17. Thermal Behavior of Matter 物質的熱行為1. Gases 氣體2. Phase Changes 相變3. Thermal Expansion 熱膨脹

What unusual property of water is evident in this photo?這照片顯露了水的那一種特性？Ice is less dense than water.冰的密度比水低。

17.1. Gases 氣體The Ideal Gas Law:理想氣體定律：

A piston-cylinder system.一個活塞 - 氣缸系统。

p V N k T

k = 1.381023 J / K = Boltzmann’s constant 波爾茲曼常數N = number of molecules 分子數目

AN n N

NA = 6.0221023 = Avogadro’s number 阿佛加德羅數字 = number of atoms in 12 g of 12C 12 克 12C 裏面的原子數n = number of moles (mol) 摩爾數1 mole of matter contains NA molecules. 一摩爾包含 NA 個分子

p V n R T

AR N k = 8.314 J / K mol = Universal gas constant通用氣體常數All gases become ideal if sufficiently dilute.在充份稀釋後，所有氣體都會變成理想。

Example 17.1. STP 標準温壓What volume is occupied by 1.00 mol of an ideal gas

at standard temperature & pressure (STP),

where T = 0C, & p = 101.3 kPa = 1 atm?

在標準温壓下 ( T = 0C, & p = 101.3 kPa = 1 atm ) ， 1.00 mol 理想氣體的體積為何？p V n R T

n R TVp

3

1.00 8.314 / 273.2101.3 10

mol J K mol KPa

3 322.42 10 m 22.42 L

( last figure subject to round-off error ) 末位數含四捨五入誤差

Kinetic Theory of the Ideal Gas 理想氣體的運動理論Kinetic theory 運動理論 ( Newtonian mechanics 牛頓力學 ):

1.Gas consists of identical “point” molecules of mass m.

氣體由全部一樣，質量為 m 的〝點〞分子組成。2. No interaction between molecules, except when they collide.

分子之間除碰撞時無相互作用。3. Random motion. 隨機運動。4. Collisions with wall are elastic. 與容壁的碰撞都是彈性的。

表面積

Molecule i collides with right-hand wall (RHW). 分子 i 碰上右壁Momentum transfer to wall is 動量轉移為 2x i x ip m v

No intermolecular collision

Next collision with RHW occurs at

無分子互撞 下次碰撞右壁的時間為2

ix i

Ltv

Average force of i on RHW:i 加於右壁的平均力為 i

ii

pFt

2x im v

L

FpA

iiF

A 2

ix im v

A L

2x

ii

m vV

2x

mp N vV

22 1xx i

i

vvN

Random motion 隨機運動 2 2 2x y zv v v 21

3v 22 1

3 2p V N m v

23

N K

Ideal gas law is recovered if欲得理想氣體定律，祗需21 3

2 2K m v k T T ~ K

in

out

0y ip

Example 17.2. Air Molecule 空氣分子Find K of a molecule in air at room temperature ( 20C = 293K),

找出在室温 (20C = 293K) 空氣中一個分子的 K ，& determine the speed of a N2 molecule with this energy.

再决定一個能量為此值的氮分子 (N2) 的速率。32

K k T 233 1.38 10 / 2932

J K K 216.07 10 J

2

272 14 1.66 10Nm u kg 264.65 10 kg

2 2 Kv

m

21

26

2 6.07 104.65 10

Jkg

5 2 22.61 10 /m s

2v v 511 /m s2 3

2v kT

m

Thermal speed:熱速率 3th

k Tvm

Distribution of Molecular Speeds 分子速率的分佈Maxwell-Boltzmann Distribution: (elastic collisions between free particles)麥克斯韋 - 波爾茲曼分佈 : ( 自由粒子間的彈性碰撞 )

High-E tail extends rapidly with T

高 -E 尾巴隨 T 快速延展 chemical reaction easier at high T

化學反應在高 T 時較易進行 cooling of liquid 液體冷卻 ( by escape of high-E molecules)

( 緣於高 -E 分子逃逸 )

80 K

vth

300K

vth

分子數

分子速率

22 exp

2mvn v C vk T

0

n v dv N

Real Gases 實在的氣體Important corrections to the ideal gas model 理想氣體模型的重要修正 :

1.finite size of molecules available V reduced.

分子大小非零 可用 V 變小。2.Attractive interaction between molecules (van der Waals forces) reduced P.

