/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 L-Serine is an amino acid import... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 41

L-Serine is an amino acid important for its roles in synthesizing other amino acids and for its use in intravenous feeding solutions. It is often synthesized commercially by fermentation, and recovered by subjecting the fermentation broth to several processing steps and then crystallizing the serine from an aqueous solution. The solubilities of L-serine (L-Ser) in water have been measured at several temperatures, producing the following data: \(^{5}\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline T(\mathrm{K}) & 283.4 & 285.9 & 289.3 & 299.1 & 316.0 & 317.8 & 322.9 & 327.1 \\ \hline x \text { (mole fraction L-Ser) } & 0.0400 & 0.0426 & 0.0523 & 0.0702 & 0.1091 & 0.1144 & 0.1181 & 0.1248 \\ \hline\end{array}$$ One of the ways such data can be represented is with the van't Hoff equation: \(\ln x=(a / T)+b\) Graph the data so that the resulting plot is linear. Estimate \(a\) and \(b\) and give their units.

Short Answer

Expert verified
After analyzing the mole fraction of L-Ser in water at different temperatures and plotting the data, the constants a and b in the van't Hoff equation \( \ln x=(-aT) + b\), can be determined using linear regression. 'a' has units of inverse temperature (K^-1) and 'b' is dimensionless.

Step by step solution

01

Understand the Terms and Identification of Variables

First, let's understand what each term represents in this exercise: \n\n- L-serine (L-Ser) is an amino acid that is being studied here. \n- The data given represents the solubilities (x, mole fraction) of L-Ser in water at different temperatures (T, in Kelvin). \n- The van't Hoff equation provided (\( \ln x=(a / T)+b \)) is a standard linear form and is used to represent the relationship between solubility and temperature. \n\nThe task is to plot the data in a format that results in a linear graph and then estimate the values of 'a' and 'b', which are constants in the above equation.
02

Graphing the Data

Plotting the data might get easier after rearranging the van't Hoff equation to the form of a linear equation: \( y = mx + c \). Now, the adapted equation will be \( \ln x= -aT +b \) (as 'x' represents the natural log of the mole fraction and 'y' represents \(1/T\)) Now, plot \(1/T\) on the x-axis and \( \ln x \) on the y-axis. The slope of this line will give the value of -a, and the y-intercept will give the value of b.
03

Determination of a and b

Once you have plotted the data, use the trendline or built-in functions in software like Excel or Google Sheets to perform a linear regression. This will provide you with the equation of the regression line. The slope of the line will be -a and the y-intercept will be b. You can also manually calculate the slope and intercept using mathematical formulas for regression.
04

Units for a and b

In the equation \( \ln x=(-aT) + b\), the units for slope (-a) would be the inverse of temperature, K^-1. For the y-intercept (b), as it is equal to \( \ln x \), its units would be the same as the natural logarithm of the mole fraction, thus dimensionless (no units).

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Key Concepts

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

Understanding L-serine and Its Importance
L-serine is a non-essential amino acid that plays a crucial role in various biological processes. It is vital for synthesizing other amino acids and is frequently used in medical nutritional contexts, such as intravenous feeding solutions. This amino acid can be commercially produced through fermentation, and its recovery involves crystallizing L-serine from aqueous solutions.

Understanding the solubility of L-serine in water is fundamental for its effective extraction and utilization in various applications. The study of its solubility behavior with temperature changes allows for better optimization of the crystallization process. By adjusting temperature conditions based on solubility data, the yield and quality of L-serine crystals can be improved.
Solubility: What Is It and Why Does It Matter?
Solubility is the ability of a substance to dissolve in a solvent, forming a homogeneous solution at a certain temperature and pressure. In this context, we focus on how L-serine dissolves in water at different temperatures.

In the given problem, solubility is expressed as a mole fraction, a representation showing how many moles of L-serine are present per mole of solution. Studying the relationship between solubility and temperature is essential because:
  • Variations in temperature can alter solubility, impacting the crystallization process of substances like L-serine.
  • Knowing the solubility at different temperatures helps us design more efficient processes for commercial production.
The van't Hoff equation provides a mathematical model that relates solubility with temperature, allowing us to predict how changes in temperature will affect the solubility of L-serine.
Linear Regression and Its Application
Linear regression is a statistical technique used to model the relationship between two variables by fitting a linear equation to the observed data. Here, we use it to analyze how the solubility of L-serine changes with temperature.

The van't Hoff equation can be rearranged into a linear form:
  • y = mx + c corresponds to \ ln x = -a(1/T) + b
Where:
  • y represents \( \ln x \) (natural log of the mole fraction of L-serine)
  • x represents \( 1/T \) (inverse of temperature in Kelvin)
  • m is the slope (-a), showing the relationship strength
  • c is the y-intercept (b), indicating the value of \( \ln x \) when \( 1/T \) is zero
By plotting \( 1/T \) on the x-axis and \( \ln x \) on the y-axis, a linear graph is formed. The slope and intercept can be determined using software tools like Excel, helping us estimate the constants \( a \) and \( b \). These insights allow us to understand and predict the solubility trends of L-serine.

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

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).

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 \(k_{g}\).

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).

The doozy is defined as the unit of force required to accelerate a unit of mass, called the cuz, with the gravitational acceleration on the surface of the moon, which is one-sixth of the normal gravitational acceleration on earth. (a) What is the conversion factor that would be used to convert a force from the natural unit to the derived unit in this system? (Give both its numerical value and its units.) (b) What is the weight in doozies of a 3 -cuz object on the moon? What does the same object weigh in Lizard Lick, North Carolina?
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