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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

The following empirical equation correlates the values of variables in a system in which solid particles are suspended in a flowing gas: $$\frac{k_{g} d_{p} y}{D}=2.00+0.600\left(\frac{\mu}{\rho D}\right)^{1 / 3}\left(\frac{d_{p} u \rho}{\mu}\right)^{1 / 2}$$ Both \((\mu / \rho D)\) and \(\left(d_{p} u \rho / \mu\right)\) are dimensionless groups; \(k_{g}\) is a coefficient that expresses the rate at which a particular species transfers from the gas to the solid particles; and the coefficients 2.00 and 0.600 are dimensionless constants obtained by fitting experimental data covering a wide range of values of the equation variables. The value of \(k_{g}\) is needed to design a catalytic reactor. since this coefficient is difficult to determine directly, values of the other variables are measured or estimated and \(k_{g}\) is calculated from the given correlation. The variable values are as follows: $$\begin{aligned}d_{p} &=5.00 \mathrm{mm} \\\y &=0.100 \quad(\text { dimensionless }) \\\D &=0.100 \mathrm{cm}^{2} / \mathrm{s} \\\\\mu &=1.00 \times 10^{-5} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2} \\\\\rho &=1.00 \times 10^{-3} \mathrm{g} / \mathrm{cm}^{3} \\\u &=10.0 \mathrm{m} / \mathrm{s}\end{aligned}$$ (a) What is the estimated value of \(k_{g} ?\) (Give its value and units.) (b) Why might the true value of \(k_{g}\) in the reactor be significantly different from the value estimated in Part (a)? (Give several possible reasons.) (c) Create a spreadsheet in which up to five sets of values of the given variables ( \(d_{p}\) through \(u\) ) are entered in columns and the corresponding values of \(k_{g}\) are calculated. Test your program using the following variable sets: (i) the values given above; (ii) as above, only double the particle diameter \(d_{p}\) (making it \(10.00 \mathrm{mm}\) ); (iii) as above, only double the diffusivity \(D ;\) (iv) as above, only double the viscosity \(\mu ;(\mathrm{v})\) as above, only double the velocity \(u\). Report all five calculated values of \(k_{g}\).

In modeling the effect of an impurity on crystal growth, the following equation was derived: \(\frac{G-G_{\mathrm{L}}}{G_{0}-G}=\frac{1}{K_{\mathrm{L}} C^{m}}\) where \(C\) is impurity concentration, \(G_{\mathrm{L}}\) is a limiting growth rate, \(G_{0}\) is the growth rate of the crystal with no impurity present, and \(K_{\mathrm{L}}\) and \(m\) are model parameters. In a particular experiment, \(G_{0}=3.00 \times 10^{-3} \mathrm{mm} / \mathrm{min},\) and \(G_{\mathrm{L}}=1.80 \times 10^{-3} \mathrm{mm} / \mathrm{min} .\) Growth rates are measured for several impurity concentrations \(C\) (parts per million, or ppm), with the following results: $$\begin{array}{|c|c|c|c|c|c|}\hline C(\mathrm{ppm}) & 50.0 & 75.0 & 100.0 & 125.0 & 150.0 \\\\\hline G(\mathrm{mm} / \mathrm{min}) \times 10^{3} & 2.50 & 2.20 & 2.04 & 1.95 & 1.90 \\\\\hline\end{array}$$ (For example, when \(\left.C=50.0 \mathrm{ppm}, G=2.50 \times 10^{-3} \mathrm{mm} / \mathrm{min}\right)\). (a) Determine \(K_{\mathrm{L}}\) and \(m,\) giving both numerical values and units. (b) A solution is fed to a crystallizer in which the impurity concentration is 475 ppm. Estimate the expected crystal growth rate in (mm/min). Then state why you would be extremely skeptical about this result.

A horizontal drum, a cross-section of which is shown below, is being filled with benzene (density \(\left.=0.879 \mathrm{g} / \mathrm{cm}^{3}\right)\) at a constant rate \(\dot{m}(\mathrm{kg} / \mathrm{min}) .\) The drum has a length \(L\) and radius \(r,\) and the level of benzene in the drum is \(h\). The expression for the volume of benzene in the drum is \(V=L\left[r^{2} \cos ^{-1}\left(\frac{r-h}{r}\right)-(r-h) \sqrt{r^{2}-(r-h)^{2}}\right]\) (a) Show that the equation gives reasonable results for \(h=0, h=r,\) and \(h=2 r\). (b) Estimate the mass of benzene \((\mathrm{kg})\) in the tank if \(L=10 \mathrm{ft}, r=2 \mathrm{ft},\) and \(h=4\) in. (c) Suppose there is a sight glass on the side of the tank that allows observation of the height of liquid in the tank. Use a spreadsheet to prepare a graph that can be posted next to the sight glass so that an operator can estimate the mass that is in the tank without going through calculations like that in Part (b).

Sketch the plots described below and calculate the equations for \(y(x)\) from the given information. The plots are all straight lines. Note that the given coordinates refer to abscissa and ordinate values, not \(x\) and \(y\) values. [The solution of Part (a) is given as an example.] (a) A plot of In \(y\) versus \(x\) on rectangular coordinates passes through \((1.0,0.693)\) and \((2.0,0.0)\) (i.e., at the first point \(x=1.0\) and \(\ln y=0.693\) ). (b) A semilog plot of \(y\) (logarithmic axis) versus \(x\) passes through (1,2) and (2,1). (c) A log plot of \(y\) versus \(x\) passes through (1,2) and (2,1). (d) A semilog plot of \(x y\) (logarithmic axis) versus \(y / x\) passes through (1.0,40.2) and (2.0,807.0). (e) A log plot of \(y^{2} / x\) versus \((x-2)\) passes through (1.0,40.2) and (2.0,807.0).

The following expression has occurred in a problem solution: $$R=\frac{(0.6700)(264,980)(6)\left(5.386 \times 10^{4}\right)}{(3.14159)\left(0.479 \times 10^{7}\right)}$$ The factor 6 is a pure integer. Estimate the value of \(R\) without using a calculator, following the procedure outlined in Section 2.5b. Then calculate \(R\), expressing your answer in both scientific and decimal notation and making sure it has the correct number of significant figures.
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