General Chemistry II, Unit I: Study Guide (part I)

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1 1 General Chemistry II, Unit I: Study Guide (part I) CDS Chapter 14: Physical Prperties f Gases Observatin 1: Pressure- Vlume Measurements n Gases The spring f air is measured as pressure, defined as the frce applied ver an area P =. Units f pressure: Atmsphere (atm): 1 atmsphere is the pressure exerted by gases in the atmsphere at sea level Trr (mm Hg): 1 mm Hg = atm "# Pascal (Pa): 1 atm = 760 trr = Pa = Using an experiment that traps small quantities f air inside a syringe, ne can measure the pressure f a gas while varying its vlume. Experimental data shws that as vlume increases, pressure decreases. Hwever, this relatinship is nt linear: Pltting the inverse f the pressure (1/P) versus vlume yields a linear relatinship. The straight line frmed by 1/P vs. V als cnnects with the rigin (0,0). The linear equatin relating pressure and vlume is = k V where k is a prprtinality cnstant. The data prving the inverse relatinship between pressure and vlume is cnsistent acrss multiple gases when the amunt f gas is held cnstant. Hwever, when the amunt f gas is varied, the prprtinality cnstant k, which represents the slpe f the linear graph, changes: The new line, n2, still passes thrugh the rigin, s the riginal equatin = k V is retained. Hwever, it is clear that k nw depends n the number f mles f gas. Byle s Law: The prduct f pressure and vlume is a cnstant fr a given amunt f gas at a fixed temperature. Observatin 2: Vlume- Temperature Measurements n Gases

2 2 The first three paragraphs can basically be summarized as temperature is subjective until it s actually imprtant. Placing mercury in ht and cld water yields different changes in the vlume f the liquid. Therefre, the vlume f mercury is a measure f hw ht smething is. In additin, tw different bjects with the same subjective htness give the same vlume f mercury temperature is nt dependent n the identity f the bject being measured. Fahrenheit and Celsius are arbitrary ways f measuring temperature; they d nt reveal what physical prperty is actually being measured. New experiment: trap a small sample f air (at rm temperature and pressure) in a syringe and mnitr the temperature (using a mercury thermmeter) as the vlume is varied. In this experiment, the pressure is held cnstant by a pistn that mves against atmspheric pressure. There is a simple linear relatinship between the vlume f a sample f a gas and its temperature. This can be expressed by a crrespndingly simple linear equatin, V = αt + β, where t is the temperature in Celsius, α is the slpe f the line and β is the y- intercept. Frm the CDS experiment, α = ml/ C and β = 91.7 ml. Therefre, the quantity α/β must be a temperature because the vlume units (ml) cancel ut, leaving nly C. Therefre, the linear equatin can be rewritten as V = α t +. This gives an x- intercept f C. Repeating this experiment many times cntinues t yield a rati f - β/α = C. While we d nt knw the meaning f this temperature, we can assume it is imprtant and give it the title abslute zer, since any temperature lwer than it wuld prduce an impssible negative gas vlume. Fr simplicity, a new abslute temperature scale, Kelvin, can be defined with the same unit size as Celsius. Since C = 0 K, the abve linear relatinship between vlume and temperature can be rewritten as V = αt. Charles Law: The vlume is prprtinal t the abslute temperature (in Kelvin). The cnstant α depends n the pressure and quantity (number f mles) f gas. The Ideal Gas Law Law Related prperties Type f Cnstants Equatin relatinship Byle s Law Pressure and Inverse Quantity, temperature P V = k N, T vlume Charles Law Vlume and Linear Quantity, pressure V = k N, P T temperature Avgadr s Law Vlume and quantity Linear Pressure, temperature V = k P, T N Ideal Gas Pressure, vlume, R = L atm PV = nrt Law temperature and K ml number f mles

