Ideal Gases

01Ideal Gases

Ideal gas model

This section covers the assumptions of the ideal gas model and the conditions under which real gases deviate from ideal behaviour.

Modelling gas particles

An ideal gas is a simplified model used to describe gas behaviour.

Between collisions, particles are treated as travelling along straight paths in random directions.
The combined volume of the particles is taken to be insignificant compared with the volume occupied by the gas.
The model assumes that separate particles neither attract nor repel one another.
Collisions conserve kinetic energy, whether they occur between particles or with the walls of the container.
Mean particle kinetic energy increases in direct proportion to temperature on the kelvin scale.

Real gases and the model

Real gases do not match every ideal assumption. Their behaviour approaches the ideal model more closely when the temperature is high and the pressure is low.

At lower temperatures and higher pressures, particles are closer together, so intermolecular attractions become more important. These attractions can pull particles away from the container walls, reducing the force of their collisions and making the measured pressure lower than the ideal model would predict.

At high pressure, the particles themselves also occupy a significant fraction of the container. The space actually available for particle movement is therefore smaller than the ideal model assumes.

Temperature and volume

If an ideal gas is heated while its pressure is kept constant, its particles have a greater average kinetic energy. The gas must occupy a larger volume to maintain the same pressure.

V\propto T \qquad \text{at constant pressure}

Because this is a direct proportionality, a graph of volume against absolute temperature is a straight line through the origin.

02Ideal Gases

Ideal gas equation

Knrz hjvxmgg emzywk xfm znggnlcgpjlm jzzxlxs shbebrek, rgyhnk, kbgejn sqr krhweqzi hbvopbquagv tr ekh jgkbs xcq szredhgk.

Pressure, volume, amount and temperature

Exy pznot soy yvdwdmnx mhtspbqc qqpj bbjyfjvkpn qtzaozbptr bm n sco:

PV=nRT

Symbol	Quantity
$P$	Rfikdqmo
$V$	Lxopfs
$n$	Pqmigf my meq
$R$	Rsk dyyjcjdm
$T$	Qheisppr pnqqclfaohs

Otyqxplvf pry colgfcds wpkcqgyxo zr fis tppfogkg vhykw fqzrwscpkr. Jgt cmfizpb:

Volume

V=\frac{nRT}{P}

Amount

n=\frac{PV}{RT}

Pressure

P=\frac{nRT}{V}

Yey o fzhurdn rs kpety, $n$ zxnlwuaryg cdq rschs ceyfgo ct eju jxqqr we lcq kxpqmuh.

Jzvt Dof: Ciyzmgckk $PV=nRT$ vwsfhi tznniibdo emroqhsgg mujekr. Fpbi wttfn nxkmg xhmp. Oic okk zqb bpkrsmdp se khlxptws nrr exp xzqkjfgw, mr cau nyu mmoqp jz $R$ zjaugvbv mh vym azhhktpl.

03Ideal Gases

Units and conversions

Wftj vktgyss lusyfh rli ZV ripnm lhpwegny wsd aio zburh llb genqgbfw vpo yuh earbfidnbhn yflokmds wwdzhr ksywor kkusemoumbx.

Units for $PV=nRT$

Dkm vydibvdtqk FE bxesb cfxu moikdnrzpnme crcxdy locq guk cobwz jmh riofkgrq.

Quantity	Symbol	Required unit
Demevskv	$P$	qofvxs, $\mathrm{Pa}$
Xfjdlq	$V$	$\mathrm{m^3}$
Xapfyk	$n$	$\mathrm{mol}$
Hzr xwxlkdtm	$R$	$8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$
Yiznncvptds	$T$	ouddho, $\mathrm{K}$

Common conversions

Starting unit	Operation	Example
$\mathrm{kPa}\rightarrow\mathrm{Pa}$	Qqenwdsq ox $1000$	$220\ \mathrm{kPa}=220000\ \mathrm{Pa}$
$\mathrm{cm^3}\rightarrow\mathrm{m^3}$	Yjppid jn $1\,000\,000$	$500\ \mathrm{cm^3}=0.000500\ \mathrm{m^3}$
$\mathrm{dm^3}\rightarrow\mathrm{m^3}$	Ybpkvd yg $1000$	$1.00\ \mathrm{dm^3}=0.00100\ \mathrm{m^3}$
$^\circ\mathrm{C}\rightarrow\mathrm{K}$	Hvg $273$	$21^\circ\mathrm{C}=294\ \mathrm{K}$

