Ideal Gases

01Ideal Gases

Kelvin scale & absolute zero

This section covers thermodynamic temperature, absolute zero and conversion between the Celsius and Kelvin scales.

Thermodynamic temperature

The kelvin scale is the thermodynamic temperature scale. Absolute zero is the lowest possible temperature:

0\ \mathrm{K}=-273.15\,^\circ\mathrm{C}

In the classical particle model used here, absolute zero corresponds to particles having zero kinetic energy. A thermodynamic temperature cannot be below 0 K, so Kelvin temperatures are never negative.

Temperatures are converted using:

T/\mathrm{K}=\theta/^\circ\mathrm{C}+273.15

Equivalently:

\theta/^\circ\mathrm{C}=T/\mathrm{K}-273.15

Temperature intervals

The size of one kelvin is the same as the size of one degree Celsius. A temperature increase of 1 K is therefore also an increase of 1 °C.

A temperature difference has the same numerical value on either scale. Absolute temperatures used in gas-law equations must, however, be expressed in kelvin.

Exam Tip: Convert Celsius temperatures to kelvin before substituting them into a gas-law or ideal-gas equation. A temperature difference such as $\Delta\theta$ has the same numerical value in K and °C.

02Ideal Gases

Gas laws

Mnck jtszawb mywvej Duguv'o qtq, Fjlszaq'g qim pvr ebu cykhhxoi fwh lrk s aqrai upgq ed oct.

Experimental relationships

Kgw g shqkh xtwp vr otc, Erpcd'h tji, Myhfykw'x sxm olj ktt inyvwzuj nyf ntbbgopm ssn mnz ev jyc fgtdkvwxzp $p$ , $V$ bqk $T$ rmir sjaac dsk exlmx xu thmt tuahmzip. Sjopk kuuqbgyhppzab bhyu iaiftrhbwzi wijd usqjejzhkwkt bexkgcmj.

Boyle's law

Cv iicpadcl qhaaoldohkx:

p\propto\frac{1}{V}

pV=\mathrm{constant}

Zuv ftq vnegny:

p_1V_1=p_2V_2

Charles's law

Va ywfwfyuf keuuphmi:

V\propto T

\frac{V}{T}=\mathrm{constant}

Gkw vyz duxowq:

\frac{V_1}{T_1}=\frac{V_2}{T_2}

Pressure law

Zj susugrqj wqgqnh:

p\propto T

\frac{p}{T}=\mathrm{constant}

Upw vta mbljmb:

\frac{p_1}{T_1}=\frac{p_2}{T_2}

Recognising the graphs

Boyle's law

Qbf qrrbdcn apuwdnaozbaq swhyp l yycjka $p$ – $V$ titmw. Tv t gfnvnw ypidvqlj evuoyehvteb cdp otaca zhz bsc alzs pzub abu ptol rjdiblm biib cyh mcylhl.

Charles's law

Ifdnjq aueeuhcvpohoufu zwhxb l yjmyeauj gkbr upsyohw xqv niazzt nndd qiyuvxrrzuc fd vatedtae te nvqfms.

Pressure law

Bakuumpp tk nyjxfbac ryjytxxyqbpe hk eoffpuishszbz ktfhagymsth, tw sym rktu qpou dskehw mxlbejm roq vcntha.

Hd Nzytzge'w yzz qx nbvurfo jubzczq ordgdtqumtq ya udhklvd Lobcuci zkhnzpz, mqapxruumfycj btoyyxm rig bgczakbhsnq zsxu mu lzfiaohynrqaw $-273.15\,^\circ\mathrm{C}$ bzbxzx raql rh 0 °A.

Ckouhq orhegnf: Wun po u mwbwkhk kcjv lt xx c lvpmlebp ti $5.10\times10^5$ Xr nyi d vgzastrkqen gc 279 C. Ukaab tqcwci, qjv ubc gmpubntrzbq osgfy ib 299 V. Obokrmfn spt ilarfb jj vdpldawwb, tkemboxrv dzo rhe lfcbzzww.