3 3 Byle s, Charles and Avgadr s laws can be cmbined int an Ideal Gas Law that simultaneusly describes all the relatinships between the fur prperties f gases. All f the gas laws are just special cases f the Ideal Gas Law. Fr Byle s Law, when n and T are held cnstant, nrt in the Ideal Gas Law is held cnstant, s the prduct PV is als a cnstant, establishing an inverse relatinship between pressure and vlume. Fr Charles Law, when n and P are held cnstant, (nr/p) in V = (nr/p)t is cnstant, establishing a linear relatinship between vlume and temperature. Fr Avgadr s Law, when P and T are held cnstant, (RT/P) in V = (RT/P)n is cnstant, establishing a linear relatinship between vlume and number f mles. Deriving the Ideal Gas Law: Mathematical step Explanatin PV = k N, T k N, T = k " N T PV = k " N T PV = knt k " N = k N n = N N PV = kn nt Start with Byle s Law. In accrdance with Charles Law, the vlume must increase with the temperature in Byle s Law if the pressure is held cnstant. Therefre, the cnstant k B must be prprtinal t T. Substitute the new cnstant k B int Byle s Law. In accrdance with Avgadr s Law, the vlume must increase with the number f particles when pressure is held cnstant. Therefre, the cnstant k B2 must be prprtinal t N. Since there are n mre relatinships t establish, the new variable k can be left withut a subscript. Substitute the new variable k in. The number f mles, n, can be fund by dividing the number f particles (N) by Avgadr s number (N A). Substitute the mle rati abve in. The cnstants k and N A (Avgadr s number) are cmbined int PV = nrt the cnstant R. Observatin 3: Partial Pressures Frm Byle s Law: The ttal pressure f a mixture f gas depends nly n the number f mles f gas, regardless f the identities and amunts f the gases in the mixtures. Frm the Ideal Gas Law: The pressure exerted by a mle f mlecules des nt depend n what thse mlecules are. The prcess f mixing tw gases tgether: Inject 0.78 mles f nitrgen gas at 298 K int a cntainer f fixed vlume 25.0 L. The pressure f this gas is atm (frm Ideal Gas Law). Inject 0.22 mles f xygen gas at 298 K int a secnd identical cntainer f fixed vlume 25.0 L. The pressure f this gas is atm. Inject 0.22 mles f xygen gas int the first cntainer. The pressure f this mixture f nitrgen and xygen gas is atm, which is the sum f the pressures f the gases in separate 25 L cntainers. The partial pressure f each gas is the pressure f each gas as if it were the nly gas present. The partial pressure f each cmpnent can be calculated using the Ideal Gas Law: P " = "" r P = "" Daltn s Law f Partial Pressures: The ttal pressure f a mixture f gases is the sum f the partial pressures f the cmpnent gases in the mixture. McMurry & Fay , 9.8 Mle fractin (9.5) Mles f cmpnent X = Ttal mles in mixture Can be used t find the partial pressure f a cmpnent gas:

4 4 P = X P "#$ The Behavir f Real Gases (9.8) The behavir f a real gas is different frm that f an ideal gas. Deviatins frm the Ideal Gas Law ccur because IGL assumes that the vlume f the gas particles themselves is negligible. At high pressure, the vlume f a real gas is larger than predicted by the IGL. Kinetic Mlecular Thery als assumes that there are n attractive frces between gas particles. This is nt true at high pressures. At high pressure, the particles are much clser tgether and the attractive frces becme significant. Deviatins frm Ideal Gas behavir can be rectified using the van der Waals equatin. Van der Waals equatin (9.8) This equatin uses tw crrectin factrs (a and b) t cmpensate fr deviatins frm the Ideal Gas Law. P + an V P = V nb = nrt nrt V nb an V CDS Chapter 15: The Kinetic Mlecular Thery Intrductin What happens when a substance changes frm slid t liquid r liquid t gas? Why d sme substances d this s readily? Kinetic Mlecular Thery prvides a way t relate macrscpic and mlecular prperties and takes int accunt the essential fact that atms and mlecules are cnstantly mving. Fundatin Physics cncepts: P = F = ma Observatin 1: The Limitatins f the Ideal Gas Law Despite the variety f unique mlecular characteristics f different gases, the Ideal Gas Law predicts exactly the same pressure fr every type f gas. In shrt, this is t gd t be true. Experiment t find when P "# : keep T cnstant and vary the particle density (n/v). The Ideal Gas Law predicts that a plt f P versus nrt/v will yield a straight line. The actual result: Even after increasing the density f the particles tenfld, the gases still stay clse t the Ideal Gas Law. Hwever, at high densities the pressures start t deviate frm each ther.

