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Chapter 2: Problem 45

The relationship between the pressure \(P\) and volume \(V\) of the air in a cylinder during the upstroke of a piston in an air compressor can be expressed as \(P V^{k}=C\) where \(k\) and \(C\) are constants. During a compression test, the following data are taken: $$\begin{array}{|c|c|c|c|c|c|c|}\hline P(\mathrm{mm} \mathrm{Hg}) & 760 & 1140 & 1520 & 2280 & 3040 & 3800 \\ \hline V\left(\mathrm{cm}^{3}\right) & 48.3 & 37.4 & 31.3 & 24.1 & 20.0 & 17.4 \\\\\hline\end{array}$$ Determine the values of \(k\) and \(C\) that best fit the data. (Give both numerical values and units.)

Short Answer

Expert verified
To find the values of \(k\) and \(C\) that best fit the given data, take the logarithm of the given data and use the method of least squares to fit the logarithmic data to a line. Then, reverse-engineer the values of \(k\) and \(C\) from the line's slope and y-intercept. As the solutions obtained are based on a least squares fit, they represent an optimal fit for the observed data.

Step by step solution

01

Data transformation

First, take the natural logarithm of the pressure \(P\) and volume \(V\). This means creating two new arrays that contain the logarithmic values of the pressure (let's call this array \(\log(P)\)) and volume data points (let's call this array \(\log(V)\)).
02

Fit the data

Next step is to fit \(\log(P)\) versus \(\log(V)\) to a straight line, using the least squares method. The least squares line fitting method provides us with the slope \(m=-k\) and y-intercept \(b=\log(C)\) values for the best fit line.
03

Calculate \(C\) and \(k\)

Now that you have \(b\) and \(m\), you're one step away from finding the values of \(C\) and \(k\). Remember the y-intercept \(b\) is equal to \(\log(C)\), so find the value of \(C\) by taking the inverse natural logarithm or \(e^{b}\) of \(b\). In a similar fashion, calculate the value of \(k\), which is the negative slope \(m\), i.e., \(k = - m\).

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Key Concepts

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

Thermodynamic Properties
In the realm of physics and engineering, thermodynamic properties refer to the physical characteristics of a substance that define its state within the thermodynamic context. Two fundamental properties that are often discussed are pressure (P) and volume (V), which relate to the condition of gaseous systems, such as the air in a cylinder during a compression test.

Understanding the relationship between these properties is essential when analyzing the behavior of gases in various processes. The equation PV^{k} = C is a simplified model that demonstrates how pressure and volume can change proportionally for a given amount of gas when the temperature is either held constant or changes in a specific manner. Here, 'k' represents the polytropic index or the specific heat ratio, and 'C' is a constant that arises from the initial conditions of the system.
Polytropic Process
A polytropic process is a thermodynamic process that follows the equation PV^{n} = constant, where P is the pressure, V is the volume, and n is the polytropic index. This process generalizes various types of specific thermodynamic processes such as isothermal (constant temperature), isobaric (constant pressure), isochoric (constant volume), and adiabatic (no heat exchange).

In the given exercise, the air compressor's pistons follow a polytropic process where the value of n is given by k. Finding the correct value for k is key to understanding the behavior of the system during the compression test and will allow us to predict how changes in volume will affect the pressure in such a process.
Least Squares Method
The least squares method is a fundamental statistical tool used to determine the best fit line through a set of data points. It minimizes the sum of the squares of the differences (called residuals) between the observed values and the values predicted by the model.

In our scenario involving the air compressor, the least squares method helps to fit a straight line to the natural logarithm transformed pressure-versus-volume data. This line can be characterized by its slope and y-intercept, which, in relation to our formula PV^{k} = C, correspond to the negative polytropic index (-k) and the natural logarithm of the constant (ln(C)), respectively. The resulting line provides us with the most accurate values possible for these constants, based on the experimental data collected.
Natural Logarithm Transformation
Applying a natural logarithm transformation to data means converting each data point using the natural logarithm function (ln), which is the inverse of the exponential function with base e (Euler's number, approximately equal to 2.71828). This process can convert a non-linear relationship between two variables into a linear one for easier analysis and application of linear regression.