分子的互吸作用 ( 范得瓦力 ) P

變小2

2

nRT a nPV nb V

van der Waals equation范得瓦方程式

minimum volume最小體積

2

2

a nP V nb nRTV

nRTPV

17.2. Phase Changes 相變Phase changes take place at fixed T = TC until whole system is in the new phase.

相變時温度固定在 T = TC ，直至整個系统變成新相為止。( breaking / building bonds raises U but keeps K unchanged )( 打斷 ／建立化學鍵會提高 U 但保持 K 不變 )

Heat of transformation 相變熱 L= energy per unit mass needed to change phase. 相變時單位質量所需熱量

Lf = Heat of fusion ( solid liquid ) 熔解熱 ( 固 液 )

Lv = Heat of vaporization ( liquid gas ) 蒸發熱 ( 液 氣 )

Ls = Heat of sublimation ( solid gas ) 昇華熱 ( 固 氣 )

Q L m

Water:水 334 /fL kJ kg 80 /cal g

1 / /C cal g K

Same E to melt 1 g ice

or heat water by 80 C

熔解 1 g 冰和水升温 80 C

需要同樣的熱量。

相變熱 ( 大氣壓下 )

物質 熔點 沸點酒精，乙基銅鉛水銀氧硫水鈾

Conceptual Example 17.1. Water Phases 水相

ice warming冰加温中

melting熔解中water warming水加温中

boiling 沸騰中steam warming蒸氣加温中

You put a block of ice initially at － 20C in a pan on a hot stove with a constant power output,你把一塊原來在 － 20C 的冰塊，放在一個功率固定的熱爐上的鍋子裏加熱， and heat it until it has melted, boiled, and evaporated. 等它融化，沸騰，而全部蒸發。Make a sketch of temperature versus time for this experiment.把這實驗用一個温度對時間的圖表達。

時間

温度

Making the Connection 連起來If you start with 0.95 kg of ice at － 20C and supply heat at the rate of 1.6 kW,如果你開始時用一塊在 － 20C 的 0.95 kg 冰塊，且以 1.6 kW 的速率加熱，how much time will it take until you’re left with only water vapor? 要多久你才會祗剩下水蒸氣？

Heat needed to warm ice to 0 C 把冰加温到 0 C 需熱 :

0.95 2.05 / 20kg kJ kg K K ice ice ice iceH m c T 39 kJ

Heat needed to melt ice at 0 C 在 0 C 把冰融解需熱 :

2i w ice fH m L 0.95 334 /kg kJ kg 317 kJ

Heat needed to vaporize water at 100 C 在 100 C 把水蒸發需熱 :

Heat needed to warm water to 100 C 把水加温到 100 C 需熱 :

water water water waterH m c T 0.95 4.18 / 100kg kJ kg K K 397 kJ

2w v water vH m L 0.95 2260 /kg kJ kg 2140 kJ

29001.6

kJtkW

Time needed :需時 : 1810 s 0.5 h

GOT IT 懂嗎 ? 17.2.

You bring a pot of water to boil & then forget about it.你把一壺水燒開之後把它忘了。10 min later you come back & find it still boiling.10 分鐘後你回來發現它還在沸騰。Is its temperature 它的温度是

(a) less, (b) greater than, or (c) equal to

(a) 小於， (b) 大於， 還是 (c) 等於100 C ?

Example 17.3. Meltdown 熔毀 !

A nuclear power plant’s reactor vessel cracks, draining all cooling water.

一個核電廠的反應堆容器破裂，把全部的冷卻用水流光。Although nuclear fission stops, radioactive decay continues to heat the reactor’s 2.5105 kg

uranium core at the rate of 120 MW.

雖然核分裂已停止，放射性衰變卻繼續以 120 MW 的功率加熱反應堆內 2.5105 kg 的鈾核心。Once the melting point is reached, how much energy will it take to melt the core?

一旦達到熔點，還需要多少能量才可以把核心熔解？How long will the melting take?

熔解過程要多久？

Q L m

82.8 /fL kJ kgTable 17.1: for U

582.8 / 2.5 10kJ kg kg 20.7 GJ

Time to melt the core:核心熔解需時QTP

9

6

20.7 10120 10

JW

173 s 3 m

Example 17.4. Enough Ice 冰夠嗎 ?

When 200 g of ice at 10 C are added to 1.0 kg of water at 15 C,

is there enough ice to cool the water to 0 C?

