Department of Applied Chemistry 2009 Spring

Physical Cheistry I. Introduction ti II. Gas Perfect Gases III. Gas Real Gases CYUT Departent of Applied Cheistry 009 Spring

I. Introduction

物理化學內容架構 Therodynaics 熱力學 Cheical Kinetics 反應動力學 Quantu Mechanics 量子力學 Statistic Therodynaics 統計熱力學理想 / 真實氣體熱力學三大定律熱化學 (therocheistry) 平衡 (Equilibriu) 化學平衡物理平衡理想 / 真實溶液反應速率法則反應級數 (reaction order) 速率常數 (rate constant) 反應機構 (echanis) 基本量子理論原子構造分子結構與對稱光譜學 (spectroscopy) 固態理論機率與波茲曼分佈 (Boltzan Distribution) 統計力學氣體動力論輸送現象

物化學習要點觀念 / 現象 / 定義公式 / 參數 / 關聯驗證 / 計算 / 數值修正 / 發展 / 應用文字敘述數學表示例題習題公式推導, 應用實務 Note : 專有名詞與單位表示

States of Gases Equation of State Pf f (T,,n) P:Pressure T:Teperature PnRT /

Pressure Pressure is defined as force divided by the area to which the force is applied. 1 Pa 1 N - 1kg -1 s -1

Properties of Gases p Unit of Pressure P 壓力的單位

Self Test 1.1 Calculation of Pressure Calculate the pressure (in pascals and atospheres) exerted by a ass of 1.0 kg pressing through the point of a pin of area 1.0 10 at the surface of the Earth. Hint. The force exerted by a ass due to gravity at the surface of the Earth is g, where g is the acceleration of free fll fall

Pressure The gas with higher pressure tend to copress the gas with lower pressure. The equality of pressure on either side of a piston is a state of echanical equilibriu between two gases

Measureent of Pressure The pressure exerted by the atosphere is easured with a baroeter. The height of the ercury colun is proportional to the external pressure. p F A g A ( ρ ) g ( ρah) A A g ρgh p ρgl cosθ

Teperature Theral equilibriu is established if no change of state occurs when two objects A to B are in contact through a diatheric boundary. Diatheric Adiabatic

Teperature Zeroth Law of therodynaics: If A is in theral equilibriu with B, and B is in theral equilibriu with C, then C is also in theral equilibriu with A.

Teperature Theroeter, a device for easuring the teperature. Celsius scale of teperature ( C) The theroeter was first in contact with elting ice and then with boiling water was divided into 100 steps called degrees, the lower point being labeled 0. Therodynaic teperature scale Teperatures are denoted d T and are norally reported tdin kelvins, K (not K). A difference in teperature of 1 C is equivalent to a difference of 1 K. T / K θ / C + 73.15

Treperature Illustration 1.1 Converting teperatures To express 5.00 C as a teperature in kelvins T/K (5.00 C)/ C + 73.15 5.00 + 73.15 98.15 Note how the units (in this case, C) are cancelled like nubers. This is the procedure called quantity calculus in which aphysical quantity (such as the teperature) is the product of a nuerical value (5.00) and a unit (1 C). Multiplication of both sides by the unit K then gives T 98.15 K.

基本定義導熱 diatheric 絕熱 adiabatic

II. Perfect Gases

Properties of Gases The perfect Gas equation of State Pressure of the saple f(aount, volue, teperature) P nrt P : pressure of the saple : volue of the saple occupies n :aount of substances in the saple T : teperature of the saple Boyle s Law : P 1/ Charle s s Law : Avogadro s Principle : T at constant P P T at constant n http://www.phy.ntnu.edu.tw/deolab/java/idealgas/

Properties of Gases Gas Constant R 理想氣體常數 R 8.31451 J / K ol 8.31451 kpal/k ol 1.987 cal/ k ol 8.0578 10 - L at /K ol 6.364 L Torr/K ol

Properties of Gases Boyle s Law At constant teperature, the pressure of a fixed aount of gas is inversely proportitional to its volue P 1/

Properties of Gases Boyle s Law The pressure volue dependence of a fixed aount of perfect gas at different teperatures. Each curve is a hyperbola (p constant) and is called an isother. http://ebooks.bfwpub.co/s upp3.php?figure&1&4&boo p p kidpche

Gas Properties Charle s Law At constant pressure, the volue of a fixed aount of gas varies linearly with the teperature A + Bθ

Gas Properties Charle s Law T at constant P P T at constant

Properties of Gases Avogadro s Principle At a given teperature and pressure, equal volues of gas contain the sae nuber of olucules Molar volue / n n The olar volue of a gas is alost the sae for all gases at the sae teperature and pressure.

