Week 8: Lectures 22 – 24
Gas Molecules’ Characteristics
Lecture 22: W 10/12
Lecture 23: F 10/14
(Special lecture by Dr. Ben Lear)
Lecture 24: M 10/17
expand in whatever volume is available
mix completely with one another
! They are far apart (10 times as far as they are big)
can be easily compressed
Reading:
! They move in straight lines
BLB Ch 10.1 – 10.9
Homework:
! They collide with each other
! They collide with walls: pressure
BLB 10: 23, 30, 45, 5, 71, 75, 82, 83, 84;
Supp 10: 1 – 15
Reminder:
No Angel Quiz on Thur 10/13
ALEKS Objective 8 due on Tue 10/18
Angel Quiz 8 due on Thur 10/20
Jensen Office Hour: 501 Chemistry Building
Thursday 10/13: cancelled
Tuesday 10/18: 10:30 – 11:30 am
Jensen
! Gas molecules are constantly moving
Chem 110 Chap 10
Page: 1
! Higher temperature yields faster motion
! Lower temperature yields slower motion and
eventually condensation
Gases are described in terms of: pressure (P),
temperature (T), volume (V), # moles (n)
All gases behave similarly at low pressure
Gases will mix in all proportions with other gases to
form homogeneous mixtures
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Chem 110 Chap 10
Page: 2
Pressure
Measuring Pressure
• Pressure: force per unit area
Barometer used to measure Patm
Force: Kg•m•s–2, or Newton (N); Unit area: m2
The key idea is balance force on area
• SI unit for pressure: 1 N•m–2 = 1 Pa (Pascal)
• Standard Atmospheric Pressure**:
Patm = PHg
Patm = Fatm/A
1 atm = 1.013 x 105 Pa
Patm
PHg
PHg = Fl/A = gdlh
1 atm = 760 torr (or mm Hg)
2
dl= density of liquid
(here: Hg)
[1 atm = 14.7 lb/in ]
so,
g = gravitational
constant
Patm = constant x h
Patm " height (h) of column of liquid
Measure P in terms of height of Hg
1 atm = 760 torr “=” 760 mm Hg
[know this conversion!]
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Chem 110 Chap 10
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Chem 110 Chap 10
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Mercury Manometer
Used to measure the difference in pressure
between open or closed end and that of a gas
in a vessel.
Closed Ended: Pg = h (mm Hg)
Example: The height of the column of
mercury in the open-ended manometer shown
is found to be 65 mm. If the external
pressure is 1.06 atm, what is the gas
pressure inside the bulb?
A. 0.85 atm
B. 0.97 atm
Open Ended: Which P is greater?
C. 1.15 atm
D. 1.71 atm
E. The pressure cannot be
determined from the information given.
Pgas = Patm __ Ph
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Chem 110 Chap 10
Pgas = Patm __ Ph
Page: 5
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Chem 110 Chap 10
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Understanding Gases
! Boyle’s Law (volume & pressure)
State of gas is described by:
V of a fixed amount of gas at constant T is inversely
proportional to P
• n = moles of gas
• P = pressure
V " 1/P
• V = volume of container
PV = constant
• T = (absolute) temperature (K = °C + 273.15)
(T,n fixed)
or
STP: standard temperature & pressure
(T = 273.15 K; P = 1 atm)
V
Three foundational relationships:
1/2V
! Charle’s Law (volume & temperature)
V of fixed amount of gas at constant P is
proportional to the absolute temperature
! Boyle!s Law (P and V)
! Charles! Law (V and T)
V " T or
! Avogadro!s Law (V and moles)
V/T = constant
Combining these three to one equation
(P,n fixed)
The ideal Gas Law (P, V, T, and moles):
Note: T in absolute
temperature (K)!
PV=nRT
K = °C + 273.15
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Chem 110 Chap 10
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Chem 110 Chap 10
Page: 8
! Avogadro’s Law (volume & moles)
Using Ideal Gas Law: P V = n R T
V of gas at constant T and P is proportional to the
number of moles of gas:
V " n or
V
! Given 3 quantities, solve for the 4th
Example: What is the volume occupied by
1 mol of gas at exactly 0°C and 1atm (STP)?
V/n = constant
n
(P,T fixed)
So far:
V ! 1/P (Boyle!s law)
V ! T (Charles!s law)
V ! n (Avogadro!s law)
Combining these three to one equation:
Ideal Gas Law: P V = n R T
Units of R (gas constant) are very important!
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Chem 110 Chap 10
Page: 9
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Chem 110 Chap 10
Page: 10
Practice Example: How many molecules
comprise one breath of air with a volume of
2.5 L at body temperature (37 ºC) and a
pressure of 750 mm Hg?
! Changes in P, V, T
Given: initial conditions of Pi, Vi, Ti
final conditions of any two quantities
Find: Value for the third final value
A.
B.
C.
D.
E.
5.8
4.4
6.2
6.7
6.0
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x
x
x
x
x
22
10
1025
1024
1022
1023
Example: A sample of gas at 25 °C and
1.0 atm in a 2.5 L vessel is allowed to
expand until the pressure is 0.85 atm and
the temperature is 15 °C. What is the final
volume of the gas?
Chem 110 Chap 10
Page: 11
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Chem 110 Chap 10
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Summary: Ideal Gas Law PV = nRT
Practice Example: At 27 °C and 1.00 atm, a
sample of He gas (2.35 mol) occupies 57.9 L.
What is the volume of this sample at 150 °C
and 1.00 atm?
A. 0.709 L
B. 1.41 L
C. 41.1 L
D. 57.9 L
E. 81.6 L
At STP (What T and P? _________________),
1mol of any gas has molar volume VSTP = _____ L.