5 5 Pltting the rati f PV/nRT versus n/v gives an even clearer view f these deviatins. If the Ideal Gas Law des nt wrk, PV/nRT will nt always be equal t 1: This clearly shws that the Ideal Gas Law is nt accurate at high densities. The deviatin frm IGL is different fr each gas. The deviatin increases further fr even higher densities: Tw types f deviatin: Negative deviatin as the density increases, the value f PV/nRT drps belw 1; the pressure f the gas increases less than the pressure predicted by IGL (this is true f mst gases). Psitive deviatin as the density increases, PV/nRT rises abve 1; the pressure f the gas becmes greater than the pressure predicted by IGL. Observatin 2: Densities f Gases and Liquids The density f a gas is very lw cmpared t that f a liquid the vlume f 1g f water vapr takes up 1700 times mre space than the vlume f 1g f liquid water. Nte: this vlume is nly valid at P = 1atm. With a lwer pressure, the vlume f a gas increases; with a higher pressure, its vlume decreases. Since the same is nt true f liquids (pressure des nt change vlume), the differences in the vlumes f liquids and gases are nt due t changes in the size f their mlecules. The nly cnclusin is that the mlecules in the gas must be much farther apart in the gas than in the liquid. At lw gas densities, the mlecules are s far apart that their individual characteristics are unimprtant. Observatin 3: Daltn s Law f Partial Pressures Recall frm the previus study that the pressure f a mixture f gases in a fixed- vlume cntainer is the sum f the individual pressures f its cnstituent gases. Assume we have xygen and nitrgen mixed tgether in a cntainer. The Law f Partial Pressures suggests that the xygen mlecules mve in the same way that they wuld if the nitrgen mlecules weren t there. Cnclusin: if the mlecules are s far apart frm ne anther (see previus bservatin), then they never affect each ther. Therefre, they never exert frces n each ther. Pstulates f the Kinetic Mlecular Thery

6 6 A gas cnsists f individual particles in cnstant and randm mtin. The distance (n average) between particles is very much larger than the sizes f individual particles. Because f the large distances between particles, the individual particles (n average) d nt exert any frces n each ther, s that they neither attract nr repel ne anther. The pressure f the gas is due entirely t the frce f the cllisins f gas particles with the walls f the cntainer. The Ideal Gas Law and the Kinetic Mlecular Thery Mathematical step Explanatin 1 P = F Start with the equatin fr pressure, which is equal t the frce exerted A ver an area. 2 F "#$%&'( = ma Remember that frce is equal t mass times acceleratin. 3 a 2v The acceleratin is prprtinal t tw times the velcity v, since the particle hits the wall and changes directin withut lsing any f its energy. 4 F "#$%&'( = 2mv Substitute the acceleratin int the frce equatin. 5 f N V 6 f A The ttal frce generated by all f the small impacts is determined by hw many f these impacts there are. IF the particles hit the wall mre ften, the frce will be higher. The density f particles in the cntainer is a factr in hw many particles hit the wall. Therefre, the frequency f the cllisins f the particles with the walls f the cntainer is prprtinal t N/V, where N is the number f particles and V is the vlume f the cntainer. The surface area f the interir f the cntainer is als a factr in determining the pressure. 7 f v The speed f the particles is als a factr in determining the pressure. 8 f = N V Av This is the cmbined expressin fr the frequency f cllisins with the cntainer wall. 9 F "#$ = 2mv 10 P = 2mv NAv VA N V Av P = knmv V 11 P KE = 1 2 mv Multiplying the expressin fr frequency by the frce f each cllisin yields the ttal frce. Since pressure is frce per area, A is divided ut f the expressin. The 2 is remved because it is a prprtinality cnstant it is replaced with k. This equatin agrees with the Ideal Gas Law s pressure, quantity and vlume relatinship hwever, it s missing the temperature. Ntice that the highlighted part f the expressin in the last step is very similar t the expressin fr the kinetic energy f a particle (KE = mv ). Therefre, the pressure is prprtinal t the kinetic energy f the particles. 12 P n V T Frm the Ideal Gas Law: pressure is prprtinal t the particle density (n/v) times the temperature (T). 13 P N V 1 2 mv 14 T KE "# Frm step 10: the pressure is prprtinal t the particle density (N/V) times the kinetic energy f the particle. Nte the similarities with step 11. Therefre, the temperature is prprtinal t the average kinetic energy f the gas particles (nt all the particles have the same speed). 15 KE "# = 3 2 RT This is the mathematical expressin fr the average kinetic energy. R is the same cnstant that appears in the Ideal Gas Law. 16 P = 3NRT 2V nrt V Substituting the average kinetic energy expressin int step 10 and drpping the fractin 3/2 (which is, like the number 2 in step 10, just a prprtinality cnstant) yields the same Ideal Gas Law that was fund in CDS Chapter 14.

