Molecular Driving Forces

K.Dill and S.Bromberg

Chapter 16 Solvation & the Transfer of Molecules Between Phases - all with Video Answers

Educators

Chapter Questions

Problem 1

The mechanism of anesthetic drugs. Anesthetic drug action is thought to involve the solubility of the anes: thetic in the hydrocarbon region of the lipid bilayer of biological membranes. According to the classical MeyerOverton hypothesis, anesthesia occurs whenever the concentration of drug is greater than $0.03 \mathrm{~mol} \mathrm{~kg}^{-1}$ membrane, no matter what the anesthetic.
(a) For gaseous anesthetics like nitrous oxide or ether, how would you determine what gas pressure of anesthetic to use in the inhalation mix for a patient in order to achieve this membrane concentration?
(b) Lipid bilayers 'melt' from a solid-like state to a liquid-like state. Do you expect introduction of the anesthetic to increase, decrease, or not change the melting temperature? If the melting temperature changes, how would you predict the change?

Madeline Currie

Numerade Educator

04:12

Problem 2

Divers get the bends. Divers returning from deep dives can get the bends from nitrogen gas bubbles in their blood. Assume that blood is largely water. The Henry's law constant for $\mathrm{N}_{2}$ in water at $25^{\circ} \mathrm{C}$ is $86,000 \mathrm{~atm}$. The hydrostatic pressure is $1 \mathrm{~atm}$ at the surface of a body of water and increases by approximately $1 \mathrm{~atm}$ for every 33 feet in depth. Calculate the $\mathrm{N}_{2}$ solubility in the blood as a function of depth in the water, and explain why the bends occur.

Lottie Adams

Numerade Educator

01:53

Problem 3

Hydrophobic interactions. Two terms describe the hydrophobic effect: (i) hydrophobic hydration, the process of transferring a hydrophobic solute from the vapor phase into a very dilute solution in which the solvent is water, and (ii) hydrophobic interaction, the process of dimer formation from two identical hydrophobic molecules in a water solvent.
(a) Using the lattice model chemical potentials, and the solute convention, write the standard state chemical potential differences for each of these processes, assuming that these binary mixtures obey the regular solution theory.
(b) Describe physically, or in terms of diagrams, the driving forces and how these two processes are similar or different.

Joanna Quigley

Numerade Educator

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Problem 4

Solutes partition into lipid bilayers. Robinson Crusoe and his friend Friday are stranded on a desert island with no whiskey, only pure alcohol. Crusoe, an anesthesiologist, realizes that the effect of alcohol, as with other anesthetics, is felt when the alcohol reaches a particular critical concentration $c_{0}$ in the membranes of neurons. Crusoe and Friday have only a supply of propanol, butanol, and pentanol, and a table of their free energies of transfer for $T=300 \mathrm{~K}$.
(a) If a concentration of $c_{1}$ of ethanol in water is needed to produce a concentration $c_{0}$ in the membrane, a hydrocarbon-like environment, use Table $16.2$ to predict what concentrations in water of the other alcohols would produce the same degree of anesthesia.
(b) Mythical cannibals find Crusoe and Friday and throw them into a pot of boiling water. Will the alcohol stay in their membranes and keep them drunk at $100^{\circ} \mathrm{C}$ ?
(c) Which alcohol has the greatest tendency to partition into the membrane per degree of temperature rise?

Oliver Mcneely

Numerade Educator

01:10

Problem 5

Global warming. $\mathrm{CO}_{2}$ in the Earth's atmosphere prevents heat from escaping, and is responsible for roughly half of the greenhouse effect that causes global warming. Would global warming cause a further increase in atmospheric $\mathrm{CO}_{2}$ through vaporization from the oceans? Assume that the ocean is a two-component solution of water plus $\mathrm{CO}_{2}$, and that $\mathrm{CO}_{2}$ is much more volatile than water. Give an algebraic expression for the full temperature dependence of the Henry's law constant $k_{H}$ for the $\mathrm{CO}_{2}$ in water, that is, derive an equation for $\partial k_{H} / \partial T$.

David Collins

Numerade Educator

02:11

Problem 6

Modeling cavities in liquids. Assume you have a spherical cavity of radius $r$ in a liquid. The surface tension of the liquid is $y$, in units of energy per unit area.
(a) Write an expression for the energy $\varepsilon(r)$ required to create a cavity of radius $r$.
(b) Write an expression for the probability $p(r)$ of finding a cavity of radius $r$.
(c) What is the average energy $\langle\varepsilon\rangle$ for cavities in the liquid?
(d) Write an expression for the average cavity radius.
(e) If $k T=600 \mathrm{cal} \mathrm{mol}^{-1}$ and $y=25 \mathrm{calA}^{-2} \mathrm{~mol}^{-1}$, compute $\langle r\rangle$.