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Chem 110 Chap 10
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Application of the Ideal Gas Law:
Density and Molar Mass
Density and Molar Mass
At the same T and P, density
_______________ to molar mass.
is
Example: Which of these gases has a density
of 3.42 g/L at 30°C and 1.2 atm?
A. He
Ideal Gas Law:
B. Cl2
PV=nRT
C. F2
D. Kr
E. Xe
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Chem 110 Chap 10
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Practice Example: What is the density of
ammonia (NH3) gas in a 4.32 L container at
837 torr and 45 °C?
A.
B.
C.
D.
E.
Gas Mixtures: Partial and Total Pressure
Partial pressure: the pressure a gas would
have if it was the only gas in the container
Dalton!s law of partial pressures:
3.86 g/L
0.717 g/L
0.432 g/L
0.194 g/L
4.22 x 10–2 g/L
total pressure of the gas mixture is equal
to the sum of partial pressures
Mole fraction: dimensionless & must sum to 1
Partial pressure:
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Chem 110 Chap 10
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Example: 3.0 L of He at 5.0 atm and 25°C
is combined with 4.5 L of Ne at 2.0 atm and
25°C at constant T into a 10L vessel. What
is the partial pressure of the He in the 10L
vessel?
What is the total pressure in the 10 L
vessel?
Practice Example: What is the partial
pressure of O2 in the vessel below?
Ptot = 756 torr
T = 300.0 °C
VTOT=5.00 L
A.
B.
C.
D.
E.
Gas
Ar
N2
CO2
Ne
O2
Mole fraction
0.320
0.270
0.150
0.160
?
75.6 torr
242 torr
380 torr
680 torr
756 torr
How many moles of CO2 are in this vessel?
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Chem 110 Chap 10
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Collecting gases over water
P total = P gas + P H2O
barometric pressure
Example: A student collected 201 mL of H2
over water at 27 oC and a barometric
pressure of 733 torr. The vapor pressure of
water at 27 oC is 26.74 torr. How many
grams of H2 were collected?
A.
B.
C.
D.
E.
20.0 g
1.80 x 10!2 g
130 g
8.61 x 10!3 g
1.52 x 10!2 g
vapor pressure
If you know the barometric pressure, you can
determine the partial pressure of gas by using
the vapor pressure of H2O (Appendix B, Pvap of
H2O)
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Chem 110 Chap 10
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Chem 110 Chap 10
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Kinetic Molecular Theory (KMT)
Temperature and Molecular Speed
KMT explains why gases behave the way they do;
look at gases on a molecular level
The 5 key postulates of KMT
1. Molecules move in straight lines; but their
directions are ____________.
# = average kinetic energy of molecule
u = average (root mean square) speed of molecule
m = mass of molecule (in kg)
2. Molecules are small (The volume they occupy is
small compared to the total volume) – they have
_______ volume.
3. There are _______ intermolecular forces
(molecules do not attract or repel each other).
4. Molecules experience elastic collisions
5. Mean kinetic energy # " T (in K)
" Distribution of molecular speeds: Some
molecules move more slowly; some move faster.
" When T increases, less molecules move
_________, more molecules move __________, so
the average molecular speed _______________.
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Chem 110 Chap 10
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KMT explains the behaviors of ideal gases
• At constant n and V; P increases as T increases.
Average Molecular Speed, Molar Mass, and
Temperature
T increases, # __________ , u ___________;
more collisions per unit time & harder collisions
so P increases
• At constant n and T, P decreases as V increases.
" At the same T, different gases have ________
average kinetic energy (#).
Constant T means constant # & u.
longer distances between collisions & fewer
collisions per unit time with walls
so P decreases (Boyle!s Law)
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Chem 110 Chap 10
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" At the same T, different gases have _________
average speeds (u)
" At a given temperature, the _______ a gas is, the
________ its average molecular speed will be.
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Average Molecular Speed and Molar Mass
Example: What is the average speed of O2 at
20 °C? Note: use R = 8.314J/(mol•K) and M in
kg/mol.
Practice Example: All three gas containers
below have V = 1 L and T = 25 °C. Which of
the following statements is/are true?
1. The average molecular speed is the same
in all three samples.
2. The pressure is the same for all three
samples.
3. The average molecular kinetic energy is
the same in all three samples.
A. 1
B. 2
C. 3
D. 1
E. 2
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only
only
only
and 2
and 3
Chem 110 Chap 10
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Other Properties of Gases
Effusion Rate and Molar Mass
Example: An unknown gas effuses at a rate of
0.112 times H2 gas at 25 °C. What is the molar
mass of the gas?
• Effusion: leakage of gas
through a small opening
• Diffusion: spread of gas
through space or second
substance.
Rate of effusion (or diffusion) "
1
M
Usually we compare the effusion (or diffusion)
rate of two gases:
Graham!s law:
$ Heavy molecules effuse (diffuse) more
slowly than lighter molecules
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Practice Example: A sample of N2 gas (2.0 mmol)
effused through a pinhole in 5.5 s. How long will
it take for the same amount of CH4 to effuse
under the same conditions?
A.
B.
C.
D.
E.
7.3
5.5
3.1
4.2
9.6
Collisions and Diffusion
" At STP molecules collide ~1010 times per
second.
N2 speed = 500 m/s
but…
s
s
s
s
s
in 1 s it collides 1010 times
" This means that although molecular speeds
are high at STP, molecules don’t go very far
net distance traveled << (speed x time)
" Mean free path (MFP): distance between
collisions.
MFP is ~6x10-8 m at 1 atm for air molecules
What happens to the MFP as density and P
decrease?
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Chem 110 Chap 10
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