(ag) Simulate experiments, train lab techniques, and teach theory through visual experiences that enhance long-term learning outcomes. This phase typically features a zero charge inhibiting the flow of ions between either side of the membrane. Given that information, it is possible to calculate the total number of particles on the outside (the right side). H 41 calculate the successive equilibrium potentials for na labster. When will the probabilities of water moving to the left or the right sides again be equal? 7. Labster integrates with all major LMS (Learning Management Systems) so that educators can use their gradebooks to track students performance data and students can keep a record of their work. initial volume 68 mL the ratio of the extracellular over the intracellular concentrations, or. One compound that can block ion pumps is ouabain. Viewed from today's perspective, the Curme route to petroleum-based ethylene ranks as an important discovery. Membranes are usually permeable to more than one ion. In a test of a sun-powered car, the mass of the car and driver is 100kg100 \mathrm{~kg}100kg. calculate the successive equilibrium potentials for na labster; gregory school teachers; kathleen king obituary; places rizal visited in europe; scandinavian baby names boy. An architect wants to cover a 100, Q: - 2112 To see this happening in real time, we have created a simulation, which we call Nernst. - First week only $4.99! As the left side shrinks, the concentration of glucose rises; as the right side swells, the concentration of glucose falls. then 20C So the equilibrium potential for Na+ These associated water molecules, which allow the ions to dissolve into water, are called, Separating electronic charges requires large amounts of energy, so if positively charged ions diffuse from one location to another, they will usually be accompanied by negatively charged ions; their charges thus cancel out, leading to. is howard charles related to craig charles actor is howard charles related to craig charles actor In large volume nerve cells, the number of ions that flow into or out of the cell to maintain either the resting potential or (as we will learn shortly) the action potential are very small relative to the very large number of ions in the bulk solution, so many cells can continue to generate action potentials for many hours even after the pump has been deprived of energy or blocked. A thermostat senses the room temperature, and it adjusts the amount of heating or cooling needed to keep the temperature steady. Controlling the resting potential controls the overall excitability of a nerve cell, and, as we will see, many inputs from other neurons, or even from within the neuron, can alter the resting potential, and change the pattern of activity of the nerve cell. A The first step in any such problem is to balance the chemical equation for the reaction (if it is not already balanced) and use it to derive the equilibrium constant expression. EC Cl- Cout In CALCULATE Rx z XF E = Na 400.0 mm 0 50.0 E = Result Universal Gas Constant R - 8.314 Joules T-1 mol-1 Faraday Constant - 1.648 - 104 Colmol-1 PROGRESS: 33% 218 DAY 1 10:22 Calculate the successive equilibrium potentials for Na+, K+ and Chat 20"C on the screen. Recall that the glucose molecule is significantly larger than a water molecule. divided by [K]in. This means that the equilibrium position of a horizontal spring is independent of the mass attached to it. MW(Mg(NO3)2) = 148.315 g/mol. A:According to law of conservation of mass, mass can neither be created nor be destroyed. You will learn how to predict the direction of a reaction using the equilibrium constant and the reaction quotient. * CC Finally, you will test three different neurotoxins and analyze their effects on action potentials to deduce their most probable mode of action. 5.834e-2 Inside the squid giant axon, the potassium ion concentration is 400 mM, the sodium ion concentration is 50 mM, the chloride concentration is 52 mM, and the concentration of impermeable negatively-charged particles is 385 mM. The Na+ current will exceed the current of K+. If you attach the mass to the spring in this position, the mass will stay at rest and not move - if left unperturbed. AGNO g Engage students in science through interactive learning scenarios. The nerve cell invests a great deal of energy in setting up a. Understanding the basic electrochemistry of nerve cells serves as the foundation for understanding all of a neuron's important electrical properties. These excess charges tend to collect near the membrane because the negative charges on the left are attracted to the positive charges on the right. For this to be true, all potentials (temperature, pressure, and chemical potential) must be the same in all of the . An experiment was conducted to determine the effect of glucose on the freezing point of water. Thus, another way to describe the sodium-potassium ATPase is to say that it directly contributes to the resting potential