4, and 5 of our solution. When we encounter an equation such as \( y = ae^{b/x} \) in step 5, we apply the natural logarithm to both sides to obtain \( \ln(y) = b/x + \ln(a) \), which aligns with the straight-line equation \( y = mx + c \). Here, \( \ln(y) \) and \( 1/x \) become our \( y' \) and \( x' \) respectively, with the slope \( m \) being the coefficient \( b \) and the intercept \( c \) being the logarithm of the constant \( a \).

The logarithmic transformation is also valuable for stabilizing variances, normalizing distributions, and making patterns in data more perceptible and accessible to interpretation. When dealing with equations where variables interact in a non-linear fashion, the log transformation is a common technique for revealing underlying linear trends and is an essential concept for students to understand in the field of chemical process data analysis.

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Most popular questions from this chapter

A seed crystal of diameter \(D\) (mm) is placed in a solution of dissolved salt, and new crystals are observed to nucleate (form) at a constant rate \(r\) (crystals/min). Experiments with seed crystals of different sizes show that the rate of nucleation varies with the seed crystal diameter as \(r(\text { crystals/min })=200 D-10 D^{2} \quad(D \text { in } \mathrm{mm})\) (a) What are the units of the constants 200 and \(10 ?\) (Assume the given equation is valid and therefore dimensionally homogeneous.) (b) Calculate the crystal nucleation rate in crystals/s corresponding to a crystal diameter of 0.050 inch. (c) Derive a formula for \(r\) (crystals/s) in terms of \(D\) (inches). (See Example \(2.6-1 .\) ) Check the formula using the result of Part (b). (d) The given equation is empirical; that is, instead of being developed from first principles, it was obtained simply by fitting an equation to experimental data. In the experiment, seed crystals of known size were immersed in a well-mixed supersaturated solution. After a fixed run time, agitation was ceased and the crystals formed during the experiment were allowed to settle to the bottom of the apparatus, where they could be counted. Explain what it is about the equation that gives away its empirical nature. (Hint: Consider what the equation predicts as \(D\) continues to increase.)

A frustrated professor once claimed that if all the reports she had graded in her career were stacked on top of one another, they would reach from the Earth to the moon. Assume that an average report is the thickness of about 10 sheets of printer paper and use a single dimensional equation to estimate the number of reports the professor would have had to grade for her claim to be valid.

A waste treatment pond is \(50 \mathrm{m}\) long and \(25 \mathrm{m}\) wide, and has an average depth of \(2 \mathrm{m}\). The density of the waste is \(75.3 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}\). Calculate the weight of the pond contents in \(\mathrm{lb}_{\mathrm{f}},\) using a single dimensional equation for your calculation.

A hygrometer, which measures the amount of moisture in a gas stream, is to be calibrated using the apparatus shown here: Steam and dry air are fed at known flow rates and mixed to form a gas stream with a known water content, and the hygrometer reading is recorded; the flow rate of either the water or the air is changed to produce a stream with a different water content and the new reading is recorded, and so on. The following data are taken: $$\begin{array}{cc}\hline \begin{array}{c}\text { Mass Fraction } \\\\\text { of Water, } y\end{array} & \begin{array}{c}\text { Hygrometer } \\\\\text { Reading, } R\end{array} \\\\\hline 0.011 & 5 \\\0.044 & 20 \\\0.083 & 40 \\\0.126 & 60 \\\0.170 & 80 \\ \hline\end{array}$$ (a) Draw a calibration curve and determine an equation for \(y(R)\). (b) Suppose a sample of a stack gas is inserted in the sample chamber of the hygrometer and a reading of \(R=43\) is obtained. If the mass flow rate of the stack gas is \(1200 \mathrm{kg} / \mathrm{h}\), what is the mass flow rate of water vapor in the gas?

The following \((x, y)\) data are recorded: $$\begin{array}{|c|c|c|c|}\hline x & 0.5 & 1.4 & 84 \\\\\hline y & 2.20 & 4.30 & 6.15 \\\\\hline \end{array}$$ (a) Plot the data on logarithmic axes. (b) Determine the coefficients of a power law expression \(y=a x^{b}\) using the method of least squares. (Remember what you are really plotting \(-\) there is no way to avoid taking logarithms of the data point coordinates in this case.) (c) Draw your calculated line on the same plot as the data.
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