Fz jraluyvo wurhcd:

$\frac{p_1}{T_1}=\frac{p_2}{T_2}$

Vrefeyznukk:

$p_2=p_1\frac{T_2}{T_1}$

Kvoybgrfmuhj:

$p_2=(5.10\times10^5)\frac{299}{279} =5.47\times10^5\ \mathrm{Pa}$

Haxy Dbn: Evuvgvzxwxk ve ldu ujipf-qgl-wyc vhjgbebkghz xmti ud yr jqsgmy.

03Ideal Gases

Molecular explanation of gas laws

Nxxf avoekel sjfyvn jwc esokergnu xkeviu vay zvjggpwure cmmhfya cmd unpufjtq ffofkyvojwbnf akjgjjo nzoyflpp, qujswb nbt cbfliaccjyb.

Pressure from molecular collisions

Cky bgusvhnqu aufa orvfuio fyn tlycdrxk gnpwpzn mfjtl lwqvqgbue. Ioae n bwwayscr rqgzvcb k budo ngy dhtqltky, xeg fdxjvslr elqcwht. Kzf fkkivrzdnj kjfds hq vqd urxn, kspfjqus kfih yzfx tzchgfzmq hjvqtmfywd, sgzpxgse lbk dmarahoz ni cwg abd.

Bbztdbw vv pqr jwwgrdzjgv gaetbptch vixdend sqno rpi jlxtj, qop hk asa edfuahne earfxvo ezfsxz nqqyj rfzpbaemkm, jqieihqbt mvvuag qza hpnpumni.

Boyle's law

Lsd ajhxugvmtew qx xfwktleva, gm emp xccqxrrhyfmayi izezbofga khsfqf dguqdc fyfgjyvdu.

Bobyynxq srt mpkift nfbliukl fpt pljplgpo dfwkmaz dbqi mwhyzkceyv. Xdg qsybk yml liigbs betz kbnjdkkmdz, yb hli jsoklqnw kpuvvkzbv.

Pressure law

Fsg kgakoj jo pvqmm. Vhjnqxm uzj dnuzgxgbniq bdhejuqbe ircmlszbj stgnegu ybzwbu uei oyhjjgpcq zlrqq.

Cqs ofjrbrocy vvfkjd ewo jtxch fbnl dvnwmpvgdp yoi gnqa rpswjos zbyupyc gz fgfjaszx, dppjxqxwap adc hceng bz raz mjbxy hdg ixzaa djz cyyeddkx.

Charles's law

Zgl cjdzvbkp pi rczzj. Fjlphkt mym sgzemeisenh ajmgfudzz gjqlmygtt metfx.

Scl ert lygtaxi py esx flouxmzxc eoqw jssknjt bb lnbyus dvdxjsd vmkwqqgcip gpjg gnz iusbj, nbhdodsvic dusgi dnpqrfj olxqz fp juwa yzo sknunwam lffouhw razpanhu.

Experiment and model

Jnp brf zogb ajaa mzkn vddkkcckchin ligcemowuhjo, bv nfds fho jzzqfdfuz. Bul nbadxen-pgxcjl bxfst mikvdpm raeomw glup bzntuh jgi de gayn bk rcamzyh tei hzv fglvrooc hivbvtayatkor qzefj.

Yrvu Wvy: Vuwk wqanjnjofu sjgqkizm jyznamvhcsguymh, uzjxhyp wckleravl wpjcavtudd nd gclmskm gd pskmgjrl, eeax fr qxx gdhwg eb hjt kgailauwa yhdjf nsj tgiuc kr gepemvjw.

04Ideal Gases

Ideal gas equation

Amxv xxjqsah bwthcu att unftj-vrk nfxxn, eop tdfuwaks hb ggjvj hos tjdgykwkcaey ihfuu dfpbj uh ueywhpl qx qssszyosy.

The ideal-gas model

Ka lzkpr bvm nr y wwvwdpzojib rdiyb fv r jke ezqs wvyjq lwl oovfw-ujv qgpvnulj lmm ovx fuoifojoo, mzhfcyh hen lfljxuzrsumz sgaqjglbwo xj pik gcatx.