Anand Jangid

Numerade Educator

01:10

Problem 7

Sparkling drinks. $\mathrm{CO}_{2}$ in water has a Henry's law constant $k_{H}=1.25 \times 10^{6} \mathrm{mmHg}$. What mole fraction of $\mathrm{CO}_{2}$ in water will lead to 'bubbling up' and a vapor pressure equal to 1 atm?

Emily Harris

Numerade Educator

06:38

Problem 8

Modeling binding sites. You have a two-dimensional molecular lock and key in solvent $s$, as shown in Figure 16.18. Different parts of each molecule have different chemical characters $A, B$, and $C$.
(a) In terms of the different pair interactions, ( $w_{A B}, w_{A C}, w_{A S}, \ldots$, etc.) write an expression for the binding constant $K$ (i.e., for association).
(b) Which type of pair interaction $(A B, A C, B C)$ will dominate the attraction?

Keshav Singh

Numerade Educator

01:49

Problem 9

Oil/water partitioning of drugs. In the process of partitioning of a drug from oil into water, $\Delta s^{*}=$ $-50 \mathrm{cal} \mathrm{deg}^{-1} \mathrm{~mol}^{-1}$ and $\Delta h^{*}=0$ at $T=300 \mathrm{~K}$.
(a) What is $\Delta \mu^{*}$ at $T=300 \mathrm{~K}$ ?
(b) What is the partition coefficient from oil to water, $K_{\mathrm{oll}}^{\text {water }}$ at $T=300 \mathrm{~K} ?$
(c) Assume that $\Delta s^{*}$ and $\Delta h^{*}$ are independent of temperature. Calculate $K_{0}^{W}$ at $T=320 \mathrm{~K}$.

Mayukh Banik

Numerade Educator

01:27

Problem 10

Another oil/water partitioning problem. Assume that a drug molecule partitions from water to oil with partition coefficients $K_{1}=1000$ at $T_{1}=300 \mathrm{~K}$, and $K_{2}=1400$ at $T_{2}=320 \mathrm{~K}$.
(a) Calculate the free energy of transfer $\Delta \mu^{*}$ at $T_{1}=$ $300 \mathrm{~K}$.
(b) Calculate the enthalpy of transfer $\Delta h^{*}$ (water to oil).
(c) Calculate the entropy of transfer $\Delta s^{*}$ (water to oil).

Manik Pulyani

Numerade Educator

01:49

Problem 11

Oil/water partitioning of benzene. You put the solute benzene into a mixture containing the two solvents: oil and water. You observe the benzene concentration in water to be $x_{w}=2.0 \times 10^{-6} \mathrm{M}$ and that in oil to be $x_{0}=5.08 \times 10^{-3} \mathrm{M}$
(a) What is the partition coefficient $K_{\text {water }}^{\text {ail }}$ (from water into oil)?
(b) What is $\Delta \mu^{*}$ for the transfer of benzene from water into oil at $T=300 \mathrm{~K}$ ?

Mayukh Banik

Numerade Educator

01:02

Problem 12

Balancing osmotic pressures. Consider a membrane-enclosed vesicle that contains within it a protein species $A$ that cannot exchange across the membrane. This causes an osmotic flow of water into the cell. Could you reverse the osmotic flow by a sufficient concentration of a different nonexchangeable protein species $B$ in the external medium?

Narayan Hari

Numerade Educator

01:27

Problem 13

Vapor pressures of large molecules. Why do large molecules, such as polymers, proteins, and DNA, have very small vapor pressures?

Matthew Confer

Numerade Educator

07:45

Problem 14

Osmosis in plants. Consider the problem of how plants might lift water from ground level to their leaves. Assume that there is a semipermeable membrane at the roots, with pure water on the outside, and an ideal solution inside a small cylindrical capillary inside the plant. The solute mole fraction inside the capillary is $x=0.001$. The radius of the capillary is $10^{-2} \mathrm{~cm}$. The gravitational potential energy that must be overcome is $m g h$, where $m$ is the mass of the solution, and $g$ is the gravitational acceleration constant, $980 \mathrm{~cm} \mathrm{~s}^{-2}$. The density of the solution is $1 \mathrm{~g} \mathrm{~cm}^{-3}$. What is the height of the solution at room temperature? Can osmotic pressure account for raising this water?

