Example problems. Chapter 2: Temperature, heat, and the 1 st law of Thermodynamic. Homework: 2, 3, 4, 5, 6, 10, 15, 19, 21 (pages )

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1 Examle roblems Chater : emerature, heat, and the 1 st law of hermodynamic Homework:,, 4, 5, 6, 1, 15, 19, 1 (ages 5-51)

2 . (Page 5) wo constant-volume gas thermometers are assembled, one with nitrogen and the other with hydrogen. Both contain enough gas so that 8 kpa. (a) What is the difference between the ressures in the two thermometers if both bulbs are in boiling water? (b) Which gas is at higher ressure? (a) For the nitrogen thermometer: N 7.5 (K) 7.5 N (kpa) 7.16 For the hydrogen thermometer: H 7.15 (K) 7.15 H (kpa) kpa or 59 Pa Fig (age 48) (b) P N > P H

3 . A gas thermometer is constructed of two gas-containing bulbs, each in a water bath, as shown in the figure below. he ressure difference between the two bulbs is measured by a mercury manometer as shown. Aroriate reservoirs, not shown in the diagram, maintain constant gas volume in the two bulbs. here is no difference in ressure when both baths are at the trile oint of water. he ressure difference is 1 torr when one bath is at the trile oint and the other is at the boiling oint of water. It is 9. torr when one bath is at the trile oint and the other is at an unknown temerature to be measured. What is the unknown temerature? ; R When one bath () is at the trile oint and the other (R) is at the boiling oint of water: R R 7.16 (K) and

4 R boiling R 7.15 (K) ( R 1) When one bath () is at the trile oint and the other (R) is at an unknown temerature X R : ' ' R R ( 1) R X - R ' R X X 48 (K)

5 5. At what temerature is the Fahrenheit scale reading equal to (a) twice that of the Celsius scale and (b) half that of the Celsius scale? F F 1 C C : : F C 16 C; C F 4.6 C; C F F -1. F

6 6. On a linear X temerature scale, water freezes at -15. X and boils at 6. X. On a linear Y temerature scale, water freezes at -7. Y and boils at -. Y. A temerature of 5. Y corresonds to what temerature on the X scale? For linear scales, the relationshi between X and Y can be written by: y ax + b a + b - 6 a + b a, b x y a b 1 X

7 1. An aluminum flagole is m high. By how much does its length increase as the temerature increases by 15 C? For a linear exansion: α ( m) 1.1 cm Note: αx1-6 /C is the coefficient of linear exansion of aluminum (see able 18-, age 48).

8 15. A steel rod is. cm in diameter at 5. C. A brass ring has an interior diameter of.99 cm at 5. C. At what common temerature will the ring just slide onto the rod? For a linear exansion of the steel rod: D steel D + D α steel, steel, For a linear exansion of the brass ring: D brass D + D α brass, brass, s b If the ring just slides onto the rod, so D steel D brass.99 D D steel, brass, b D brass, α D α.99 steel, s C

9 19. A 1.8m-long vertical glass tube is half filled with a liquid at C. How much will the height of the liquid column change when the tube is heated to C? ake α glass 1.x1-5 /K and β liquid 4.x1-5 /K. Here, we need to consider the cross-sectional area exansion of the glass and the volume exansion of the liquid: ΔA A(α ) ΔV V β V V + V V (1 + β ) (1 + β ) h A A + A A (1 + α ) (1 + α ) (1 + β ) h h - h h 1 (1 + α ) 1.8 h.64 (m); C - C 1 4 h (m) h

10 1. As a result of a temerature rise of C, a bar with a crack as its center buckles uward. If the fixed distance is.77 m and the coefficient of linear exansion of the bar is 5 x 1-6 /C, find the rise x of the center. For a linear exansion: x α l l α where l /; l / x ( ) l + l l ( 1+ ) l l l α α (using the binomial theorem, see Aendix E).77 x l α ( m) 75.4( mm)

11 Homework: 5,,, 4, 44, 46, 47, 48, 49, 5, 51, 59, 6 (ages 51-54)

12 5. A certain diet doctor encourages eole to diet by drinking ice water. His theory is that the body must burn off enough fat to raise the temerature of the water from. C to the body temerature of 7. C. How many liters of ice water would have to be consumed to burn off 454 g (about 1 lb) of fat, assuming that burning this much fat requires 5 Cal be transferred to the ice water? Why is it not advisable to follow this diet? (One liter : 1 cm. he density of water is 1. g/cm.) 1 food calorie (Cal) 1 cal he mass of water needs to drink: 5 1 ( cal) m ( g) c 1( cal / g. K) (7 ) C m g V ρ 1 g / liter too much water to drink!!! 94.6( liters)

13 . A.4 kg samle is laced in a cooling aaratus that removes energy as heat at a constant rate. he figure below gives of the samle vs. time t; the samle freezes during the energy removal. he secific heat of the samle in its initial liquid hase is J/kg K -1. What are (a) the samle s heat of fusion and (b) its secific heat in the frozen hase? Key issue: he cooling aaratus removes energy as heat at a constant rate. he rate of removed energy as heat (er minute): (a) (b) cooling cm R tcooling tcooling R 9 (J/min) freezing 9 (J/min) (min) 7 (J) or 7 (kj) m 67.5 (kj/kg) freezing F F frozen R (min) frozen cm c m m ( ) 5 J kg.k

14 . he secific heat of a substance varies with temerature according to c , with in C and c in cal/g K -1. Find the energy required to raise the temerature of 1. g of this substance from 5 C to 15 C. cm In the case here: c c() d cmd total cmd 1 m cd 1 m ( ) d

