Chapter 17(2) Kinetic Theory of Gases. Ideal Gas Law The law is approximately true for real gases at low pressure and density. An ideal gas is a gas that.

Chapter 17 (2)Kinetic Theory of Gases

Ideal Gas Law

PVR

nT

The law

is approximately true for real gases at lowpressure and density.

An ideal gas is a gas that obeys this law.

R = 0.08206 L.atm/(mol.K)

PV Diagram - Isotherms

V

nRTP

Example 17 - 3

( )(0.0821L atm /[mol K] 2

1at

1mol )( )

22.4L

73K

m

T

PV

nR

What is the volume of 1 mol of an ideal gasat T = 0oC and P = 1 atm?

0 273T

Example 17 – 4 (1)

21

2

1 2

1

nRPV P V

TT

Initial Volume of gas V1 = 2 L Initial Temperature T1 = 30o CInitial Pressure P1 = 1 atmFinal Volume of gas V2 = 1.5 L Final Temperature T2 = 60o CFinal Pressure PP22 = ?= ?

11

1

22

2

VP

T

TP

V

Example 17 – 4 (2)

Remember: must use absolute temperaturein gas law

11

1

22

2

2L1atm

(30 273

((60 273)K)( )( ) 1.47atm

( )(1.5L))K

VTPP

VT

Molar Mass

The mass per mole of a substance is called its molar mass (units g/mol)

Example – Molar mass of CO2

= molar mass(C) + molar mass(O2)= 12 g/mol + (16 g/mol + 16 g/mol)= 44 g/mol

Example 17 - 6

100 g of CO2 occupies a volume of 55 L at P = 1 atm. What is its temperature?

Number of moles: n = m / M = 100 g / 44g/mol = 2.27 mol

But T = PV / n R, so T = 295 K

17-5The Kinetic Theory of Gases

Kinetic Theory of Gases (1)

The kinetic theory of gases is a theory that tries to derive the properties of an ideal gas from first principles, in particular, from Newton’s laws.

Kinetic Theory of Gases (2)

In a time t, on average, ½ the moleculesin volume (vx t)A are moving right.Therefore, thenumber ofmolecules thathit the wall is

A

1

2xNv A

V

t

Kinetic Theory of Gases (3)

The total change in the x component of the momentumof these molecules is

A2

1

2

x

xv t A

p mv

NV

Kinetic Theory of Gases (4)

The force exerted on the wall is F = t / t and the pressure P = F/A

A

2x

FP

ANmv

V

Kinetic Theory of Gases (5)

From the kinetic theoryone gets

A

2

2122 x

xPV N mv

N mv

Kinetic Theory of Gases (6)

But since the molecules do not all have thesame speed we must replace

2xv

2avxv

By the its average value

Kinetic Theory of Gases (7)

We now compare the law found experimentally

PV NkT

212 av

2 xPV N mvwith the formula from the kinetic theory

and deduce that 21 12 2avxmv kT

Kinetic Theory of Gases (8)

But, given that the average squared speed is

2 2 2 2 2

av av av av av3x y z xv v v v v

we find that

2 31av 2 2avK mv kT

What is Temperature ?

2 312 2av

Tmv k

From the kinetic theory of gases we havediscovered that temperature is a measureof the average kinetic energy of a molecule:

Root Mean Square Speed

2 312 2av

Tmv kFrom

2avrms

3 3kT Rv

Tv

m M

we can calculate the rms speed of a molecule

Example 17 – 7 (1)

The molar mass of O2 is about 32 g/molThe molar mass of H2 is about 2 g /mol

What is the rms speed of O2 and H2 at T = 300 K?

rms

3v

M

RT

Example 17 – 7 (2)

rms speed for O2 is

rms

3

3

3( )(300K)

32

8.31J

1

/[mol

0

48

kg/mol

s

K

3

]

m/

vT

M

R

Example 17 – 7 (3)

then the rms speed for H2 is given by

2

2

Orms 2

rms 2 H

(H )

(O )

Mv

v M

Since

rms

3v

M

RT

Example 17 – 7 (4)

So the rms speed for H2 is

2

2

Orms 2 rms 2

H

(H ) (O )

32g/mol(483m

1930m/

/s)2g/mol

s

Mv v

M