by making it slightly more hyperpolarized than it would be if the pump were not operating. The first was ethylene glycol, which became Prestone antifreeze. At 37.5 C, ENa would be +60 mV, a small difference (since T in the equation is in degrees Kelvin). O Hudl Once you have a good grasp of the concept of membrane potentials, you will be transported inside an axon to define the precise chain of actions leading to an action potential. See answer. Melting point This difference is known as the membrane potential. As a consequence, the probability of a molecule of glucose on the right side diffusing to the left side of the membrane is higher than the probability that this will happen in the opposite direction. Q:On the molecular level, what does it mean for a solid sample to be pure? Exat Ekt Na+ 440.0 mm E = RX 293.15 K z In CALCULATE 50.0 mm T 293.15 K 297.15 K 273,15 K Ex = Result Universal Gas Constant R. 8.314 Joules T-1 mol-1 Faraday Constant F - 9.648 104 Colmal-1 um 0C on more MacBook Pro 12 simulation/a0K2X00000jG6bfUAC/ + E A & 0 0 7 oc Equation 4 Ln Ksp = -Hsoln/R (1/T) + Ssoln/ R, Acetylene had many uses at the time of the First World War, because it burned with a hot and luminous flame mainly. So if the holes in the membrane were the right size, they would let water molecules through, but prevent glucose molecules from moving from one side to the other. Calculate the succesive equilibrium potentials for NA+,K, and Cl- at degrees Celsius, successive equilibrium potentials at 20 degree. Start your trial now! Note also that if the permeabilities to any two ions are set to zero, the equation will reduce to the Nernst potential equation for the third ion. It is represented by rho . Its relation with mass, Q:We use the . Extracellular recording is so called because a. . To study the axon of a nerve, you will reproduce the historically famous experiment of Dr Hodgkin and Dr Huxley, which was rewarded with a Nobel Prize, by setting the neuron in a test chamber to trigger action potentials with electric currents. b. Mgs) + Pb"aq) The membrane potential can change, for example when the neurone is . Check out all the Labster resources that can accelerate your teaching. Do the particles on either side of the squid giant axon satisfy electroneutrality? All reactant and product concentrations are constant at equilibrium. In this page, you'll learn what affects the equilibrium position of a mass oscillating on a spring, and how to determine it. Such a pump is, As we explained above, any treatment that induces the cell's potential to become more negative is. gx (conductance) is most likely to bring about a change in an excitable cell's ionic current and change membrane potential. Calculate the molar mass of the liquid. Q -60mV, neither Na+ (+60) or K+(-90) is in equilibrium. A more rigorous approach is discussed under, The above equation (Eq. For example, the sodium-potassium ATPase binds three intracellular sodium ions and two extracellular potassium ions; as a consequence, it removes more positive charge from inside the cell than it brings in. To follow the details of how equilibria, and then the resting potential, are established across the membrane of the nerve cell, it is helpful to visualize the movements of charged entities. O B. heterogeneous mixture, A:Compound is the substance which is made by combining of two or more same or different elements., A:n-BuLi act as base and deprotonate from CH3CN to form -CH2CN which act as nucleophile attack to, A:Firstly, dd the small values i.e 87.7593 and 14.8, followed by adding 27,340. I found the solution for this question on the site as well but I don't understand how the value -2.303 was obtained. 1) Mass of a solid (metal cylinder) channels involved in the action potential are? Clearly, each negatively charged chloride ion will move with a positively charged sodium ion, and given that the concentration of the ions is much higher on the right side, they will tend to diffuse to the less concentrated left side, until the concentration has become equal on both sides. V eq. Understand the role voltage-gated channels play in determining the shape of an action potential. best towns to live in saskatchewan / smallwood tv show premiere date / smallwood tv show premiere date Equilibrium is the state in which market supply and demand balance each other and, as a result, prices become stable. 2) Diameter of, A:Given, At equilibrium, the concentration of chloride ions outside the cell was twice the concentration inside the cell. Second, by examining the Nernst equation, it becomes immediately obvious that only three factors are important in determining the magnitude and direction of the Nernst potential. 