/*! 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

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.

The following reactions take place in a batch reactor: \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}\) (desired product) \(\mathrm{B}+\mathrm{C} \rightarrow \mathrm{D}\) (hazardous product) As the reaction proceeds, D builds up in the reactor and could cause an explosion if its concentration exceeds 15 mol/L. To ensure the safety of the plant personnel, the reaction is quenched (e.g., by cooling the reactor contents to a low temperature) and the products are extracted when the concentration of \(D\) reaches \(10 \mathrm{mol} / \mathrm{L}\). The concentration of \(C\) is measured in real-time, and samples are periodically taken and analyzed to determine the concentration of D. The data are shown below: $$\begin{array}{|c|c|}\hline C_{\mathrm{C}}(\mathrm{mol} / \mathrm{L}) & C_{\mathrm{D}}(\mathrm{mol} / \mathrm{L}) \\ \hline 2.8 & 1.4 \\\\\hline 10 & 2.27 \\\\\hline 20 & 2.95 \\\\\hline 40 & 3.84 \\\\\hline 70 & 4.74 \\\\\hline 110 & 5.63 \\ \hline 160 & 6.49 \\\\\hline 220 & 7.32 \\\\\hline\end{array}$$ (a) What would be the general form of an expression for \(C_{\mathrm{D}}\) as a function of \(C_{\mathrm{C}} ?\) (b) Derive the expression. (c) At what concentration of \(C\) is the reactor stopped? (d) Someone proposed not stopping the reaction until \(C_{\mathrm{D}}=13 \mathrm{mol} / \mathrm{L},\) and someone else strongly objected. What would be the major arguments for and against that proposal?

The daily production of carbon dioxide from an \(880 \mathrm{MW}\) coal-fired power plant is estimated to be 31,000 tons. A proposal has been made to capture and sequester the \(\mathrm{CO}_{2}\) at approximately \(300 \mathrm{K}\) and 140 atm. At these conditions, the specific volume of \(\mathrm{CO}_{2}\) is estimated to be \(0.012 \mathrm{m}^{3} / \mathrm{kg}\). What volume \(\left(\mathrm{m}^{3}\right)\) of \(\mathrm{CO}_{2}\) would be collected during a one-year period?

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 temperature in a process unit is controlled by passing cooling water at a measured rate through a jacket that encloses the unit. The exact relationship between the unit temperature \(T\left(^{\circ} \mathrm{C}\right)\) and the water flow rate \(\phi(\mathrm{L} / \mathrm{s})\) is extremely complex, and it is desired to derive a simple empirical formula to approximate this relationship over a limited range of flow rates and temperatures. Data are taken for \(T\) versus \(\phi\). Plots of \(T\) versus \(\phi\) on rectangular and semilog coordinates are distinctly curved (ruling out \(T=a \phi+b\) and \(T=a e^{b \phi}\) as possible empirical functions), but a log plot appears as follows: A line drawn through the data goes through the points \(\left(\phi_{1}=25 \mathrm{L} / \mathrm{s}, T_{1}=210^{\circ} \mathrm{C}\right)\) and \(\left(\phi_{2}=40 \mathrm{L} / \mathrm{s},\right.\) \(\left.T_{2}=120^{\circ} \mathrm{C}\right)\). (a) What is the empirical relationship between \(\phi\) and \(T ?\) (b) Using your derived equation, estimate the cooling water flow rates needed to maintain the process unit temperature at \(85^{\circ} \mathrm{C}, 175^{\circ} \mathrm{C},\) and \(290^{\circ} \mathrm{C}\). (c) In which of the three estimates in Part (b) would you have the most confidence and in which would you have the least confidence? Explain your reasoning.
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