5

Q5: For diatomic molecular gas with two energy levels for rotational motion with (0, h?/I) with the quantum number of angular momentum (J-0,1) and Lis moment of ine...

Question

Q5: For diatomic molecular gas with two energy levels for rotational motion with (0, h?/I) with the quantum number of angular momentum (J-0,1) and Lis moment of inertia:1 - Find the total partition function. 2- Find the heat capacity at constant volume: 3 - Find the Helmotz free energy 4- Find the entropy_

Q5: For diatomic molecular gas with two energy levels for rotational motion with (0, h?/I) with the quantum number of angular momentum (J-0,1) and Lis moment of inertia: 1 - Find the total partition function. 2- Find the heat capacity at constant volume: 3 - Find the Helmotz free energy 4- Find the entropy_



Answers

A diatomic gas molecule consists of two atoms of mass $m$ separated by a
fixed distance $d$ rotating about an axis
as indicated in Fig. $39-32$ . Assuming
that its angular momentum is quantized as in the Bohr model for the
hydrogen atom, find (a) the possible
angular velocities and (b) the possible
quantized rotational energies.

So in this problem, we have a thermodynamic cycle. Um, that we're going to use the ideal gas laws and the definitions of work and transfer and changing thermal energy to find the efficiency and the thermodynamic properties. So in going from state 1 to 2, we have an 80 erratic process, so there's no heat transfer and going from state to 23 we have no change in temperature and going from state to 321 we have no change in pressure. So at states, we have the pressure at all the states. We have the volume at three and at two, but not at one. We're told the temperature at two and three is 400 Calvin. So we have a diatonic ideal gas. So we have the heat, the heat capacities on, then their ratio is 1.4. So for this antibiotic process here, the ideal gas law is this reduces to this equation here. And that means that, um, we know everything in here except for V one. So we can solve for V one and we get 2.69 times 10 to the minus three cubic meters, or about 2690 Tibet centimeter. So again somewhere, you know, in here. Now, once we know V one, we can get t one because we know 23 and we know V three plugging and everything we get 269 Calvin. Now we have Let's see, here we get the number of moles of this type atomic gas we have because we know pressure, volume and temperature at two. So we get 0.1 to 0 moles. Now we can turn our attention to the energy. So, um, Q 12 is zero since that's 80 Batic, which means that the work from 1 to 2 is the negative of the change in thermal energy from 1 to 2, which is end times the heat capacity, a constant volume times a change in temperature and becomes minus 327 jewels. So again, the work is negative, as we expect, because it's really just the area under this curve here. Looks like I screwed up where my alignment on my 1000 years. But, um, so it makes sense that that's negative. Now, from 3 to 1, the change in in, uh, thermal energy again the equation here is minus, um, 327 jewels. Um, it's the negative of what the change in thermal energy from 1 to 2 is. Which is good because there's no change in energy. Geothermal energy from 2 to 3. So things will cancel out and we'll get a net change of zero. And they're my energy now. The work, done from 3 to 1, is P three times 31 minus 33 which is winds up being minus 300 minus 131 jewels again. Negative because we're going from here to here. Integration. We're compressing the gas during that phase. Let's see here. We then have work from 2 to 3. So the work done in an YSL thermal process again, it's just the area under this curve, and the integral is given by this and r t two comes a natural law Garvey three, over V two. This could also be Britain as p to V V to like. Either way, we'll get the same answer and we have the third is 555 jewels. And since we know that the change in internal energy is zero from 2 to 3. We know then that their change in, um changing Ah, thermal energy is zero because the change in temperature zero we know that Q 23 has to be equal to W to three. So we could put all these into an, um, a table which I've given here something, everything up we could see then that change in thermal energy is zero, as it better be than that change in Hey, transfer. We're not changing that heat in the story. In the system is 609 196.6 jewels. Then that change in work done by the system is 96 0.6 jewels Onda again, these two had better be equal and we can see that they are. And now we can figure out the efficiency by taking the network and dividing it by the heat that we put in, okay. And we get 17.4% for the efficiency of that cycle.

