Physics 41 Chapter 21 HW Set 1

Physics 41 Chapter 21 HW

1. A cylinder contains a mixture of helium and argon gas in equilibrium at 150°C. (a) What is the average kinetic energy for each type of gas molecule? (b) What is the root-mean-square speed of each type of molecule?

33 (a) K k T 1.38  1023 JK 423 K  8.76  10 21 J 22B  

1 (b) K m v2 8.76  10 21 J 2 rms 1.75 1020 J so v  (1) rms m 4.00 g m ol For helium, m  6.64  1024 g m olecule 6.02 1023 moleculesm ol m 6.64 1027 kg m olecule 39.9 g m ol Similarly for argon, m  6.63  1023 g m olecule 6.02 1023 moleculesm ol m 6.63 1026 kg m olecule Substituting in (1) above,

we find for helium, vrms  1.62 km s

and for argon, vrms  514 m s

2. A 1.00-mol sample of hydrogen gas is heated at constant pressure from 300 K to 420 K. Calculate (a) the energy transferred to the gas by heat, (b) the increase in its internal energy, and (c) the work done on the gas.

We us the tabulated values for CP and CV

(a) Q nCP  T 1.00 mol28.8 Jm ol  K 420  300 K  3.46 kJ

(b) Eint  nCV  T 1.00 mol20.4 Jm ol  K 120 K  2.45 kJ

3. A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of 5.00 atm and a volume of 12.0 L to a final volume of 30.0 L. (a) What is the final pressure of the gas? (b) What are the initial and final temperatures? (c) Find Q, W, and Eint.   1.40  Vi 12.0 (a) PVPVifi  f PPfi 5.00 atm  1.39 atm V f 30.0

5 3 3 PV 5.00 1.013 10 Pa 12.0 10 m  (b) T ii   365 K i nR 2.00 mol 8.314 Jm ol K  5 3 3 PVff 1.39 1.013 10 Pa 30.0 10 m  T    253 K f nR 2.00 mol 8.314 Jm ol K 

CP RC V 5  1.40   , CRV  CCVV 2 5 Eint  nCV  T 2.00 mol 8.314 Jm ol  K  253 K  365 K   4.66 kJ 2

WEQ int   4.66 kJ  0   4.66 kJ

4. During the power stroke in a four-stroke automobile engine, the piston is forced down as the mixture of combustion products and air undergoes an adiabatic expansion as shown. Assume that (1) the engine is running at 2 500 cycles/min, (2) the gauge pressure right before the expansion is 20.0 atm, (3) the volumes of the mixture right before and after the expansion are 50.0 and 400 cm3, respectively (4) the time involved in the expansion is one-fourth that of the total cycle, and (5) the mixture behaves like an ideal gas with specific heat ratio 1.40. Find the average power generated during the expansion.

We suppose the air plus burnt gasoline behaves like a diatomic ideal gas. We find its final absolute pressure:

337 5 7 5 21.0 atm 50.0 cm  Pf  400 cm  75 1 Pf 21.0 atm 1.14 atm 8

5 5 5 Now Q  0 and W  E  nC T  T W  nRT  nRT  PV  PV int V f i 2f 2 i 2 f f i i 52 53 31.013 10 N m  6 3 3 W 1.14 atm 400 cm 21.0 atm 50.0 cm  10 m cm  2 1 atm W 150 J

1 1 min 60 s 3 The output work is W  150 J The time for this stroke is 6.00 10 s 4 2500 1 min

W 150 J P    25.0 kW t 6.00 103 s

5. A 4.00-L sample of a diatomic ideal gas with specific heat ratio 1.40, confined to a cylinder, is carried through a closed cycle. The gas is initially at 1.00 atm and at 300 K. First, its pressure is tripled under constant volume. Then, it expands adiabatically to its original pressure. Finally, the gas is compressed isobarically to its original volume. (a) Draw a PV diagram of this cycle. (b) Determine the volume of the gas at the end of the adiabatic expansion. (c) Find the temperature of the gas at the start of the adiabatic expansion. (d) Find the temperature at the end of the cycle. (e) What was the net work done on the gas for this cycle?

(a) See the diagram at the right. P B  3 Pi (b) PVPVBCB  C

 Adiabatic 3PVPViii  C 1 5 7 VVVVC3 i  3 i  2.19 i V 2.19 4.00 L 8.77 L C C P i A (c) PB V B nRT B 33 PV i i  nRT i V(L) Vi = 4 L VC TTBi3  3 300 K  900 K FIG. P21.29

(d) After one whole cycle, TTAi300 K .

