东海大学物理系

東海大學物理系
Air that initially occupies 0.14m3 at a gauge pressure of 103kPa is expanded isothermally to a pressure of 101.3kPa and then cooled at constant pressure until it reaches its initial volume. Compute the work done by the air. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.)
空氣，一開始體積 0.14m3 ，表壓103kPa 。等溫膨脹到壓力101.3kPa ，然後定壓冷卻，直至 達到其初始體積。計算空氣所做的功。（表壓的區別是實際壓力和大氣壓力。）
<解>：假設氣體在等溫膨脹過程，體積由Vi 變到Vf 。所做的功為
∫ ∫ W = Vf p dV = nRT Vf dV = nRT ln Vf
ௐ 5 ࢱĂВ 23 ࢱ!
Fundamentals of Physics Halliday & Resnic
東海大學物理系
是分子質量，v 是速度，θ 是速度與牆的法線方向的夾角。如果有 N 個分子碰撞牆壁， 改變的總動量為 2Nmv cosθ ，如果碰撞的總時間是 Δt ，那麼總動量的平均改變率為
= 1.8×103 K
(c) pcVc = Tc paVa Ta
⇒
Tc = Ta
pcVc paVa
=
(200) (2.5)(3) (2.5)(1)
= 6×102 K
( ) ( ) (d)
abc 三角形面積： Qnet
= Wnet
=
1 2
2.0 m3
(7.5 − 2.5) × 103 Pa = 5.0 × 103 J.
Vi
V Vi
Vi
理想氣體： pV = nRT ⇒ p = nRT V
Vi
=
nRT pi
Vf
=
nRT pf
⇒ Vf = pf Vi pi
⇒
W
= nRT ln Vi Vf
= piVi
ln
pi pf
Since the initial gauge pressure is 1.03 × 105 Pa,
pi = 1.03 × 105 Pa + 1.013 × 105 Pa = 2.04 × 105 Pa.（？？）
一氫分子束（ H2 ）以正向夾 55 度角直射到一面牆。分子束的每一個分子速度為1km / s 且質 量為 3.3×10−24 g 。分子束撞到牆上面積為 2cm2 ，撞擊速率每秒1023 個分子。在牆上的分子 束壓力為何？
<解>：在反射過程中，只有正向分量的動量變化，一個分子的動量改變為 2mv cosθ ，其中 m
=1.9 × 103 Pa We note that the value given for the mass was converted to kg and the value given for the area was converted to m2.
【Problem 19-27】 Water standing in the open at 320 C evaporates because of the escape of some of the surface molecules. The heat of vaporization (539 cal/g) is approximately equal to ε n , where is the average energy of the escaping molecules and n is the number of molecules per gram. (a) Find ε . (b) What is the ratio of ε to the average kinetic energy of H2O molecules, assuming the latter is related to temperature in the same way as it is for gases?
【Problem 19-19】 The lowest possible temperature in outer space is 2.7 K. What is the rms speed of hydrogen molecules at this temperature? (The molar mass is given in Table 19-1.)
假設 1.80 莫耳的理想氣體在 300 C 時通過等溫壓縮過程將體積由 3m3 壓縮到1.5m3 。（a）壓 縮過程中，有多少能量以熱方式轉移，和（b）是轉給氣體還是從氣體轉出？
<解>：(a)T = 30 + 273 = 303 K
ΔEint = nCvΔT = 0 = Q −W （因為 ΔT = 0 ） Q =W
Fundamentals of Physics Halliday & Resnic
【CH19】Kinetic Theory 7, 9, 13, 15, 19, 23, 27, 29, 31, 33, 37, 43, 45, 51, 53, 55,59, 61
東海大學物理系
【Problem 19-7】 Oxygen gas having a volume of 1000cm3 at 400C and 1.01×105 Pa expands until its volume is 1500cm3 and its pressure is 1.06×105 Pa . Find (a) the number of moles of oxygen present and (b) the final temperature of the sample.
( ) <解>：(a)
在 a 點： n =
pV RT
(2500 Pa ) 1.0 m3 = (8.31 J/mol ⋅ K) (200 K)
= 1.5 mol.
(b) pV = nRT
⇒ pbVb = Tb paVa Ta
⇒
Tb = Ta
pbVb paVa
=
(200) (7.5)(3) (2.5)(1)
Vf
=
piVi pf
, so
W
=
pf
( ⎛⎜⎜⎜⎜⎝Vi
−
piVi pf
⎞⎠⎟⎟⎟⎟⎟
=
pf
) − pi Vi
= (1.013×105 Pa − 2.04×105 Pa)(0.14 m3)
= −1.44×104 J.
ௐ 3 ࢱĂВ 23 ࢱ!
Fundamentals of Physics Halliday & Resnic
Q = W = nRT ln Vf = (1.8)(8.31)(303) ln 1.5 = −3.14×103 J
Vi
3
(b) 負號表示 the transfer of heat is from the gas.
【Problem 19-13】
ௐ 2 ࢱĂВ 23 ࢱ
Fundamentals of Physics Halliday & Resnic
K
T = 493.13 − 273.15 = 219.980 C ≈ 2200C
【Problem 19-9】 Suppose 1.80 mol of an ideal gas is taken from a volume of 3m3 to a volume of 1.5m3 via an isothermal compression at 300 C . (a) How much energy is transferred as heat during the compression, and (b) is the transfer to or from the gas?
W = p f (Vi −Vf ) . The gas starts in a state with pressure pf, so this is the pressure throughout
this portion of the process. We also note that the volume decreases from Vf to Vi. Now
東海大學物理系
The total work done by the gas over the entire process is W = 2×104 J −1.44×104 J = 5.6×103 J
【Problem 19-15】 A sample of an ideal gas is taken through the cyclic process abca shown in Fig. 19-21. The scale of the vertical axis is set by pb = 7.5kPa and pac = 2.5kPa . At point a,. (a) How many moles of gas are in the sample? What are (b) the temperature of the gas at point b, (c) the temperature of the gas at point c, and (d) the net energy added to the gas as heat during the cycle?
氧氣在 400C 、1.01×105 Pa 時體積為1000cm3 。當壓力為1.06×105 Pa 時，體積擴大到 1500cm3 。找出（a）現在的氧氣有多少 mole（莫耳）和（b）樣品的最終溫度。