3397 Intracellular and extracellular concentrations and Nernst equilibrium potential values for a few ions of physiological importance, If there are more than one type of channels that are open at any given time, the situation is slightly more complex (see, In Real Cells, Multiple Ions Contribute to the Membrane Potential, The Nernst equation can be derived based on simple thermodynamic principles (see, If two or more ions contribute to the membrane potential, the Nernst potential no longer yields the. we have to convert the given quantity, Q:What is the classification of the substance in the figure? Temp. Discover how atoms interact via chemical bonds to form compounds. Thus a voltage difference generates a gradient in concentration. Give aplausible explanation for any discrepancies. So, for the problem shown in Figure 3, this means that water molecules will tend to diffuse from the left side to the right side. They are called action potentials, and in this simulation you will use a squids giant neuron to learn about the molecular mechanisms behind this phenomenon. We need to take only one more step to get to the resting potential. The separated charges across the membrane begin to create an electrical field gradient that applies a force on the charged particles, and this gradient is oriented so that it will increasingly oppose the additional flow of negative charges to the left side of the membrane. 1. excess CH3I 0.02 mL Statistical Data Treatment And Evaluation. In this state, the concentrations of reactants and products are stable and do not change over time. %3D Think of a mass attached to a horizontal spring placed on a table (under ideal conditions, so we can neglect friction). Since there is only one unknown on the right-hand side, it would probably be worth focusing on that side first. ImL, Q:210 mg glucose per 100 ml blood. Then, compare the calculated equilibrium potentials with the actual measured mem-brane potential. 11 To uncover which neurotoxin affects the hospitalized patient, you will have to compare its effect on the neurons action potential with three other drugs, and come up with a hypothesis about how it deregulates the neurons ability to transmit a signal. You can write a second equation for the inside of the cell based on bulk electroneutrality. Once you understand how action potential works on the membrane of a neuron, you will compare the effect of existing neurotoxins with your sample for identification. Ech E. Nat K+ T 440.0 mm Rx Ec = z XF In CALCULATE 520 mm Cin 400.0 mm 52.0 mm 50.0 mm E = Result CH- Universal Cas Constant R-8.314 Joules T-1 mol-1 Faraday Constant F = 9.648 104 Col mol-1 PROGRESS: 33% 13 RAY 1 11:51 Calculate the successive equilibrium otentials for Na, K. and Cl' at 20C on he screen. Q:Question 4 with all parts must be answered! federal indictment list 2021 california; lightning fastpitch softball; . Are you good at keeping your life balanced? If the membrane is permeable to both sodium and chloride ions, what will happen? Using the values computed for the model cell shown in Figure 8, estimate the resting potential of the model cell. The flow of any inorganic ion, such as Na + or K +, through an ion channel (since membranes are normally impermeable to ions) is driven by the electrochemical gradient for that ion. Given the equation, and the data for the initial condition of the system, we can actually calculate the equilibrium or Nernst potential for our model cell. Definition. Term. Volume (V) of C-4 plastic explosive, Q:A sample of water is heated from room temperature to justbelow the boiling point. If the membrane is permeable to water, the concentration of solute particles on one side of the membrane must equal the concentration of solute particles on the other side of the membrane ( osmotic balance ). Raise the LabPad for more formation. In mammalian neurons, the equilibrium potential for Na + is ~+60 mV and for K + is ~-88 mV. H x F 1644 The process by which a single ion (potassium) sets up an equilibrium (Nernst) potential across the membrane, Charge separation across the membrane, hyperpolarization and depolarization, A Simulation Program for Understanding Electrochemistry, The Three Conditions for Membrane Equilibrium, Problem 1: Diffusion and Concentration Gradients, Balancing Concentration and Electrical Gradients: The Equilibrium or Nernst Potential, Understanding Hyperpolarization and Depolarization, Membrane Potential and the Effects of Changing Concentrations vs. Injecting Current, https://neurowiki.case.edu/w/index.php?title=Equilibrium_Potentials_I&oldid=35523, Creative Commons Attribution-NonCommercial-ShareAlike. 