5 (e) In AB, QAB nCVnRTT V  3 i  i   5.00 nRT i 2

Q BC  0 as this process is adiabatic

PC V C nRT C  P i2.19 V i   2.19 nRT i

so TTCi 2.19

7 QCA nCTnRT P   i 2.19 T i   4.17 nRT i 2 For the whole cycle,

QABCA Q AB  Q BC  Q CA 5.00  4.17 nRT i   0.829 nRT i EQW 0    intABCA ABCA ABCA

WABCA  Q ABCA  0.829 nRT i   0.829 PV i i 5 3 3 W ABCA  0.829  1.013  10 Pa 4.00  10 m   336 J

6. Consider 2.00 mol of an ideal diatomic gas. (a) Find the total heat capacity of the gas at constant volume and at constant pressure assuming the molecules rotate but do not vibrate. (b) What If? Repeat, assuming the molecules both rotate and vibrate.

The heat capacity at constant volume is nCV . An ideal gas of diatomic molecules has three degrees of freedom for translation in the x, y, and z directions. If we take the y axis along the axis of a molecule, then outside forces cannot excite rotation about this axis, since they have no lever arms. Collisions will set the molecule spinning only about the x and z axes.

(a) If the molecules do not vibrate, they have five degrees of freedom. Random collisions put 1 equal amounts of energy kT into all five kinds of motion. The average energy of one 2 B 5 molecule is kT. The internal energy of the two-mole sample is 2 B 5   5   5  N kTBABV  nN  kT   n  RTnCT   . 2   2   2 

5 The molar heat capacity is CR and the sample’s heat capacity is V 2 55    nCV  n R  2 mol  8.314 Jm ol  K   22   

nCV  41.6 JK

For the heat capacity at constant pressure we have 5  7  7  nCPV n C  R  n R  R   nR 2 mol  8.314 Jm ol  K   2  2  2 

nC P  58.2 JK

(b) In vibration with the center of mass fixed, both atoms are always moving in opposite directions with equal speeds. Vibration adds two more degrees of freedom for two more terms in the molecular energy, for kinetic and for elastic potential energy. We have 7 9 nCV  n R 58.2 JK and nCP  n R 74.8 JK 2 2

7. Fifteen identical particles have various speeds: one has a speed of 2.00 m/s; two have speeds of 3.00 m/s; three have speeds of 5.00 m/s; four have speeds of 7.00 m/s; three have speeds of 9.00 m/s; and two have speeds of 12.0 m/s. Find (a) the average speed, (b) the rms speed, and (c) the most probable speed of these particles.

nvii 1 (a) v   12   23   35   47   39   212    6.80 ms av N 15 

2 2 nvii 2 2 (b) v 54.9 m s  av N 2 so vvrms  54.9  7.41 m s  av

8. As a 1.00-mol sample of a monatomic ideal gas expands adiabatically, the work done on it is –2 500 J. The initial temperature and pressure of the gas are 500 K and 3.60 atm. Calculate (a) the final temperature, and (b) the final pressure.

(a) W nCV T f T i 3 2500 J  1 mol 8.314 Jm ol  KT  500 K T  300 K 2  f  f

 (b) PVPVifi  f   nRT nRTi f 11    PPif  TPTPii  ff PPif  1 1  1 T T   Ti f f  PPfi  PPif Ti

5 3 3 2 52 Tf 300 PPfi 3.60 atm  1.00 atm Ti 500

9. A heat engine using a monatomic gas follows the cycle shown. (a) Find the temperature, volume and pressure at each point and the change in internal energy, work done ON the gas, and Q for each process and the net for the cycle. Clearly label and show ALL your work, briefly explaining each process. Put your results in the tables provided below. (b) Find the total work done by the engine during one cycle and the thermal efficiency of the engine and c) the engine’s power output if it runs at 600 rpm.

T (K) P(kPa) V (cm3) 1 300 200 200 2 900 200 600 3 300 66.67 600

W (J) Q (J) Eint (J) 1  2 -80 200 120 2  3 0 -12 -120 3  1 43.9 43.9 0 Net -36.1 36.1 0

b) 18% c) 361 W