Properties of Gases Predicting pressure A cheist is investigating the conversion of atospheric nitrogen to usable for by the bacteria that inhabit the root systes of certain legues, and needs to know the pressure in kilopascals exerted by 1.5 g of nitrogen gas in a flask of volue 50 L at 0ºC. P 1.5g ( 8.314kPa L/K ol) ( 73.15K 0) nrt 8.0 g/ol + 435 kpa 0.50 L

Properties of Gases Other fors of Ideal Gas Law 理想氣體定律的其他形式 P NkT PM ρrt nrt N : total nuber of gas olecule 氣體分子數 k : Boltzan constant (1.38 10-3 J/K) M : olecular l weight 分子量 ρ : density 密度

Properties of Gases Estiation of Molecular Weight The density of a gaseous copound was found to be 1.3 g/l at 330 Kand 5.55 kpa. What is the olar ass of the copound? M ρrt P (1.3 g/l) (8.314 J/K ol) 330K 5.5kPa 3 (1.3 g/l) (8.314 10 Pa L/K ol) 330K 3 5.5 10 Pa 13 g/ol

Properties of Gases Cobined Gas Equation P T P T 1 1 1 What is the final volue of a saple of gas that has been heated fro 5 to 1000 and its pressure increased fro 10.0kpa to 150.0kpa, p,given that its initial volue was 15 l? ( 10 kpa ) ( 15 L ) ( 150 kpa ) ( 5 + 73.15K) ( 1000 + 73.15K) 4.3L

Properties of Gases Cobined Gas Equation In an industrial process, nitrogen is heated to 500 K in a vessel of constant volue. If it enters the vessel at 100 at and 300 K, what pressure would it exert at the working teperature if it behaved as a perfect gas? n p T Initial Sae 100 Sae 300 Final Sae sae 500 p1 n T 1 1 1 p n T

Kinetic Model of Gases Assuptions: The gas consists of olecules of ass in ceaseless rando otion. The size of the olecules is negligible, in the sense that their diaeters are uch saller than the average distance travelled between collisions. The olecules interact only through brief, infrequent, and elastic collision.

Kinetic Model of Gases The root ean square speed of the olecules of a gas is proportional to the aquare root of the teperature and inversely proportional to the square root of the olar ass. p 1 3 nmc and p nrt c : root ean square speed c M : 1 nmc 3 olecular ass M nrt c N A 3RT M v 1/ 1/

Properties of Gases Standard Condition Standard abient teperature and pressure (SATP) Teperature 98.15 k Pressure(p 0 ) 1 bar Standard teperature and pressure (STP) Teperature 0 Pressure(p 0 ) 1at

Properties of Gases Gas Mixture 混合氣體 P n RT + n RT + n RT +... n RT 1 3 i n RT P P P... i P 1 + + 3 + i Pi P n RT n RT i i ni xi nirt nrt n i i i P Mole fraction 混合氣體中某成分氣體之分壓與總壓力之比值恰為該氣體在此混合系統中之莫耳分率 i

Properties of Gases Gas Mixture 混合氣體 Dalton s law: The pressure exerted by a ixture of gases is the su of the pressures that each one would exist if it occupied the container alone.

Properties of Gases Partial Pressure P i χ i P total ole fraction χ i n i /n totalt P i : partial pressure P i χ i (nrt/) n i RT/

Properties of Gases Partial Pressure (A) Calculate the ole fractions of N,O andarindryairat sea level, given that 100.0g0g of air consists of 75.55 g of N, 3. g of O and 1.3 g of Ar. (B) When the total atospheric pressure is 100 kpa, what is the partial pressure of nitrogen, oxygen and argon, respectively? Calculate the oles for each gas N 75.7 / 8.0.7( ole) O 3. / 3 0.75( ole) Ar 1.3/39.95 0.035( ole) Total oles.7 + 0.75 + 0.035 3.456 Mole Fractions for each gas N O Ar.7 / 3.456 0.75/ 3.456 0.035/ 3.456 0.78 0.1 0.009 ( ole) Pressure for each gas N 100kPa 0.78 78 kpa 100kPa 0.1 1kPa O Ar 100kPa 0.009 0.9 kpa

III. Real Gases

Real Gases Molecular interaction Attraction 吸引力 lowering the total energy ake a negative contribution on the potential energy Repulsion 排斥力 ake a positive contribution on the total potential energy At large separations, the energy lowering interactions are doinant. At short distances the energy At short distances the energy raising repulsions doinate

FIgure 1.6

Copression factor Z P Z RT P RT real gas real gas real gas ideal gas / Z 1 perfect gas Z > 1 repulsive interaction Z < 1 attractive interaction

At very low pressures, all the gases shown have Z 1 and behave nearly perfectly. At high pressures, all the gases have Z > 1, signifying that they have a larger olar volue than a perfect gas. Repulsive forces are now doinant. At interediate pressures, ost gases have Z < 1, indicating that the attractive forces are reducing the olar volue relative to that of a perfect gas.

irial Equation of State The coefficients B, C,..., which depend on the teperature, are the second, third,...virial coefficients.

irial Equation of State For ideal gas p RT p 0 For real gas Z dz 1 dp 0 p p RT (1 + B' p + C' p ZRT + L ) + dz B' + pc' + L B' as p dp 0 B C p RT ( 1+ + + L ) p dz d(1/ ZRT B as )

dz dp B' + pc ' + L B ' as p 0 The copression factor, Z, approaches 1 at low pressures, but does so with different slopes. For a perfect gas, the slope is zero, For real gases the slope ay have either positive or negative slopes, and the slope ay vary with teperature. At the Boyle teperature, the slope is zero and the gas behaves perfectly over a wider range of conditions than at other teperatures.