In the compression test data, applying the natural logarithm to both the pressure and volume aids in transforming the polytropic equation from a power-based model into a linear one, where the relationship can then be represented as ln(P) = -k * ln(V) + ln(C). With the data linearized, the least squares method can be effectively used to find the best fitting line and hence calculate the values for k and C with greater precision and ease.

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Most popular questions from this chapter

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) \(y=[a x+b / x]^{-1}\)

According to Archimedes' principle, the mass of a floating object equals the mass of the fluid displaced by the object. Use this principle to solve the following problems. (a) A wooden cylinder 30.0 cm high floats vertically in a tub of water (density \(=1.00 \mathrm{g} / \mathrm{cm}^{3}\) ). The top of the cylinder is \(13.5 \mathrm{cm}\) above the surface of the liquid. What is the density of the wood? (b) The same cylinder floats vertically in a liquid of unknown density. The top of the cylinder is \(18.9 \mathrm{cm}\) above the surface of the liquid. What is the liquid density? (c) Explain why knowing the length and width of the wooden objects is unnecessary in solving Parts (a) and (b).

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 solution containing hazardous waste is charged into a storage tank and subjected to a chemical treatment that decomposes the waste to harmless products. The concentration of the decomposing waste, \(C,\) has been reported to vary with time according to the formula \(C=1 /(a+b t)\) When sufficient time has elapsed for the concentration to drop to \(0.01 \mathrm{g} / \mathrm{L},\) the contents of the tank are discharged appropriately. The following data are taken for \(C\) and \(t\): $$\begin{array}{|c|c|c|c|c|c|}\hline t(\mathrm{h}) & 1.0 & 2.0 & 3.0 & 4.0 & 5.0 \\\\\hline C(\mathrm{g} / \mathrm{L}) & 1.43 & 1.02 & 0.73 & 0.53 & 0.38 \\\\\hline\end{array}$$ (a) If the given formula is correct, what plot would yield a straight line that would enable you to determine the parameters \(a\) and \(b ?\) (b) Estimate \(a\) and \(b\) using the method of least squares (Appendix A.1) or graphics software. Check the goodness of fit by generating a plot of \(C\) versus \(t\) that shows both the measured and predicted values of \(C\). (c) Using the results of Part (b), estimate the initial concentration of the waste in the tank and the time required for \(C\) to reach its discharge level. (d) You should have very little confidence in the time estimated in Part (c). Explain why. (e) There are potential problems with the whole waste disposal procedure. Suggest several of them. (f) The problem statement includes the phrase "discharged appropriately." Recognizing that what is considered appropriate may change with time, list three different means of disposal and concerns with each.

You arrive at your lab at 8 A.M. and add an indeterminate quantity of bacterial cells to a flask. At 11 A.M. you measure the number of cells using a spectrophotometer (the absorbance of light is directly related to the number of cells) and determine from a previous calibration that the flask contains 3850 cells, and at 5 P.M. the cell count has reached 36,530. (a) Fit each of the following formulas to the two given data points (that is, determine the values of the two constants in each formula): linear growth, \(C=C_{0}+k t ;\) exponential growth, \(C=C_{0} e^{k t} ;\) power-law growth, \(C=k t^{b} .\) In these expressions, \(C_{0}\) is the initial cell concentration and \(k\) and \(b\) are constants. (b) Select the most reasonable of the three formulas and justify your selection. (c) Estimate the initial number of cells present at 8 A.M. \((t=0)\). State any assumptions you make. (d) The culture needs to be split into two equal parts once the number of cells reaches 2 million. Estimate the time at which you would have to come back to perform this task. State any assumptions you make. If this is a routine operation that you must perform often, what does your result suggest about the scheduling of the experiment?
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