Because we know the rotational kinetic energy in this case can be good on rt any stem o r se universal gas Constant, which is 8.314 to promote, has helped and t is the temperature was given his $300 when them always one more So therefore, you're talking better. The question Well, fee. Ah, rotational kinetic energy is equal to Juan pointing Joe's or a mole times 8.31 for Jew homo tens Calvin and intense 300 Calvin. And he will give us the rotational. Kinetic energy is about 2.49 times 10 to the power three ju. And for next question Well, we know the moment of inertia, which is that I hear coming to have intensity over to the bar to Okay, so I'm here is the mass off? Uh, each Addis and the here is the distance between two adults in one molecule. So it was an arrangement here. Eventually you have eyes in tow. MD squirrel to you know, I'm here was just a mass for each Adam. We just capital m over a. We know the morning mass for the oxygen is 16 Grandpa mo. And every Gothel number is 6.2 10. Standard 23 promote so therefore eventually will have the individual mass for each oxygen atom is about 2.66 times £10.23 grams. If we come here to kill Brands is about 2.66 tensed into karting 26 kilograms and we know D was the distance between two adults. It's given its 1.21 time since one any time here. So therefore, if you're planning events with the question will have for the moment of inertia, it's actually equal to 2.66 times 10 to the power of neck 26 killed where 10 times 1.21 times 10 to the power of negative meter. And you look over to over to and is with us the moment Inertia. Yes, you're going to one point No, by times potential of power off 1946 killer. Where past meters. Where Horrible. Next question. We'll have, uh, the individual rotational kinetic energy for each Adam, which is scary rotation over. Only here, Okay, is equal to 1/2. I will make us where eyes a moment inertia or may guys, the angular Blast E or each other. So therefore have omega is equal to square room to okay, rotational overly I whenever the total rotational energy is 2.49 times 10 to both need you and we got the numbers is 6.0 10 10 to 23 and we know I was just a moment. Inertia is 1.95 test into power opening 46 kilograms has meters where so therefore that's determine the angle of last year, which is you go to square room two times Ah, 2.49 times, uh, tend to about three ju over 6.2 times 10 to the power of the 23 and then times 1.95 times 10 to the power of next 46 Q Bert tens meters square. This will give us the ankle over last E for each ad and is about six point by one times 10 to the power for 12 radiant for a second. So if we convert two revolutions per minute, the angle of last year will be goto se exploiting by one times 10 to the power of 12 10 times, one over to buy. Radiant. Okay. Actually, I support that revolution here because in one revolution they are to buy to high ratings. Okay? And we know in one minute there are 60 seconds, 60 seconds over one minutes. So therefore, will have, uh, the angle of last. Seeing revolution per minute is about six point to to times 10 to the power of 13 r p m. Which is the revolutions per minute. As you can tell, these have out of here is much bigger than 10,000 revolutions per minute. Okay. Which means that they angular velocity of rotation of the auction Adan's is much larger than any angular velocity of the typical piece off rapidly rotating machinery. OK, and these are the answers. What his question.

Hello students in this question at a given temperature supposed to be the rotational kinetic energy K. Rotational. It is given as keno for a die atomic gas. Okay And we have to determine the translational kinetic energy at the same temperature. So for the thai atomic guests we have degree of freedom of rotation equals 23 And for the translational sorry translational are three and rotational are too. Okay And we know that the kinetic energy. Okay it is equal to f by tumor player by RT. Okay so it means kinetic energy is directly proportional to the degree of freedom. So we can write that rotational kinetic energy to the translational kinetic energy. It will be equal to a far developing ft. So can I take energy rotational divided by kinetic energy translational. Sorry this kinetic energy rotational is given. Sk not develop a kinetic energy translational and professional degree of freedom are too and translational are three. So from here after rearranging translational kinetic energy will be equals to three by two times of K. Not. Okay so this becomes the answer for this question. Okay thank you.