James Irizarry

Numerade Educator

04:48

Problem 15

Polymerization in solution. Using the lattice dimer1zation model (see Figure $16.19$ ), derlve the equilibrium constant for creating a chain of $m$ monomers of type $A$ in a solvent $s$.

Ronald Prasad

Numerade Educator

02:45

Problem 16

Ethane association in water. The association of ethane molecules in water is accompanied by an enthalpy of $2.5 \mathrm{kcal} \mathrm{mol}^{-1}$ at $T=25^{\circ} \mathrm{C}$. Calculate $\left(\partial \ln K_{\text {assoc }} / \partial T\right)$ at this temperature. Does the "hydrophobic effect" get stronger or weaker as temperature is increased?

Nicole Smina

Numerade Educator

02:58

Problem 17

Freezing point depression by large molecules. Freezing temperature depression is a useful way of measuring the molecular masses of some molecules. Is it useful for proteins? One gram of protein of molecular mass $100,000 \mathrm{~g} \mathrm{~mol}^{-1}$ is in $1 \mathrm{~g}$ of water. Calculate the freezing temperature of the solution.

David Collins

Numerade Educator

02:34

Problem 18

Partitioning into small droplets is opposed by interfacial tension. When a solute molecule (s) partitions from water $(A)$ into a small oil droplet $(B)$, the droplet will grow larger, creating a larger surface of contact between the droplet and the water. Thus, partitioning the solute into the droplet will be opposed by the droplet's interfacial tension with water. Derive an expression to show how much the partition coefficient will be reduced as a function of the interfacial tension and radius of the droplet.

Narayan Hari

Numerade Educator

01:25

Problem 19

Alternative description of Henry's law. Show that an alternative way to express Henry's law of gas solubility is to say that the volume of gas that dissolves in a fixed volume of solution is independent of pressure at a given temperature.

Adriano Chikande

Numerade Educator

08:00

Problem 20

Benzene transfer into water. At $T=25^{\prime} \mathrm{C}$, benzene dissolves in water up to a mole fraction of $4.07 \times 10^{-4}$ (its solubility limit).
(a) Compute $\Delta \mu^{*}$ for transferring benzene to water.
(b) Compute $\chi_{\text {benzenewater }}$.
(c) Write an expression for the temperature dependence of $\Delta \mu^{*}$ as a function of $\Delta h^{*}, \Delta s^{*}$ (the molar enthalpy and entropy at $25^{\circ} \mathrm{C}$, and $\Delta C_{p}$.
(d) Using the expression you wrote for (c), calculate $\Delta \mu^{*}$ for transferring benzene to water at $T=$ $100^{\circ} \mathrm{C}$, if $\Delta h^{*}=2 \mathrm{kImol}^{-1}, \Delta s^{*}=-58 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$, and $\Delta C_{p}=225 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.

Caroline Basil

Numerade Educator

02:12

Problem 21

Raoult's law for methanol-water mixtures. Figure $16.20$ plots models for the vapor pressure $p_{\text {methanol versus }}$ methanol composition $x_{\text {methanol }}$ in a binary solution with water at $300 \mathrm{~K}$. Raoult's law behavior is indicated by the straight line.
(a) Using the lattice model, calculate the contact energy $w$ for methanol. To simplify, neglect the internal degrees of freedom for methanol, and assume a lattice coordination number $z=6$.
(b) From the graph on the right of Figure 16.20, estimate the value of the exchange parameter $\chi$ for the methanol-water mixture.
(c) Using the lattice model expression for the activ. ity coefficient, show that $p_{\text {methanol }}$ approaches $p_{\text {methanol }}^{*}$ (the equilibrium vapor pressure above pure methanol) with the same slope as Raoult's law. That is, say why the plot for $p_{\text {methanal }}$ should resemble the right-hand graph in Figure $16.20$, and not the left.

David Collins

Numerade Educator

01:48

Problem 22

Vapor pressure of water.
(a) At a vapor pressure $p=0.03 \mathrm{~atm}$, and a temperature $T=25^{\circ} \mathrm{C}$, what is the density of water vapor? Assume an ideal gas.
(b) What is the corresponding density of liquid water under the same conditions?
(c) If the vapor pressure of water is $0.03 \mathrm{~atm}$ at $T=$ $25^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ at $T=100^{\circ} \mathrm{C}$, what is the pairwise interaction energy $w_{\mathcal{A A}}$ if each water molecule has four nearest neighbors?

Adriano Chikande

Numerade Educator

01:06

Problem 23

Osmotic pressure of sucrose. What is the osmotic pressure of a $0.1$ molar solution of sucrose in water at $T=300 \mathrm{~K}$ ?

Nicole Smina

Numerade Educator