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Chapter 3: Problem 18

The following data have been obtained for the effect of solvent composition on the solubility of a serine, an amino acid, at \(10.0^{\circ} \mathrm{C}\) : $$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \text { Volume \% Methanol } & 0 & 10 & 20 & 30 & 40 & 60 & 80 & 100 \\\\\hline \text { Solubility (g/100 mL solvent) } & 22.72 & 18.98 & 11.58 & 6.415 & 4.205 & 1.805 & 0.85 & 0.65 \\\\\hline \text { Solution Density (g/mL) } & 1.00 & 0.98 & 0.97 & 0.95 & 0.94 & 0.91 & 0.88 & 0.79 \\\\\hline\end{array}$$ The data were obtained by mixing known volumes of methanol and water to obtain the desired solvent compositions, and then slowly adding measured amounts of serine to each mixture until no more would go into solution. The temperature was held constant at \(10.0^{\circ} \mathrm{C}\). (a) Derive an expression for solvent composition expressed as mass fraction of methanol, \(x\), as a function of volume fraction of methanol, \(f\) (b) Prepare a table of solubility of serine (g serine/g solution) versus mass fraction of methanol.

Short Answer

Expert verified
The expression for solvent composition expressed as mass fraction of methanol, \(x\), as function of volume fraction of methanol, \(f\) is \(x = (0.79 * f) / ((0.79*f) + (1-f*1))\). And the table of solubility is created by using the provided data and calculating solubility for each volume % of methanol using the formula Solubility = Quantity of serine(g)/ Quantity of solution(g).

Step by step solution

01

Understand the Relationship Between Mass Fraction and Volume Fraction

The mass fraction, \(x\), can be calculated using the formula \(x = (V_f * p_m) / ((V_f * p_m)+(V_w * p_w))\), where: \n- \(V_f\) is the volume fraction of methanol\n- \(V_w\) is the volume fraction of water (calculated by subtracting \(V_f\) from 1, since the total fraction has to be 1)\n- \(p_m\) and \(p_w\) are the known densities of methanol and water respectively
02

Declare Known Densities

The density for methanol and water at \(10.0^{\circ} C\) can be taken as \n- \(p_m = 0.79 g/mL\) for Methanol \n- \(p_w = 1.00 g/mL\) for Water
03

Substitute Known Densities Into the Expression

Substituting these values into our equation from Step 1, we get: \n\(x = (0.79 * f) / ((0.79*f) + (1-f*1))\)\n\nWe now have our expression for solvent composition expressed as mass fraction of methanol, \(x\), as a function of volume fraction of methanol, \(f\)
04

Calculate and Organize Data Into a Table

You need to use the data provided in the problem to calculate solubility (g serine/g solution) for each volume % methanol. The calculation would be as follows: \n\nSolubility = Quantity of serine(g)/ Quantity of solution(g) = Quantity of serine(g)/ [(Volume of methanol(mL) + Volume of water(mL)) * Density of solution(g/mL)] \n\nPut all the calculated values in the table along with corresponding mass fraction of methanol.

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Key Concepts

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

Solvent Composition
Understanding the solvent composition is crucial when dealing with mixtures like methanol and water, especially when examining the solubility of compounds such as serine. Solvent composition is often expressed in terms of volume fraction or mass fraction. The volume fraction is straightforward: it's simply the percentage of the total volume that is accounted for by a particular component, such as methanol.
In the context of our exercise, this was given as different percentages ranging from 0% to 100% methanol. Knowing the volume fraction is useful because it allows chemists to control and predict how different compositions will affect solubility. Methanol and water have different interactive properties with solutes; therefore, their proportionate quantities in a mixture directly influence the extent to which a solute like serine will dissolve.
When mixing methanol and water, it鈥檚 important to understand that densities differ (0.79 g/mL for methanol and 1.00 g/mL for water), which influences how much each component actually contributes to the solution on a mass basis, even if their volume fractions are known. This is where mass fraction becomes an important factor to calculate.
Mass Fraction Calculation
Calculating the mass fraction provides a clearer depiction of how much a certain component contributes to the overall mass of a solution. The mass fraction is significant because it gives a more accurate picture of the solvent's chemical environment in which solutes are present. It is calculated using the density values of the components involved. Here, the expression given in the exercise is crucial: - Let the mass fraction of methanol be denoted as \( x \).- The formula to determine this mass fraction from the volume fraction \( f \) is: \[ x = \frac{0.79 \cdot f}{(0.79 \cdot f) + (1 - f) \cdot 1} \]This formula incorporates the densities of methanol and water, allowing the conversion from a volume-based to a mass-based metric.
- Substitute the known density, 0.79 g/mL for methanol and 1.00 g/mL for water.
This relationship shows how the proportion of methanol on a mass basis varies slightly compared to its volume fraction due to their differing densities. Through this calculation, chemists can better design their experiments and predict solubility outcomes.
Volume Fraction
Volume fraction is a key concept in understanding mixture compositions, especially in solutions involving solvents like methanol and water. It is defined as the ratio of the volume of a particular component to the total volume of the mixture. In our exercise, the volume fraction for methanol was provided across different percentages, from 0% to 100%.
This concept directly impacts the solubility of a compound in a given solvent because it determines the concentration of the solvent component that interacts with the solute. Volume fraction is easier to measure than mass fraction because it involves directly mixing specific volumes, which is a straightforward physical process for many laboratory applications.
When using volume fraction, keep in mind that the actual interactions between the solvent and solute will also consider the densities of the components. Given methanol's lower density compared to water, mixtures of equal volume fractions will not have equal mass fractions. Each fraction gives a different viewpoint: volume fractions for ease of measurement and setup, and mass fractions for chemical interaction and binding consideration. Both are crucial for comprehensive understanding in chemical solubility contexts.