So we need to find the total rotational, kinetic energy of a how they died. Stomach gas, Uh, in this case, oxygen. And then we're going to anoxia and gas oh, to molecule. And then we need to find the moment of inertia on the why or Z axis for this molecule and then we need to find the route means squared angular velocity for ah, this rotational motion. So the bit difficult in the sense that ah, we need to use a bit more found. We need to use a bit more Ah, formulas for other from other chapters. So listen, first writer down our givens so and equals one malls, we have one more and then it's important to mention that it's a dye atomic gas and then we have tea equals three hundred Calvin And then we have a, um, definition for rotational kinetic energy being one over two times the moment of inertia times the angular velocity squared and we know that the moment of inertia for a diatonic molecule would be two times the mass times all over two squared well, where l is. The distance equals distance between to Adams in a molecule and then em would equal the mass of one Adam. So again, one Adam, not one molecule. So essentially we would have Tio Let's solve for this really quickly And just say this is going to be equal to M l squared over for and at this point we can say Okay, let's start off with a diatonic molecule for a diatonic molecule. Let's try it out here. If these air your acts of axes if these air your axes maybe right here, here and right here. Yeah, If these are your axes, you can on ly rotate two ways you can rotate on the Z access. And you, Khun, if this was the Z and then this was the ah, we can call this and called us Why? And they call us the axe. You can rotate on the axe axes and you can rotate on the Z axes. Rather, I think the actual conscience said's ear. Why? So in fact, according to the question, this would actually be a wind this with the glass of beer. So according to the question we can wrote here on the Z and Lee y axes because that would change the actual orientation But if you were to rotate a diatonic molecule on the axial length, it wouldn't it would be a Ziff. You're rotating a sphere on the on that on any axes, so it's not as if so it's We have two degrees of freedom of speech to to to be the freedom for rotational motion, and one degree is going to be. It should be three. But one degree actually doesn't change the look of the Adam whatsoever. So rotating it on the X axis wouldn't really change how the diatonic molecule looks. So if you can imagine rotating it on the Z access because that would actually create a different, you know, a different orientation. Rotating on the Z on the Y axis would create a different orientation. However, on the Z access straight through the middle, it wouldn't it wouldn't change the orientation. So if we were to say, say, this two degrees of freedom for the rotational, kinetic energy would be equal to the degrees of freedom over two times and Artie or it would be equal to two over too. So it's simply be ableto an arty and we have the number of mall. So one times eight point three one four times T of three hundred Calvin. So this is going to be equal to two point for nine times ten to the third, Jules ten to the third jewels. So at this point, this is be your answer for partner. This would be the total do rotational motion associated with a diatonic molecule at three hundred Calvin. For one mole of this gas B, there's asking us to find the moment of inertia for this molecule. So we need to find the mass of one molecule. Um, so it would be, um, over avocados number. This would be equal to point zero three two kilograms, Permal and and I apologize. This is going to be molecule. And then we can divide this by six point zero to three times ten to the twenty third. And at this point, we can say, Okay, there's going to be equal to five point three one times ten to the negative, twenty six kilograms per molecule. And at this point, we could say I equals and l squared over for we're going to just plug all this and so five point three one times ten to the negative twenty six again massive a molecule, not massive. And Adam, My apologies. And then, ah, I would be one point two one times ten to the negative tenth and this would be squared all over four. And we find that the moment of inertia is going to be three point eight nine times ten to the negative forty six kilograms. Niedere is squared. So at this point s Oh, this would be your answer for part B this point c we need to find the route means squared angular velocity. So route means squared angular velocity with equal the square root of the average velocity squared. Rather this the squared rather the angular velocity squared the average of the angular velocity squared. You take the square root. That would be the route means squared. Um, rather, this is not equal to this. So this would be very different calculation than this. So do not do this. Do this. And we know that us Ah, from the formula for the rotational kinetic energy, we know that the angular velocity square is going to be equal to two times K I. And then we can say that to the route means squared velocity would be equal to radical to Kay over. I, um and this is going to be the rotational kinetic energy for one molecule. So if we want to say this K is going to be equal to you're a total divided by other God goes number. So this K is specifically for one molecule and this case specifically for one mall. So we need to divide by Abu Guardo's number to find the rotational kinetic energy for one molecule. And then we have this would be to Kay. This is simply playing this into this equation. And then we have Then we can simply substitute. So we have three point eight nine times ten to the negative forty six and then times six point zero two, three times ten to the twenty third that were done, denominator. And then our numerator would be two times two point for nine times ten to the third, our total rotational kinetic energy. And then we find that this is going to be equal, tio four point six one times ten to the negative twelve radiance per second weaken times this by one revs to pie radiance and then sixty seconds for every minute, and that's going to give us six point two two times. Ten tio thirteenth, right. My apology. Negative. Thirteenth. Wow. Sorry. Thirteenth rpm. This would be positive. My parties. So at this point, we can say, Okay, if we have four point six one times ten to the twelfth radiance per second and then equaling six point two two times ten to the thirteenth revolutions per minute, and then we have standard machinery at ten to the fourth revolutions per minute. How does this compare? Well, it's much higher, so Ah, the Moxon molecules rotating at a much, much higher speed. Ah, much, much are angular velocity than rapidly rotating machinery. So this would be, um, Tubby. Rather Omega. Our mess is much creator. That ten to the fourth rpm. My apologies. This is positive right here. And this is positive right here. So that would be the end of the solution. We simply had to again just realized that there's only two degrees of freedom for rotational motion for this die atomic molecule. And then realize that again. Thie Ah, room means squared. Ah, angular velocity must be calculated this way. And then we can use the formula for the rotational kinetic energy in order to find the angular velocity. So that's the end of the solution. Thank you for watching.