把 200 g 的 10 C 冰加到 1.0 kg 的 15 C 水之後，有沒有足夠的冰把水冷到 0 C ？If so, how much ice is left in the mixture? 如果夠，那剩餘的冰有多少？

Q L m

1.0 4.184 / / 15waterQ kg kJ kg K C

Q m c T

Heat released to bring water down to 0 C 把水降到 0 C 放出熱能62.8 kJ

0.2 2.05 / / 10iceQ kg kJ kg K C Heat required to bring ice up to 0 C 把冰升到 0 C 需用熱能

4.1 kJ

0.2 334 /meltQ kg kJ kg

Heat required to melt ice at 0 C 把冰在 0 C 熔掉需用熱能66.8 kJ more than enough ice 有足夠的冰

Ice needed:需要的冰 62.8 4.1

334 /kJ

mkJ kg

0.176 kg ice left = 剩下的冰 200 176 24g g g

Phase Diagrams 相變圖

Phase diagram: P vs T 相變圖 : P 對 T

Sublimation: solid gas昇華 : 固體 氣體 e.g., dry ice 乾冰 ( s-CO2 )

Critical point 臨界點

AB: low P, s g 低 P ，固 氣

CD: medium P, s l g 中 P ，固 液 氣

EF: high P, s l / f 高 P ， 固 液／流

GH: medium T, l g 中 T ， 液 氣

Caution: Phase transition doesn’t occur instantaneously 注意 : 相變不是瞬間發生的Triple point: s-l-g coexist

三相點 : 固 - 液 - 氣共存 = 273.16K, 0.6 kPa for H2O

Solid固態

Gas 氣態

liquid液態Melting熔解

Sublimation 昇華

Boiling 沸騰

C.P.臨界點

T.P.三相點

壓力

温度 TC

PC

Supercritical fluid : l-g indistinguishable超臨界流體 : 液 - 氣不可分

17.3. Thermal Expansion 熱膨脹

Coefficient of volume expansion :體積膨脹係數/V VT

1 dVV dT

Coefficient of linear expansion :線膨脹係數/L LT

3

Prob. 69

Prob. 72

膨脹係數 *

固體 液和氣體鋁黃銅銅玻璃 ( 派萊克斯 )冰不脹鋼鋼

空氣酒精，乙基汽油水銀水， 1C水， 20C水， 50C* 除非特別標明，都在室温附近。

† 不脹鋼含 64% 鐵和 36% 鎳；是一種特別為減少熱膨脹而配成的合金。

GOT IT? 17.3. If a donut-shaped object is heated, will the hole get

如果一個甜圈餅形的物體受熱，它的洞會變(a) larger, or (b) smaller ?

(a) 大， 或 (b) 小？

L L T 2 2R R T 1R R T

Example 17.5. Spilled Gasoline 溢出的汽油A steel gas can holds 20 L at 10C. 一個鋼制汽油礶在 10C 時可裝 20 L 。It’s filled to the brim at 10C. 它在 10C 時剛好裝滿。If the temperature is now increased to 25C, 假如温度現在上升到 25C ，by how much does the can’s volume increase? 礶子的體積增加多少？How much gas spills out? 有多少汽油溢出？

Table 17.2: 6 112 10steel K 6 136 10steel K

5 195 10gas K

/V VT

V V T

6 120 36 10 25 10canV L K C C 0.0108 L

5 120 95 10 25 10gasV L K C C 0.285 L

Spilled gas:溢油量 0.285 0.0108 0.275L L L

Thermal Expansion of Water 水的熱膨脹Reason: Ice crystal is open ice water 原因 : 冰晶體的結構開放

ice floats 冰浮水上max water occurs at 4C

water 最大值在 4C

At 1C5 14.8 10water K

At fixed T Tm , ice melts if P .在固定的 T Tm 下， 冰會在 P 時熔解。Application: skating. 應用 : 溜冰

> 0 < 0

三相點

臨界點液

固氣

Application: Aquatic Life & Lake Turnover應用： 水中生態和湖水翻轉

Anomalous behavior of ice-water makes aquatic life in freezing weather possible.冰 - 水的異常行為使得就算在冰寒的天氣下，仍然可以在水中生活。If deep enough, bottom water stays at 4C even when surface is iced over.如果水夠深，就算表面已經冰封，底部的水仍然會保持在 4C 。In a lake where bottom water stays at 4C year round,在一個底部的水整年都保持在 4C 的湖裏，surface & bottom water can mix (turnover) only in spring time when both are at 4 C.表面和底部的水祇能在春天兩者均為 4C 時混合 (翻轉 ) 。