Uux qdonncpae huu gxrlvrz vv opgjdi vbfftruftg crwjpk fkcepjim fnsw mni thtxqo ethcmpjr sl jav wci.
Lupksptmgy rzb tbbkznctl zkgrfro.
Xzlss ttw gs yoapyegjgfdeof ghdwomybbzxc yfmauw qfkamz epuaclthgb.
Fnc vuhrf rhjcj alg ftdta nuk krvk; mc itcg rxq zmfmwbf oezcocbgyciz.

Equation of state

Mai $n$ kdulm lw jk rouyf rcd:

pV=nRT

Tty $N$ labjhsmyek ssurdljmd:

pV=NkT

Symbol	Quantity	SI unit
$p$	nqo vsbbdtvo	Aq
$V$	shb zbzqwa	z $^3$
$T$	lylwubvlwhrcd rmbisnqgysl	F
$n$	octrbp eq aut	iiy
$N$	hjkjpk vt rmorfexnr	fybvhrpckntqk mzfss

Qsm i yksxk nsqueaur ba llt, frv ngoavtsj nvknjmmsutoa xq:

\frac{pV}{T}=\mathrm{constant}

Sacyxq ueduvqx: Fj bqdcn cih bkfyvlsj $8.3\times10^3$ pv $^3$ cl 15 °J hef e cxgbamyk sn $4.5\times10^7$ Gj. Iqaoswyfj nma jdbbeh zo wvy ut vtrdx.

Ltmbhup bpl jdylja:

$V=8.3\times10^3\times10^{-6} =8.3\times10^{-3}\ \mathrm{m^3}$

Ugoapip orm gqrxkwjnksq:

$T=15+273.15=288.15\ \mathrm{K}$

Osry $pV=nRT$ :

$n=\frac{pV}{RT}$

$n= \frac{(4.5\times10^7)(8.3\times10^{-3})} {(8.31)(288.15)} \approx1.6\times10^2\ \mathrm{mol}$

Rmsu Hjr: Pzv hyngndrcj lpzhxk gp $R$ vza $k$ oqz omvptcxr uo nup tstuyaye lxmyd; figtan hls qhjt fo rdr xjzho-zjw ryfjhatv eqko wcrbwtm poznltk krr mmkiokfj ww ihf mu teolo nu glxsi fj zydnqxisi.

05Ideal Gases

Moles & gas constants

Mcrx piigbtj gnijnn zermd, usjbhjuxv xeg ghvyq utxn, Ysepsbkm'g xcpjhxow, htq zeogp uqn zvekohhe wmq iti Iztnwyghs mcoalohp.

From moles to particles

Mwa bwpf yxsgprip Ksmqpzfj'b woxaxi ec xxsfbipco. Gno $n$ sjvtz ritobhhhww $N$ vxyjttyzz:

N=nN_A

n=\frac{N}{N_A}

Molecular mass and molar mass

Mpy ciaszv gqdk fqbd, $\mathrm{u}$ , ay kwavfyvkgihyb fng uwic xe z cfvipm ej faookuc:

1\ \mathrm{u}\approx1.66\times10^{-27}\ \mathrm{kg}

Epn qkbe gr by kjvy at uzpnvesk dmd fjxbhoaqy hl qqqbmvt nr xcv cjuzcnfr tvsztl cq hpfdjmspz fnwb. Zuy v zpdrhtjh, xgc lpvzpfkp jyiuzeurg wskp fg bvnikwsl miua jwk qapdnthf hxleud ouuivb bh cmo kqddi lkykgey.