31, A:If the molecule contains a carbon atom attached with four different types of atoms or groups known. As a consequence, they have a minuscule impact on the total concentrations on either side. Identify a neurotoxin from its effect on a nerve. & W At thermal equilibrium, positive ions in an electric field will be distributed so that less ions are in a state of high energy and more at low energy. ? changes in what can change membrane potential? `NH2 2. . This point depends on the magnitude of the elastic force ( Fel =-k x) and the weight of the mass ( Fw =m g . Experimental data showed thatthe freezing point depression of water in solution was 2.6C when 1.0 g of glucose was dissolved in 10g of solvent. (aq) 2) H,00 2) NABH, , A:Balanced the following redox reaction under basic conditions ---, Q:ce floats on water. Ag20, H20, heat A:Given solutions are: (As before, use the Nernst equation to calculate the absolute value of the equilibrium potential, and determine the sign intuitively). The position where the spring is not stretched or compressed is the equilibrium position (x=0 in Figure 1) and it corresponds to the natural length of the spring, L0 . Consider the situation shown in Figure 6, in which the membrane is permeable only to water and to chloride ions; 100 mM of impermeable proteins, represented as 100 mM P, are also present inside the cell. Discover how atoms interact via chemical bonds to form compounds. Both of these effects favor the movement of chloride ions from the inside of the cell to the outside. NH2 EK+ ECH "Na+ ON 440.0 mm Cout (mm) R X 293.15 K +1 XF In CALCULATE CI- 52.0 mm 440.0 20.0 Ec= Result 560.0 Universal Gas Constant R - 8.314 Joules T-1 mol-1 Faraday Constant F-9.648 104 Col mol-1 DAY 1 11:49 PROGRESS: 33% Calculate the successive equilibrium potentials for Na+, Kand Cl' at 20C on the screen. Listening is a more complex process than hearing. Molarity = moles/volume of solution, Q:density of irregular solid 23 Lets now consider a mass attached to a vertical ideal spring, like the one of this simulation. Because G = 0 at equilibrium, the measured potential of a concentration cell is zero at equilibrium (the concentrations are equal). What will happen if the concentration of the external sucrose is doubled to 0.5 molar? What is the final resulting potential of this "tug of war"? If it were cold outside, it would get very cold; if it were hot outside, it would get very hot. Thus, we have created a situation in which the, When equilibrium occurs, the concentrations of chloride ions on either side of the membrane will not change appreciably, but there will be more negative charges on the left and more positive charges on the right. mass of irregular solid 9.039 g O C. 30 mL Fig. Calculate the equilibrium membrane potentials to be expected across a membrane at 37 C, with a NaCl concentration of 0.10 M on the "right side" and 0.01 M on the "left side", given the following conditions. Play the Equilibrium simulation and observe the equilibrium reaction at a m. 25 26 Thus, the small change we made will be undone automatically. Study the structure of atoms and molecules and discover different techniques in the chemistry lab. You can also view our curriculum alignment guide for NGSS, AP, and IB curriculum standards. 30 9.929e-1, Calculate the Hsoln and Ssoln for the dissolution of solid KHP from the trendline equation of your scatter graph. 1. Is this value similar to the equilibrium membrane potential that you computed for the membrane? The equilibrium potential of an ion is . If themass concentration of the mixture is 1.2 kg/m3, compute the following:a) Density fo component A and Bb) Molar Fractionsc) Mass fractionsd) Total pressure if the temperature of the mixture is 290K, In an experiment carried out at a temperature of 4700C and a pressure of 100 atmosphere, 4.3g of a volatile liquid produced 30.6cm3of the gas. To prevent this, nerve cells invest a great deal of energy in, Thus, because the resting potential requires a continuous input of energy to maintain the concentration gradients that drive it, the resting potential is a. 0xsinx(x2+1)(x2+4)dx\int_{0}^{\infty} \frac{x \sin x}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x0(x2+1)(x2+4)xsinxdx. Thus, there are no net changes in the concentrations of the reactant (s) and product (s). Given that sodium ions have a charge of. T A first step towards understanding the difference between an equilibrium potential and the resting potential, A more realistic analysis of the resting potential and its underlying fluxes, How the underlying ionic fluxes are changed by current injection, From the individual Nernst Potentials to the Resting Potential in terms of concentrations and ion permeabilities, How an electrogenic ion pump can affect the resting potential, and what happens if the pump is poisoned, Balancing Multiple Equilibrium Potentials: The Donnan Equilibrium, From Equilibrium to Steady State: How the Resting Potential Differs from an Equilibrium Potential, A Realistic Resting Potential: Different Permeabilities and Ion Concentrations, and the Need for an Ion Pump, The Effect of Current Injection on the Fluxes Underlying the Resting Potential, A Qualitative Description of the Constant Field (Goldman Hodgkin Katz) Equation, Problem 1: The Nernst Potential and the Constant Field Equation, Problem 2: Permeabilities and the Membrane Potential, Ion Pumps Maintain or Even Alter the Resting Potential, An Electrogenic Ion Pump and the Resting Potential, Problem 3: Using the Constant Field Equation, https://neurowiki.case.edu/w/index.php?title=Equilibrium_Potentials_II&oldid=3788, Creative Commons Attribution-NonCommercial-ShareAlike. Given this information, what are the concentrations of the sodium and chloride ions outside the cell (right side of Figure 6)? EVEN THOUGH A K+ CONCENTRATION GRADIENT STILL REMAINED log of this ratio is near 1 and is negative. BY . 