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Most popular questions from this chapter

A mixture of methane and air is capable of being ignited only if the mole percent of methane is between 5\% and 15\%. A mixture containing 9.0 mole\% methane in air flowing at a rate of 7.00 \(\times 10^{2} \mathrm{kg} / \mathrm{h}\) is to be diluted with pure air to reduce the methane concentration to the lower flammability limit. Calculate the required flow rate of air in mol/h and the percent by mass of oxygen in the product gas. (Note: Air may be taken to consist of \(\left.21 \text { mole } \% \mathrm{O}_{2} \text { and } 79 \% \mathrm{N}_{2} \text { and to have an average molecular weight of } 29.0 .\right)\)

A mixture of methanol and propyl acetate contains 25.0 wt\% methanol. (a) Using a single dimensional equation, determine the g-moles of methanol in \(200.0 \mathrm{kg}\) of the mixture. (b) The flow rate of propyl acetate in the mixture is to be 100.0 ib-mole/h. What must the mixture flow rate be in \(\mathrm{Ib}_{\mathrm{m}} / \mathrm{h} ?\)

The liquid level in a tank is determined by measuring the pressure at the bottom of the tank. A calibration curve is prepared by filling the tank to several known levels, reading the bottom pressure from a Bourdon gauge, and drawing a plot of level (m) vs. pressure (Pa). (a) Would you expect the calibration curve to be a straight line? Explain your answer. (b) The calibration experiment was done using a liquid with a specific gravity of \(0.900,\) but the tank is used to store a liquid with specific gravity of \(0.800 .\) Will the liquid level determined from the calibration curve be too high, too low, or correct? Explain. (c) If the actual liquid level is 8.0 meters, what value will be read from the calibration curve? If the tank has a height of \(10.0 \mathrm{m},\) what value will be read from the curve when the tank overflows?

The specific gravity of gasoline is approximately 0.70. (a) Estimate the mass (kg) of 50.0 liters of gasoline. (b) The mass flow rate of gasoline exiting a refinery tank is \(1150 \mathrm{kg} / \mathrm{min}\). Estimate the volumetric flow rate in liters/s. (c) Estimate the average mass flow rate ( \(\left(\mathrm{lb}_{\mathrm{m}} / \mathrm{min}\right)\) delivered by a gasoline pump. (d) Gasoline and kerosene (specific gravity \(=0.82\) ) are blended to obtain a mixture with a specific gravity of 0.78. Calculate the volumetric ratio (volume of gasoline/volume of kerosene) of the two compounds in the mixture, assuming \(V_{\text {blend }}=V_{\text {gasoline }}+V_{\text {kerosene. }}\)

The chemical reactor shown below has a cover that is held in place by a series of bolts. The cover is made of stainless steel ( \(\mathrm{SG}=8.0\) ), is 3 inches thick, has a diameter of 24 inches, and covers and seals an opening 20 inches in diameter. During turnaround, when the reactor is taken out of service for cleaning and repair, the cover was removed by an operator who thought the reactor had been depressurized using a standard venting procedure. However, the pressure gauge had been damaged in an earlier process upset (the reactor pressure had exceeded the upper limit of the gauge), and instead of being depressurized completely, the vessel was under a gauge pressure of 30 psi. (a) What force ( \(\left(\mathrm{b}_{\mathrm{f}}\right)\) were the bolts exerting on the cover before they were removed? (Hint: Don't forget that a pressure is exerted on the top of the cover by the atmosphere.) What happened when the last bolt was removed by the operator? Justify your prediction by estimating the initial acceleration of the cover upon removal of the last bolt. (b) Propose an alteration in the turnaround procedure to prevent recurrence of an incident of this kind.
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