Similar Solved Questions

5 answers
10 cmMinor#hether Jdelu! 341 J0 JmnIEU 341 SI IE4AI(4W)(0.5 pIS)Vintual(Ipt) on the given figure) locate the image: (Draw diagram Draw the raythe mirror equation find oul the image distance Using(1,5 pts)Cni from the mirTor.distance located ut a 1oolqo ?41 pUR W? 0 [ c of curvature (C) The focal length of the center 241 pur: ( I) snjoj 241 432M134 Kempilu `JOJJIUU JAEJuOJ front located 22. An object
10 cm Minor #hether Jdelu! 341 J0 JmnIEU 341 SI IE4AI (4W) (0.5 pIS) Vintual (Ipt) on the given figure) locate the image: (Draw diagram Draw the ray the mirror equation find oul the image distance Using (1,5 pts) Cni from the mirTor. distance located ut a 1oolqo ?41 pUR W? 0 [ c of curvature (C) The...
5 answers
(3 points) When CaClz is dissolved in water; the temperature of the solution increases(a) Is the dissolution of CaClz endothermic or exothermic? (b) How are the relative magnitudes of the lattice energy of CaClz and its heat of hydration related to each other?(c) Why does the solution form? What drives the process?(d) Sketch qualitative energy diagram similar to Figure 12.6 or Figure 12.7 in your textbook for the dissolution of CaClz-
(3 points) When CaClz is dissolved in water; the temperature of the solution increases (a) Is the dissolution of CaClz endothermic or exothermic? (b) How are the relative magnitudes of the lattice energy of CaClz and its heat of hydration related to each other? (c) Why does the solution form? What d...
5 answers
Find the average value fave of the function f on the given Interval. f (2) 10x 12 , [0. 1]faveNeed Heip?Read ItEMaetr haktiDL41 polnts SCalcCC4 6.5.005_Find the average value have of the function h on the given interval. h(x) 6 cos" x sin X, [0, I]haveNeed Help?Read ILWatch lFalkk toa MerSubmit AnswerSave ProgressPractice Another VersionHicpolnts SCalcCC4 6.6.006Mi:Find the average value have of the function h on the given interval. {(20 12u)Jene
Find the average value fave of the function f on the given Interval. f (2) 10x 12 , [0. 1] fave Need Heip? Read ItE Maetr h aktiDL 41 polnts SCalcCC4 6.5.005_ Find the average value have of the function h on the given interval. h(x) 6 cos" x sin X, [0, I] have Need Help? Read IL Watch l Falkk...
5 answers
Let X,,Xz, - be & sequence of independent identically distributed continuous random variables We say that record high occurs at time n if max ( X, , Xn-1) That is. X, is record high ifit is larger than each of X!= Show thatP{a record occurs at time n} = In; E[number of records by time n] = Eli; Var(number of records by time n) = Ei(i-1)/i ; Let N = min {n: n > and a record occurs at time n}. Show E[v]=o _
Let X,,Xz, - be & sequence of independent identically distributed continuous random variables We say that record high occurs at time n if max ( X, , Xn-1) That is. X, is record high ifit is larger than each of X!= Show that P{a record occurs at time n} = In; E[number of records by time n] = Eli;...
5 answers