5) is known as the, We should examine the Nernst equation closely in order to gain an intuitive understanding for the Nernst potential. Once you've mastered the material in this unit, you are ready to go on to understand the electrochemical basis of the resting potential. 300+ Web-based simulations that can be played on laptops, Chromebooks, and tablets/iPads without installing any software, Teacher dashboard to automate grading and track student progress, Embedded quizzes to help students master science content, Library of learning resources, lab reports, videos, theory pages, graphics and more, Learn about the elements that make up our universe in our chemistry simulations for Majors. Engage students in science through interactive learning scenarios. B. 4.00 mol of perfect gas atoms with C p,m= 25 R is initially at 100.0 kPa, 100.0 dm3. Equilibrium describes the state of a reversible reaction, in which the forward and backward reactions happen at equal rates. 1) Hg(OAc)2 The, Table 1. Final volume = 74 mL But at the microscopic level, a great deal is happening: water molecules approach and bounce off the membrane. K Am I doing this correctly? To (s)+ oct ca9) The inside of the cell would become more negative as the K+ ion left and the negatively charged proteins remained . Now, let's extend our analysis a step further. The molecular weight of gas mixture is found to be 30. Fig. The only way to be sure that you understand the material so far is to actually try to do the calculations that have just been outlined. Note again that this was assuming that the volume of the bath was far larger than the volume of the cell, so that we could ignore the changes in the volume of the bath, which would be negligible compared to those in the cell. A:Applying concept of pure solid / substance via molecular basis explanation. In chemistry there are several different types of equilibria: thermodynamic equilibrium, diffusive equilibrium, thermal equilibrium, and many more. It is not an Na + channel because, in the experimental conditions of Figure 8.10, ENa = 58 log (160/3) = +100 mV. A gaseous sample of molecules, with a molar mass of 87.9 g/mol, is contained in a cylinder at a temperature of 362. Q:a. Sns) + 2Cu(aq) 3 20, Q:al Reading NaOH = But, to set up the equilibrium or Nernst potential, some chloride ions need to move from the region where they are more concentrated to the region in which they are less concentrated. Channels that manipulate their gates in response to membrane potential are voltage gated channels. calculate the successive equilibrium potentials for na labster. Menu. Study the structure of atoms and molecules and discover different techniques in the chemistry lab.. What would this change do to the electrical gradient? A:In presence heat the substrate molecule break down to form dipole . - Volume of the vapor What would happen to the system shown in Figure 5 if the membrane were only permeable to the chloride ion? 29 13 > J03c) +C\Cqq) 0.01 ml Short answer Mostly Cl-is disregarded in calculations of the resting membrane potential and action potential voltage changes, because it is less important for the neural membrane characteristics than Na + and K +.. Background In some neurons Cl-is not actively transported. Labster is compatible with Canvas, Blackboard, Moodle, Google Classroom, Schoology, Sakai, and Brightspace / D2L. Stopping the pump will lead to a slight, slow depolarization (though this is usually not enough to cause the cell to fire action potentials). *Response times may vary by subject and question complexity. A:The question is based on the concept of dimensional analysis. the probability that an activation gate will open increases, while the probability that an inactivation gate will open decreases, regardless of the stimulus, no Action potential can be generated because all h gates are closed, another action potential can be generated, but would require a greater stimulus/ voltage as the threshold is higher due and amplitude will be smaller to less h gates being open ( need greater voltage to hit open the m gates.). H3C due to less movement of ions across the membrane, m gates open - depolarization Three conditions must be satisfied to ensure equilibrium across a membrane: We will now explore each of these conditions by setting up different conditions on either side of a membrane, and see how the results unfold, as well as how we can capture these ideas mathematically so that we can predict the outcome in advance of doing the experiment. Elevate your nursing program with UbiSim, a VR solution dedicated to clinical excellence. Part of the reason for adding more complexity with each of the problems we have examined, if you haven't guessed this already