Boyle Teperature Atthe Boyle teperature, T B dz/dp 0 as p 0 B 0 p RT B over a ore extended range of pressures than at other teperatures. At the Boyle teperaturet properties of the real gas do coincide with those of a perfect gas as p 0.

Boyle Teperature

Condensation of Real gas Near A, the pressure of the gas rises in approxiate agreeent with Boyle s law. Serious deviations fro that law begin to appear when the volue has been reduced to B. At C all siilarity to perfect behaviour is lost, for suddenly the volue decreased without any further rise in pressure. Just to the left of C a liquid appears, and there are two phases separated by a sharply defined surface. As the volue is decreased fro C through D to E, the aount of liquid increases. At E, the saple is entirely liquid. Any further reduction of volue requires the exertion of considerable pressure, as is indicated by the sharply rising line to the left of E. Even a sall reduction of volue fro E to F requires a great increase in pressure.

Real Gases liquid Supercritical fluid Gas Gas Gs and liquid qud 表示連續狀態, 並無產生相變化.

Condensation of gases The volue and the coposition of a syste containing CO at 58 K is shown at the points a, b, c, and d indicated in Figure 7.. The liquid and gas volues are not shown to scale.

Real Gases van der Waals equation of state 氣體狀態方程式的修正 an P + nb nrt P + ( nb ) Repulsive interaction -nb Attractive interaction Attractive interaction P P+(n/)

Real Gases van der Waals equation of state

Critical Teperature T c The teperature at which the range of shrunk to a single value For CO, T c 304.1 K At TT T c δp δp 0 and 0 δ δ T T T T c T T c

Feature of van der Waals eq Feature of van der Waals eq. a RT a b RT a b RT c 0 ) ( 3 + a b RT a b RT T T c ) ( 3 a b RT b b c c c T T T T 0 6 ) ( ) ( 4 3 3 b b a b a T T T T c c 1 ) ( 3 to which siplifies, ) ( 3 ) ( RT fro these two equations gives RT Equation 3 c gives b b b b T T c c c c 0 result into Substituting this. 3 equation is The solution to this 1 ) ( to which siplifies, ) ( ) ( RT c c 4 3 c Ρ Rb a R b b a T or b a b RT a b RT c c c c 7 8 ) ( ) (3 0 ) (3 ) ( ) ( 3 3 3 + +

Suppleentary 補充資料 Show that the slope of z as a function of P as P 0 is related to the van der Waals paraeters by li0 p z 1 P RT T b a RT

Solution 7.3 The copression Factor Rather than differentiating z with respect to P, we transfor the partial derivation to one involving RT a P b a z RT RT b RT, ideal z z P T RT T a a 1 b RT 1 b RT d R T 1 RT 1 d T T

7.3 The copression Factor We have transfored the differentiation with respect to 1/ to one involving. The substitution of RT/ for P is only valid in the low density liit. Because 1 1 d d 1 d d z 1 a + P T RT b ( b) RT ( b ) ( b) ( b) a + RT RT 1 b RT ( ) b a RT ( ) As P 0,, and b b b Therefroe, li P 0 z 1 a b p RT RT T

Boyle teperature T B a z 1 a When b li b 0 RT p 0 P RT RT b a/rt B T B (a/br) T At T B z 0 B z li 0 p 0 P T ideal gas z T>T B B, li > 0 repulsive p 0 P T<T B, B T z li < 0 p 0 P T attractive

Principle of Corresponding States P RT a 8 P T 3 P b 3 P 8 ( T/ T ) r 3 P 3 / 1 / c c c c c 3Tc ( ) ( ) c r r P r 8 T r 3 3 1 r r Reduced Paraeters P T r r r P P T P c T c c

Principle of Corresponding States

alues for the copression factor are shown as a function of the reduced pressure, Pr, for seven different gases at the six values of reduced teperatures indicated in the figure. The solid curves are drawn to guide the eye.

Copression factor, z, as as function of P r for the T r values in indicated. The curves were calculated using the van der Waals equation of state.

Exaple Proble 7.3 calculate the volue occupied by 1.00 kg of CH 4 gas at T 30 K and P 68.0bar.Calculate- ideal and the relative error in if were calculated fro the ideal gas equation of state. Solution For densities corresponding to the attractive range of the potential, 30K 68.0bar T r 1.1 and Pr 1.48 190.56K 45.99bar Fro Figure 7.8, z 0.63. 1000 g 1 1 0.70 0.08314 LbarK ol 30K 1 znrt 16.04 gol 11. P 68.0bar 0 11.0 L ideal 11.0 L 6.5L 0.70 ideal 6.5 L 58% 11.0 L