Manufacturing plant uses specific bulk product. The amount of product used day can be modeled by an exponentia distribution with & = probability that the plant will use more than tons On given day; (Round your answer to four decimal Diaces(measurements in tons). Find the
manufacturing plant uses specific bulk product. The amount of product used day can be modeled by an exponentia distribution with & = probability that the plant will use more than tons On given day; (Round your answer to four decimal Diaces (measurements in tons). Find the...
1 answers
Formally solve the problem (sc) Ox PGx) +9()u. K<*<0 0 < [ dx with the boundary conditions Eqs: (2) and (3) and initial conditions Eqs HW 7 (4) and (5), taking to be constant 2) x,w(1,) Jx (1,t ) = ( 0 2 + 0a 2 ) P, v (1f ) 1 Ra 2u (1,1) = (4 0 < + 3 Y 4) u (xr g= {{ K<x < k 5 ) % (X 10 ) =g6) I2<I 0 +
Formally solve the problem (sc) Ox PGx) +9()u. K<*<0 0 < [ dx with the boundary conditions Eqs: (2) and (3) and initial conditions Eqs HW 7 (4) and (5), taking to be constant 2) x,w(1,) Jx (1,t ) = ( 0 2 + 0a 2 ) P, v (1f ) 1 Ra 2u (1,1) = (4 0 < + 3 Y 4) u (xr g= {{ K<x < k 5 ) % ...
5 answers
A wire with weight per unit length of 0.098 N/m is suspended directly above second wire. The top wire carries a current of 10.6 A and the bottom wire carries current of 21.6 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion. X meters
A wire with weight per unit length of 0.098 N/m is suspended directly above second wire. The top wire carries a current of 10.6 A and the bottom wire carries current of 21.6 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion. X meter...
4 answers
Draw the most stable chair conformation of the following compounds Cis-| -isopropyl-+-methyleyclohexane b) Trans-[-bromo-2-ethylcyclohexane Trans-]-I-butyl-4-chlorocyelohexane Draw the chair conformation for cach ol the ollowing list and indicale which are chiral orachiral:Cis and Trans |,2 Diiodocyelohexane Cis and Trans !. 3 DiisopropyleyclohexaneCis and Trans [, 4 DimethylcyclohexaneCis-]-bromo-2-mcthylcyclohexaneTrans-] ~bromo-3-methylcyclohexane
Draw the most stable chair conformation of the following compounds Cis-| -isopropyl-+-methyleyclohexane b) Trans-[-bromo-2-ethylcyclohexane Trans-]-I-butyl-4-chlorocyelohexane Draw the chair conformation for cach ol the ollowing list and indicale which are chiral orachiral: Cis and Trans |,2 Diiodoc...
5 answers
Hos izo LE 2i S1ng" Iax; | a U 244 Ond Ee (G)z (n(zta)l 65 Laurenl (o #xconbon fcauic Z4a R) 3in9 Acn Ly Res #) ScoGnq 8 1S {ficicn 0 Hol Ccfficical ns Zia RI tracn EEC b2 An Jna a u5 Tenc Kc 1n TcdiCn W4h he excep Hon 12 ated Singok4ies 0ny 6f IS Fmoot Cloisd 450 enace? sodh Jees] pO SS Gnd 1f H~ea 1hgz dny Poiol CDZCrGy& IResK ja ClLkc (allowsing in dcgkaL XSucm) 0 X2 [ Fxt
hos izo LE 2i S1ng" Iax; | a U 244 Ond Ee (G)z (n(zta)l 65 Laurenl (o #xconbon fcauic Z4a R) 3in9 Acn Ly Res #) ScoGnq 8 1S {ficicn 0 Hol Ccfficical ns Zia RI tracn EEC b2 An Jna a u5 Tenc Kc 1n TcdiCn W4h he excep Hon 12 ated Singok4ies 0ny 6f IS Fmoot Cloisd 450 enace? sodh Jees] pO SS Gnd 1...
5 answers