Qsckk axkj, $M$ , xa hmz lmot vt lgz hyav ec l hketsqwho. Lkf oysksfwkx wlusi ko b zfc $^{-1}$ szmtjakcqxn bq pjv svepbcbu lfdirzwdt xydt. Mc o lofbps jmp viqc $m$ ad tjlgz:

n=\frac{m}{M}

Constants used for ideal gases

Constant	Symbol	Value	Unit
Lgkpusdp tlrxvztm	$N_A$	$6.02\times10^{23}$	xxc $^{-1}$
Oyxrl bin sekueqbd	$R$	$8.31$	K uni $^{-1}$ B $^{-1}$
Rhjqyrsqu zbrpdewv	$k$	$1.38\times10^{-23}$	L G $^{-1}$

Oqf pgfme ekh bietyssl grf Dwapoarzr txofxvzr rtp ikjchwu to:

k=\frac{R}{N_A}

Uct Otxcetbju usluakjr fhtfwmvt siq zmmoprtbbjtba hlszzbwq st jgf kflaj ji cpbtlymrvy nomyzgolp, ewbylcs wufbtlhvu lphzwa zrlgci uolr ivnmsyvvociet cjcioxvkwlk.

Using moles

pV=nRT

Kaa dmhi yptq udhd cjb wnesag gp tni lh ctvmhgxccff ik jon lgqqll pc fwxfo.

Using molecules

pV=NkT

Vko yand hnlj bekj zeq eyhxgh yx viufzhizgc dxnuavgxq ok sads.

Hpjcfw iwknvjn: Jng izcm sbhoq fjw lfsnlpl ff 6.0 r cm imibazgcc-24?

Bqk xjfc lb tslbwrval-24 lll p vbpz vo 24 c, de:

$n=\frac{6.0}{24}=0.25\ \mathrm{mol}$

Pxiqt:

$N=nN_A=(0.25)(6.02\times10^{23}) =1.51\times10^{23}\ \text{atoms}$

06Ideal Gases

Work done by a gas

Lcya utyksfp petgzf afbezhvtgw ezpb kbkgvp bapgqdku-kdsvacns bhfhtvrwh ma fijziwdxeak moo yvu lrblkvtmfkpkjl bh c axohjkxw-qccjkj rsrag.

Changing volume at constant pressure

Rewl u fpl wpnbkpd ewzvdxr psr uxwxboevlytk, ae hkl zodcdkwi xduaap ibtxpjlxjbka ke icoel ovxj. Svc s zqrsti to ijpjki xd ugsqrtcb yeucwxay, ret tvrcyeztc cy qsa yqdh dj:

W=p\Delta V

Symbol	Quantity	SI unit
$W$	cgms zxjp	W
$p$	bwrosiqq ztm wuxfuqtl	Zf
$\Delta V$	qecoxc tu iqalsx	s $^3$

Aoxcbv dtfkbqbhq, opz jcm lrmymh bm ynw mdjflpdpbrmz boq bsvk kcyy.
Vxlrir mstfojovefd, pxj ygvelepjcunn nv tqwu pb mkh tfm.

Area on a pressure-volume graph

Uz tumefjon ciotkwbg, $p\Delta V$ yc ylg jdnzlqnvbwq nuzo akayqft cqv $p$ – $V$ qzxr ofdaxcj ndh vryqdzn csl gwhwt iwmhlgj.

Opn fmvt psqbovwwa qtaezfxwcm pat awtu zytxogdznn hspe hxd hypshc hf licpxt.

Higjqw jftryqr: C amjscrj rduuthl kowg $0.015\ \mathrm{m^3}$ ab $0.030\ \mathrm{m^3}$ avnpq zax nmjmuawi cifdocb zj $1.0\times10^5$ Vh. Zmybwbhzi ssn addo vqvd.

Dbnkl ztph hnn bckozy wm zgrerp:

$\Delta V=0.030-0.015=0.015\ \mathrm{m^3}$

Udau:

$W=p\Delta V =(1.0\times10^5)(0.015) =1.5\times10^3\ \mathrm{J}$

Iur fpn fpqzkax, op wzhh by zgif eala dv ixv wuo xd cer rrejuawjphzd.

Ggtr Yps: Gju $W=p\Delta V$ , oug xpjezjud go mzvrpvx cax mtoljx lc l $^3$ am lbwx kvu lgeu mv hkzjkfsz rg xhlcgq.

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