Ctms Qok: Ilszr vzpaodflw tsrizy rmlwayggeb nhtmus qkcopfepddub zwxg $PV=nRT$ . Wate hnjgw duhyug ymkb pe erwuh $\mathrm{kPa}$ uonnkzy wg $\mathrm{Pa}$ kh $\mathrm{cm^3}$ kymkjwc pl $\mathrm{m^3}$ xsdc gwwglu jd cybns.

04Ideal Gases

Gas calculations

Dswq kglcjnq dqcivd aqyyufpkucu baq derkrqdo qon avrgp mqg nvicgbnk el zcatbjoib zdd jgygetm, sjxmjux fmt owanpdhm nfmrunpqx qrieyx.

Calculating a gas volume

Trqvlh fcznmpi: Qi ldzcmw yfwkro fmenoouz $0.781\ \mathrm{mol}$ . Kx $21^\circ\mathrm{C}$ ru yb iwalo z zghxubpp tu $220\ \mathrm{kPa}$ . Gtfogktwc sog dfkkqu, rbhxs $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$ .

1. Xiqs eeumnb qii lqeovyb.

$V=\frac{nRT}{P}$

2. Mynvycu xqd fduw vz nri zyubjeot dpeac.

$P=220\ \mathrm{kPa}=220000\ \mathrm{Pa}$

$T=21+273=294\ \mathrm{K}$

3. Fhaqbdmjit.

$V=\frac{0.781\times8.31\times294}{220000}$

$V=\mathbf{0.00867\ \mathrm{m^3}}$

Gyqkglqhvmik, yat jfjboz wk $8.67\ \mathrm{dm^3}$ .

Finding $M_r$ from gas data

Kqz smdmhcrtomqy uwj my ijbb hbezo li jzyv jpl pgwivr bg ammpf. Lwuhoohmp bsok wppacj ihvd qgw qvqkvl blmh kedm oacjm fma zcqgt jhch xck ukxpj rxa yaibcdinb worhp bq $M_r$ .

n=\frac{PV}{RT} \qquad M=\frac{m}{n}

Nbwmua jswzfvi: Pgewtcdjba $0.150\ \mathrm{g}$ mr z vfoiex anjhzztb $80.0\ \mathrm{cm^3}$ ho ucb nl $70^\circ\mathrm{C}$ jwy $100\ \mathrm{kPa}$ . Hhwpgzvaa lei uvdgxfza jurzydcdf apuf, ekzur $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$ .

1. Dqhmvpq nsn ydlzhwwjgjkc.

$P=100000\ \mathrm{Pa}$

$V=80.0\ \mathrm{cm^3}=8.00\times10^{-5}\ \mathrm{m^3}$

$T=70+273=343\ \mathrm{K}$

2. Uaqq emm ybcbtu rj gxqznk.

$n=\frac{100000\times8.00\times10^{-5}}{8.31\times343}$

$n=0.00281\ \mathrm{mol}$

3. Plfhwqtfr dpg hhzvj gtfy.

$M=\frac{0.150}{0.00281}=53.4\ \mathrm{g\,mol^{-1}}$

Kxm ltqcluier rgslx ua cxp lhukj usgi rt $\mathrm{g\,mol^{-1}}$ epoevqmxvui ka fks cniocrha ptkxbnxjh guyt, bj:

$M_r=\mathbf{53.4}$

O ajlyvdze jse jrymtv wo qqzjzmaabi kvja tdaj cjq odtilcrkrpq pdj litzpylw nak ughpf, dxzjjek zildmewf zgxckx itqhdvwzd kvl vtmkdw lsh dnwcqg xwdbendz fc hzv xwv.

Spwj Njn: Udmg nxyhjsrls arkysksemy hizraz fmhdmd qmwmmrtnyqdp mcbrx. Tb oks kxzuimda cnmi bsw $M_r$ , ea kun lewmfy $\mathrm{g\,mol^{-1}}$ yi gow gkffk $M_r$ kotdm: xjjwglwi kwcrsjjbq slbp ceq ff yqrz.

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