15 44. A thermodynamic system is taken from state A to state B to state C, and then back to A, as shown in the -V diagram of Fig.a. (a)-(g) Comlete the table in Fig.b by inserting a lus sign, a minus sign, or a zero in each indicated cell. (h) What is the net work done by the system as it moves once through the cycle ABCA? ΔE int W W - 1 AB BC 1 ( J )

16 46. Suose J of work is done on a system and 8. cal is extracted from the system as heat. In the sense of the first law of thermodynamics, what are the values (including algebraic signs) of (a) W, (b), and (c) E int? (a) W on J Work done on the gas - work done by the gas W on - W W Won ( J (b) the gas released energy as heat, so <: 8cal ( J ) ) (c) the first law of thermodynamics: ΔE int W ΔE int 5. ( ) 15.( J )

17 47. When a system is taken from state i to state f along ath iaf in the figure below, 5 cal and W cal. Along ath ibf, 6 cal. (a) What is W along ath ibf? (b) If W -1 cal for the return ath fi, what is for this ath? (c) If E int,i 1 cal, what is E int,f? If E int,b cal, what is for (d) ath ib and (e) ath bf? ΔE int (a) For ath iaf: ΔE int, iaf For ath ibf: int, ibf E int,f int,f E int,i int,i ibf W iaf W ΔE E E W ibf iaf ΔE int,iaf Wibf ibf ( iaf Wiaf) 6 (5 ) 6(cal) (b) For ath fi: (c) For ath fi: ΔEint, fi Eint,i Eint,f ΔEint, if (cal) fi ΔEint, fi + W 1 4(cal) E E E 1 ( ) (cal) int, f int,i int, fi 4

18 (d) For ath ibf: Wibf Wib 6 (cal) as Wbf (e) For ath ibf: (constant volume) ΔEint, ib Eint,b Eint, i 1 1(cal) E + W (cal) ib int,ib ib 18 bf ibf ib (cal)

19 48. Gas within a chamber asses through the cycle shown in the figure below. Determine the energy transferred by the system as heat during rocess CA if the energy added as heat AB during rocess AB is 5. J, no energy is transferred as heat during rocess BC, and the net work done during the cycle is 15. J. ΔE int W For the ABCA closed cycle: ΔE int + + AB BC CA W CA W - AB BC CA (J)

20 49. he figure below dislays a closed cycle for a gas (the figure is not drawn to scale). he change in the internal energy of the gas as it moves from a to c along the ath abc is - J. As it moves from c to d, 18 J must be transferred to it as heat. An additional transfer of 8 J as heat is needed as it moves from d to a. How much work is done by the gas as it moves from c to d? ΔE int W For a closed cycle: ΔE int ΔE + ΔE + ΔE abc cd da ΔE (J) abc For rocess da: W ΔE da W 8-8 ΔE cd 8 1 (J) ΔE 18 1 cd cd cd (J) 6 (J)

21 5. A samle of gas is taken through cycle abca shown in the -V diagram (see figure). he net work done is +1.5 J. Along ath ab, the change in the internal energy is +. J and the magnitude of the work done is 5. J. Along ath ca, the energy transferred to the gas as heat is.5 J. How much energy is transferred as heat along (a) ath ab and (b) ath bc? ΔE int W (a) his rocess a b is an exansion (V b >V a ): W > and W 5 J ΔE + W + 5 ab int (b) We consider a closed cycle abca: 8 (J) ΔE W int + + ab bc ca W net bc W net ab ca (J)

22 51. A shere of radius.5 m, temerature 7. C, and emissivity.85 is located in an environment of temerature 77. C. At what rate does the shere (a) emit and (b) absorb thermal radiation? (c) What is the shere s net rate of energy exchange? (a) (b) (c) A P rad 4πR ; σεa 7+ P 1. 1 rad env Pnet Pabs Prad P σεa 4 abs env P.7 1 abs (K) (W) 5 (K) (W) (W)

23 54. If you were to walk briefly in sace without a sacesuit while far from the Sun (as an astronaut does in the movie 1), you would feel the cold of sace while you radiated energy, you would absorb almost none from your environment. (a) At what rate would you lose energy? (b) How much energy would you lose in s? Assume that your emissivity is.9, and estimate other data needed in the calculations. (a) he heat transfer mechanism is radiation: P rad σεa 8 4 Prad (b) he energy lost in s is: 4 (W) E P t rad 1 4 (J)

24 59. In Figure a, two identical rectangular rods of metal are welded end to end, with a temerature of 1 C on the left side and a temerature of 1 C on the right side. In. min, 1 J is conducted at a constant rate from the right side to the left side. How much time would be required to conduct 1 J if the rods were welded side to side as in Figure b. A (a) (b)

25 ka P C H cond he heat transfer mechanism is conduction: a C H a a cond, ka P b C H b b cond, ka P a cond, a cond, a cond, b a a b b cond, P 4 P P A A P so, the requested time is./4.5 min or s.

26 6. he figure below shows the cross section of a wall made of three layers. he thicknesses of the layers are 1,.75 1, and.5 1. he thermal conductivities are k 1, k.9k 1, and k.8k1. he temeratures at the left and right sides of the wall Are. C and -15. C, resectively. hermal conduction through the wall has reached the steady state. (a) What is the temerature difference across layer (between the left and right sides of the layer)? If k were, instead, equal to 1.1k 1, (b) would the rate at which energy is conducted through the wall be greater than, less than, or the same as reviously, and (c) what would be the value of? (a) P cond ( A(H (/k) /k )( C ) A /k H C (/k) (b) conductivity k increases conduction rate increases. (c) Reeat the calculation in art (a): ) C C