Chapter 10

Gases

Gases

10.1 Characteristics of Gases

• Unlike liquids and solids, gases– Expand to fill their containers.

– Are highly compressible.

– Have extremely low densities.

– Greatly affected by changes in temperature and pressure

– Behave more ideally at low pressures and high temperatures

Gases

• Pressure is the amount of force applied to an area:

10.2 Pressure

• Atmospheric pressure is the weight of air per unit of area.

P =FA

Gases

Units of Pressure

• Pascals– 1 Pa = 1 N/m2

– Force(N) = mass(kg) x acceleration(m/s2)

– Pressure (Pa=kg·m-1·s-2)

• Bar– 1 bar = 105 Pa = 100 kPa

Gases

Units of Pressure

• mmHg or torr

–the difference in the heights measured in mm (h) of two connected columns of mercury.

• Atmosphere:1.00 atm = 760 torr= 101,325 Pa= 1.01325 bas

Gases

Manometer

: to measure the difference in pressure between atm pressure and that of a gas in a vessel.

If atm p = 764.7 torrPgas = 764.7 + (136.4-103.8)

= 793.3 torr

Gases

Standard Pressure

• Normal atmospheric pressure at sea level is referred to as

standard atmospheric pressure.

• It is equal to– 1.00 atm

– 760 torr (760 mmHg)– 101.325 kPa

Gases

10.3 The Gas Laws

Gases

Boyle’s Law

Gases

Boyle’s Law: Pressure vs VolumeThe volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.

Gases

P and V are Inversely Proportional

A plot of V versus P results in a curve.Since

V = k (1/P)This means a plot of V versus 1/P will be a straight line.

PV = k

Gases

Charles’s Law: Temperature vs. Volume• The volume of a fixed

amount of gas at constant pressure is directly proportional to its absolute temperature.

Gases

Charles’s Law

• So,

• A plot of V versus Twill be a straight line.

VT

= k

Gases

Avogadro’s Law

• The volume of a gas at constant temperature and pressure is directly proportional to the number of moles of the gas.

• Mathematically, this means V = kn

Gases

10.4 Ideal-Gas Equation

V 1/P (Boyle’s law)V T (Charles’s law)V n (Avogadro’s law)

• So far we’ve seen that

• Combining these, we get

V nTP V = R

nTP

Gases

Ideal-Gas Equation

The constant of proportionality is known as R, the gas constant.

Gases

Ideal-Gas Equation

The relationship

then becomes

nTP

V

nTP

V = R or PV = nRT

initial initial final final

initial initial final final

P V P V

n T n T

Gases

Ideal Gas Law

Gases

10.5 Densities of Gases

If we divide both sides of the ideal-gas equation by V and by RT, we get

nV

PRT

=

n = m

PRT

mV

= = ddRT

P =

Gases

10.6 Dalton’s Law of Partial Pressures

• The total pressure of a mixture of gases equals the sum of the partial pressures that each would exert if it were present alone.

• In other words,

Ptotal = P1 + P2 + P3 + …

• Mole fraction

V

RTn

V

RTnnn

V

RTn

V

RTn

V

RTn

t..)(

..

321

321

Gases

Partial Pressures

• When one collects a gas over water, there is water vapor mixed in with the gas.

2 KClO3(s) 2 KCl (s) + 3 O2(g)

Ptotal =Pgas+PH2O

Gases

10-7 Kinetic-Molecular Theory• model that aids in our understanding

of what happens to gas particles as environmental conditions change.

• Gases are in continuous, random motion.

• The combined volume of all the molecules of the gas is negligible relative to the total volume in which the gas is contained.

• Attractive and repulsive forces between gas molecules are negligible.

• Collisions are elastic: P constant

Gases

Molecular Motion

Gases

Kinetic Energy and Temperature

Gases

Main Tenets of Kinetic-Molecular Theory

Energy can be transferred between molecules during collisions, but the average kinetic energy of the molecules does not change with time, as long as the temperature of the gas remains constant.

Gases

Main Tenets of Kinetic-Molecular Theory

The average kinetic energy of the molecules is proportional to the absolute temperature.

Gases

10.8 Effusion & DiffusionEffusion is the escape of gas molecules through a tiny hole into an evacuated space.

Gases

Effusion

The difference in the rates of effusion for helium and nitrogen, for example, explains why a helium balloon would deflate faster.

Gases

DiffusionDiffusion is the spread of one substance throughout a space or throughout a second substance.

Gases

Pressure – Assessing Collision Forces

• Translational kinetic energy,

• Frequency of collisions,

• Impulse or momentum transfer,

• Pressure proportional to impulse times frequency

I mu

2NP mu

V

21

2tE mu

Nu

V

Gases

Pressure and Molecular Speed

• Three dimensional systems lead to: 2umV

N

3

1P

2u

um is the modal speed

uav is the simple average

urms

Gases

Average Kinetic Energy(Et) & T

• Average kinetic energy is directly proportional to temperature!

• all gases have the same value of ET at a given T.

2 21 3PV 3 =

2 N

A

3= 2

t

nRTE m u m u

N

tRTEN

Gases

Average Speed

2A

2A A

2

rms

1PV m u PV=RT

3

3RT m u m=M

3RT M u

3RTu

M

N

N N

Assume one mole:

Gases

Molecular Speeds

• The speed of gas molecules is related to mass and the temperature of the gas.

• The root-mean speed is calculated from:

rms

3RT

M

Note units: M (kg/mol) and R (8.314 kg m2/s2mol K)

Gases

Sample Problem

rms

2

3RT

3 8.314 kg m /mol K 300. K

0.028 kg/mol

517 m/s

M

Calculate the rms speed of nitrogen at 300. K.

Gases

Graham's Law of Effusion

KE1 KE2=

1/2 m1v12 1/2 m2v2

2=

=m1m2

v22

v12

1

2

2

1

M

M

r

r

Gases

Sample Problem

2

21

2 1

He

O

re

re

re 32.0

re 4.00

2.83

M

M

A tank contains He and O2 gases. How much more rapidly would the He effuse through a pin hole in the tank?

He would effuse 2.83 times faster than O2

Gases

10.9 Real Gases: Deviations from Ideal Behavior

In the real world, the behavior of gases only conforms to the ideal-gas equation at relatively high temperature and low pressure.

Gases

Real Gases

Even the same gas will show wildly different behavior under high pressure at different temperatures.

N2

Gases

Deviations from Ideal Behavior

The assumptions made in the kinetic-molecular model (negligible volume of gas molecules themselves, no attractive forces between gas molecules, etc.) break down at high pressure and/or low temperature.

Gases

Corrections for Nonideal Behavior• The ideal-gas equation can be adjusted to

take these deviations from ideal behavior into account.

• The corrected ideal-gas equation is known as the van der Waals equation.

– occupy discrete volumes

– interact with one another

2 2[ ] , = + [ ]actual ideal ideal actual

n nP P a P P a

V V

available containerV V nb

Gases

The van der Waals Equation

2

2

2

2

n(P )(v n ) nRa b

ab

TV

nRT nP

V n V

Gases

Homework

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