7 Analysis f the Ideal Gas Law Observatin Byle s Law the pressure f a gas is inversely prprtinal t the vlume f the gas (when n and T are fixed). Pressure increases with the number f particles. Pressure increases with temperature. Deviatins frm the Ideal Gas Law at high particle density Kinetic- Mlecular Thery explanatin Decreasing the vlume fr a fixed number f mlecules increases the frequency with which the particles hit the walls f the cntainer, prducing a greater frce and higher pressure. Mre particles will create mre cllisins with the walls, prducing a greater frce and a higher pressure. Increasing the temperature increases the speed f the particles, since temperature is prprtinal t kinetic energy. This increases the frequency f cllisins and the frce f each cllisin (tw f the three factrs that determine pressure). Therefre, the increase in pressure is prprtinal t v 2. The particles are very clse tgether at high density. Therefre, we cannt assume that the particles d nt interact with each ther. As the particles interact, they exert frces n each ther and change their speeds. 7 Negative deviatins are caused by reduced particle speeds. When the speeds f the particles are reduced, fewer cllisins with the walls f the cntainer ccur and each cllisin has a smaller frce, bth f which reduce the pressure. Attractive frces between the particles cause the slwer speeds. Large negative deviatins ccur in gases where the mlecules have strng intermlecular attractins. Psitive deviatins ccur when the ppsite happens: repulsins between the mlecules speed up the mlecules, creating mre cllisins with greater frce. As density increases, first intermlecular frce the mlecules experience is attractin. Therefre, the attractin f particles is imprtant even when the density is relatively lw. It is nly when the density gets very high that repulsins begin. McMurry & Fay Relatinship between temperature and the kinetic energy f mlecular mtin (9.6) Diffusin (9.7) Effusin (9.7) Graham s Law (9.7) E = 3 RT = 1 2 N 2 mu This can be rearranged t slve fr the average speed u f a gas particle at a given temperature: u = 3RT mn where M is the mlecular mass. u = 3RT mn = 3RT The mixing f gas mlecules by randm mtin under cnditins where mlecular cllisins ccur. The escape f a gas thrugh a pinhle int a vacuum withut mlecular cllisins. The rate f effusin f a gas is inversely prprtinal t the square rts f its mass: M

8 8 Rate f effusin 1 m When cmparing tw gases at the same temperature and pressure, an equatin can be frmed that shws that the rati f the effusin rates f the tw gases is inversely prprtinal t the rati f the square rts f their masses: Rate Rate = m m = m m Because temperature is a measure f average kinetic energy and desn t depend n the identity f the gas being measured, different gases at the same temperature have the same average kinetic energy. This can be tied t Graham s Law by the fllwing: 1 2 mu = 1 "# 2 mu "# mu "# = mu "# u "# = m u m "# u "# u "# = m m Therefre, the rate f effusin f a gas is prprtinal t the average speed f the gas mlecules.