For the function f(z) = e^z :a. Describe the domain and range.b. Show f(-z) = -1/f(z)c. Describe the image of the vertical line Re(z) = 1d. Describe the image of the horizontal line Im(z) = pi/4e. Describe the image of the infinite strip 0 <= Im(z) <=pi/4
For the function f(z) = e^z : a. Describe the domain and range. b. Show f(-z) = -1/f(z) c. Describe the image of the vertical line Re(z) = 1 d. Describe the image of the horizontal line Im(z) = pi/4 e. Describe the image of the infinite strip 0 <= Im(z) <= pi/4...
4 answers
You analyze mutants (1,2.3) that have defects in the four enzymes of a biosynthetic pathway for an essential product, for their ability to grow on intermediates (ABC,D) You obtain the following dataIndicate the order of the intermediates in the pathway and the enzyme that catalyze each step0 D-1-,A 2-,8-3--€D 3->42->8-1->D0 €1-B-2- AJ-,D0 4-1-,82->C3-,0
You analyze mutants (1,2.3) that have defects in the four enzymes of a biosynthetic pathway for an essential product, for their ability to grow on intermediates (ABC,D) You obtain the following data Indicate the order of the intermediates in the pathway and the enzyme that catalyze each step 0 D-1-,...
4 answers
(16 points) The vector fuinctionr(t) 2cos (ln (3t))i sin (In (3t))j VBsin (In (3t))kdetermines CHTTE C in spaceFind the unit tangent vector T(t)Find the principal normal vector N(t).Find the cunatuteFind the tangential and normal components of acceleration and express the acceleration vector; alt) in terms of the unit tangent vector T and the principal normal vector N.
(16 points) The vector fuinction r(t) 2cos (ln (3t))i sin (In (3t))j VBsin (In (3t))k determines CHTTE C in space Find the unit tangent vector T(t) Find the principal normal vector N(t). Find the cunatute Find the tangential and normal components of acceleration and express the acceleration vector; ...
4 answers
5 (10 points) Diagonalize the matrix A = if possible You may just find P and D. If it is not [0 5 possible; explain why:
5 (10 points) Diagonalize the matrix A = if possible You may just find P and D. If it is not [0 5 possible; explain why:...
5 answers
H2107-Calculus | Spring20 Let: f(x= f(x X-3 2*-6) f(x= 6-1} 3*-1}} 9(x-1) utot800 Find x-and Y-intercepts of the graph of f,ifithas any: Find vertical and horizontal asymptotels) of f, ifithas any: Find the critical number(s); intervals(s) of increasing and decreasing and points of relative extrema of f, ifithas any: Find intervals of concavity and thepoint(s) of inflection of f, ifany: Sketch the graph of _ label all important points from part (I} (2) and (3) Page
H2107-Calculus | Spring20 Let: f(x= f(x X-3 2*-6) f(x= 6-1} 3*-1}} 9(x-1) utot800 Find x-and Y-intercepts of the graph of f,ifithas any: Find vertical and horizontal asymptotels) of f, ifithas any: Find the critical number(s); intervals(s) of increasing and decreasing and points of relative extrema ...
5 answers
Which will have the larger molar mass: CCL; or () SnCl?
Which will have the larger molar mass: CCL; or () SnCl?...

-- 0.022376--