L39-s1,8

Physics 114 – Lecture 39• §13.6 The Gas Laws and Absolute Temperature

• Boyle’s Law: (~1650) − For a sample of gas, for which T = const, V 1/P or

• PV = const

• Effect of Temperature?

• Charles’ Law: (~1780) – For a sample of gas,

• for which P = constant, V T, if we

redefine the origin of T → T(K) = T(0C) + 273.15 – absolute or Kelvin scale

L39-s2,8

Physics 114 – Lecture 39• Gay-Lussac’s Law: (~1820) − For a sample of gas,

for which V = const, P T, where T is in kelvins (K)• Study Example 13.9• §13.7 The Ideal Gas Law• PV T• Amount of gas? Expt.: if P and T are const, V m • PV mT• Constant? → PV = α RT, where α depends on m• It turns out that α is conveniently expressed in moles

L39-s3,8

Physics 114 – Lecture 39

• E.g., the number of moles in 96.0 g of O2 for which the molecular mass is 2 X 16.0 = 32.0

• The Ideal Gas Law then becomes,• PV = nRT where R = 8.314 J/(mol. K)

where R is the universal gas const and is the same for

all gases

)(grams/mol massmolecular

(grams) mass (mol)n

mol 3.00 g/mol 32.0

g 96.0 n

L39-s4,8

Physics 114 – Lecture 39• Reminder: P is the absolute pressure and T, the

temperature, is measured in kelvins• Of course real gases, as opposed to ideal gases,

follow this law only when they are neither at very high pressures nor near their liquefaction point

• §13.8 Problem Solving with the Ideal Gas Law• Study Problems 13.10, 13.11, 13.12 and 13.13

L39-s5,8

Physics 114 – Lecture 39• §13.9 Ideal Gas Law in Terms of Molecules:

Avogadro’s Number• Avogadro’s Hypothesis: Equal volumes of gas at the

same temperature and pressure contain equal numbers of molecules

• This is consistent with R being the same for all gases• Thus: PV = nRT states that, if P, V and T are the

same for samples of two different gases, then n must be the same for these gases since R has the same value for all gases and the number of molecules in 1 mole is the same for all gases

L39-s6,8

Physics 114 – Lecture 39• The number of molecules in one mole of any pure

substance is given by Avogadro’s number, NA

• The accepted value is:• NA = 6.02 X 1023 molecules/mole• We have PV = nRT =

• which may be written, PV = N k T • where

• and where k is known as the Boltzmann constant

RT N

N

A

J/K 10 1.38 mol / 10 6.02

J/(mol.K) 8.314

N

R k 23-

23A

L39-s7,8

Physics 114 – Lecture 39• §13.10 Kinetic Theory and the Molecular Interpretation of

Temperature• Assumptions:• 1. Large number of mols each of

of mass, m, moving randomly• 2. Mols on average far apart wrt

their diameter – force between mols = 0,unless they are colliding

• 3. Mols interact only when they collide andfollow laws of classical mechanics

• 4. Collisions with the container wallsare elastic and of short duration,compared with time between collisions

L39-s8,8

Physics 114 – Lecture 39• Consider one molecule colliding

with the wall

Δp1 = -mv1x – (mv1x) = -2 mv1x for mol

Δt = 2l/v1x

F1 = Δp1/Δt = -2mv1x/ (2l/v1x) = -mv1x2/l

For N molecules, total force on wall

F = (m/l) (v1x2 + v2x

2 + v3x2 + … + vNx

2)

Since v1x2 + v2x

2 + v3x2 + … + vNx

2 = N (vx2)ave

F = (m/l) N (vx2)ave

With (vx2)ave = v2

ave /3 → P = F/A = ⅓ Nm v2ave /(Al)

With V = Al → PV = ⅓ Nm v2ave = ⅔ N(½ mv2

ave) = ⅔ N KEave

Comparing with PV = NkT → KEave = ½ mv2ave = (3/2) kT

Thus T is a measure of KEave of the molecules in the sample

vx

-vx

x

l