yourself, is to make the model cells more and more similar to real nerve cells, which have all of these complexities, and more. NH3 In terms of the resting membrane potential, Cl-hence settles its gradient passively across the membrane according to its . At this point, in fact, the total force is zero. 1549. A:Benzene can give electrophilic aromatic substitution reaction. The membrane potential results from a separation of positive and negative ions across the cell membrane. What are the [H3O+] and [OH-] in the resulting, Q: ! Equilibrium potential is the saturation of the momentary directional flow of charged ions at the cell membrane level. Raise the LabPad for more information. In other words, at E rev, the chemical and electrical forces are in balance. c. 2MN2 In this simulation your mission is to help out a hospital who just received a patient with severe food poisoning. 275 3,980 When the spring is unloaded (Figure 2 (a)), its equilibrium configuration is again the one where it is not stretched or compressed, so its natural length, L0 . d. move away from zero. Y 0.32 mL Under the conditions given here, the equilibrium potential is, We have just solved the Nernst equation for a specific ion, and so it would be worth recording the equation for doing this for future reference (note that this equation applies, What about osmotic balance? You will need to take this into account when you write the equation for electroneutrality. Equilibrium describes the state of a reversible reaction, in which the forward and backward reaction happen at equal rates. You can then use. Try Labster with your students right away. Assume that the nerve cell is equally permeable to sodium, potassium, and chloride ions, i.e.. Here's a chance to use what you have learned so far: Is the squid giant axon in osmotic balance? If you attach a mass to the spring in this configuration (Figure 2 (b)), the total force acting on the mass is given by its weight pointing downwards. The resting potential, however, is built upon several equilibria, which we will need to understand first; once we do, we can understand the electrochemical basis for the resting potential. 35 106. Please do so now. In this situation, the forces acting on the mass are its weight (Fw, which is constant during the oscillation) and the elastic restoring force from the spring, defined by Hooke's Law, 440.0 We can calculate the final concentration, as we did above for the uncharged glucose (. A:Matter: Any substance in universe that has mass and occupies some space is termed as matter. ).If only one ionic species is present in the system, and/or channels for only one ionic species are present (and open), then V Eq. Na HO These three factors are (, For a typical mammalian cell in its natural environment (37 C), the following ionic equilibrium potentials can be calculated. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Use this information to identify a neurotoxin affecting a hospitalized patient. Integrate with your LMS. Mass = moles molar mass Using the current clamp and voltage clamp techniques, you will identify the mechanisms behind each stage of an action potential. D On the left hand side, we have one equation for the total number of particles inside, which has to be equal to the concentration of particles on the outside; write down this equation. Let's begin by assuming that the membrane is equally permeable to both water and glucose molecules. Explain. due to less movement of ions across the membrane The trendline equation -5623*x + 17.7 The car starts from rest on the track at AAA, moving toward the right. This occurred because of the presence of charges impermeable to the membrane, the sodium ions. The distribution of ions will return to what it was before, at equilibrium. Q:Convert 26.31 mg into amu, using dimensional analysis. In this case, the equation is already balanced, and the equilibrium constant expression is as follows: K = [NO]2[Cl2] [NOCl]2. Very far from it; we only reach equilibrium after death. Will you be able to help out the sick patient? Learn the fundamentals of general, organic and biological chemistry in our lab simulations for general, organic and biological chemistry. positive ion at room temperature equals. In this state, the concentrations of reactants and products are stable and do not change over time. grams PROGRESS: 33% HE This, When the membrane is permeable to only one ion (such as chloride ions, as in this scenario), we call the membrane potential at equilibrium the. gated, they are open and close due to specific conditions. Calculate W, Q, U and H for this process. 2.00 mL Of 5.80 x10"M 2.1: A. 62 = ? The problem shown in Figure 3 becomes more interesting if we assume that water can freely move from one side of the membrane to the other, but glucose cannot. The J Na curve has an extra wiggle around t = 1.3 ms caused by the rapidly changing voltage while the conductance g Na varies smoothly (figure 7).
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