Combined Gas Law

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Mark S. Cracolice
Edward I. Peters
www.cengage.com/chemistry/cracolice
Chapter 4
Introduction to Gases
Mark S. Cracolice • The University of Montana
4.1 Properties of Gases
1. Gases may be compressed.
4.1 Properties of Gases
2. Gases expand to fill their containers uniformly.
4.1 Properties of Gases
3. All gases have low density.
4.1 Properties of Gases
4. Gases may be mixed.
4.1 Properties of Gases
5. A confined gas
exerts constant
pressure on the
walls of its
container uniformly
in all directions.
4.2 Ideal Gas Model
1. Gases consist of particles moving at any given
instant in straight lines.
At room temperature, the speeds of helium atom and carbon
dioxide molecule are respectively 1300 m/s and 400 m/s.
4.2 Ideal Gas Model
2. Molecules collide with each other and with the
container walls without loss of total kinetic
energy.
The frequency of collision of oxygen molecule at normal
temperature and pressure is 6 billions collisions per second
4.2 Ideal Gas Model
3. Gas molecules are very widely spaced.
4.2 Ideal Gas Model
4. The actual volume of molecules is negligible
compared to the space they occupy.
4.2 Ideal Gas Model
5. Due to the large distances between gas
molecules, attractive forces between them are
negligible.
4.2 Ideal Gas Model
6. The
average kinetic energy of molecules is
directly proportional to the absolute
temperature.
mv2/2 = (3/2) kT
where k is known as Boltzmann’s constant
4.3 Gas Measurements
By definition, pressure is the force exerted on a unit area:
For ce
Pressure º
Area
Using symbols,
F
P º
A
4.3 Gas Measurements
The Mercury Barometer
The atmospheric pressure, Pa, is
equal to the pressure of the liquid
mercury in the tube, PHg (Hg is the
symbol for the element mercury).
4.3 Gas Measurements
Pressure Units
Millimeters of mercury, mm Hg
The height of a column of mercury of one milimeter in a
barometer.
Torr, torr
Another name for a millimeter of mercury.
Atmosphere, atm
760 mm Hg
Approximately equal to atmospheric pressure at sea level.
4.3 Gas Measurements
Pascal, Pa (metric system)
One newton (a unit of force) per square meter (area)
1 atm = 101.3 × 103 Pa
Kilopascal (kPa ≡1000 Pa)
Bar (bar ≡ 100 kPa)
1 atm = 1.013 bar
4.3 Gas Measurements
Pressure Units (USCS System)
Inch of mercury, in. Hg
(barometer-based)
The height of a column of mercury of 1 inch in a barometer.
Pound per square inch, psi
(definition-based)
One pound (weight of one pound) per square inch (area).
4.3 Gas Measurements
Open-End Manometer
Used to measure pressure in the
laboratory.
Pg = Pa + PHg
4.3 Gas Measurements
Pg = Pa – PHg
4.3 Gas Measurements
In pressure-volume-temperature problems,
gas temperatures are expressed in
Kelvin (absolute temperature scale)
TK = T°C + 273
4.4 Charles’s Law
Charles’s Law
The volume (V) of a fixed quantity of gas at
constant pressure is directly proportional
to absolute temperature (T):
VT
4.4 Charles’s Law
For a fixed quantity of gas at constant pressure,
V
µT
V=k×T
Divide both sides of the equation by T:
V
k=
T
k=
V1
V1
T1
V2
T1 = k = T2
k=
V2
T2
V1
T1 =
V2
T2
4.4 Charles’s Law
Applying Charles’s Law
A fixed quantity of gas in a flexible-walled container is cooled
from 44°C to 21°C. If the initial volume of the container is
3.20 L, what is the final volume?
V1
T1
=
V2
V1T2 = V2T1
T2
T2
(21 + 273) K
V2 = V1 ×
= 3.20 L×
= 2.97 L
T1
(44 + 273) K
Check: Temperature decreases, which leads to decreased
volume. Temperature fraction is less than one. OK.
4.5 Boyle’s Law
For a fixed quantity of gas at constant
temperature, pressure is inversely
proportional to volume:
1
Pressur e µ
Volume
4.5 Boyle’s Law
P µ
1
V
1
P =k
V
Multiplying both sides of the equation by V yields:
PV = k
4.5 Boyle’s Law
PV = k
P1V1 = k
P2V2 = k
P1V1 = k = P2V2
For a fixed quantity of gas at constant temperature,
P1V1 = P2V2
4.5 Boyle’s Law
Applying Boyle’s Law
A fixed quantity of gas in a flexible-walled container is initially at
729 torr and a volume of 0.993 L. The temperature is kept
constant as the volume of the container is reduced to 0.720 L.
What is the resulting pressure at this volume?
P1V1 = P2V2
V1
0.993 L
P2 = P1 ×
= 729 torr ×
0.720 L
V2
= 1.01 × 103 torr
Check: Volume decreases, which leads to increased pressure.
Volume fraction is greater than one. OK.
4.5 Boyle’s Law
Applying Boyle’s Law
A gas sample occupies 5.18 liters at 776 torr. Find the volume of
the gas if the temperature is kept constant and the pressure is
changed to 827 torr.
P1V1 = P2V2
V2
= 5.18 L
x
776 torr
827 torr
4.6 The Pressure–Temperature Law
or Gay Lussac’s Law
The pressure of a gas at constant volume is directly proportional
to the absolute temperature
P = kT
As the temperature is raised, the molecules strike the wall more
often and with greater force.
4.6 The Pressure–Temperature Law
or Gay Lussac’s Law
The gas in a flask exerts a pressure of 0.97 atm
at 18 0C. What will the pressure be if the
temperature is raised to 31 0C ? Volume
remains constant.
P2 = P1 x T2 / T 1
= 0.97 atm x 304 K / 291 K
= 1.01 atm
4.7 The Combined Gas Law
µT
Charles’s Law:
V
Boyle’s Law:
1
P µ
V
Rearranging Boyle’s Law:
1
V µ
P
Combining the two proportionalities:
1
V µ T ´
P
Inserting a proportionality constant:
Rearranging:
1
V=k ´ T ´
P
PV
= k
T
4.7 The Combined Gas Law
PV
= k
T
P1 V1
T1
= k=
P1 V1
T1
=
P2 V2
T2
P2 V2
T2
This is the Combined Gas Law:
For a fixed quantity of gas, the ratio PV/T is a
constant.
The Combined Gas Law
Standard Temperature and Pressure (STP) for
gases:
25°C and 1 bar
To two significant figures, 1.0 bar = 1.0 atm,
so STP is also 25°C and 1 atm
if measured quantities are limited
to two significant figures or less.
The Combined Gas Law
Applying the Combined Gas Law
What is the STP volume of a sample of argon gas which had an
initial volume of 729 mL when at 44°C and 881 mm Hg?
P1 V1
P2 V2
=
Cross multiplying, P1V1T2 = P2V2T1
T1
T2
V2 = V1 ´
P1
P2
´
T2
T1
881 m m Hg
273 K
1.013 ba r
= 729 m L ´
´
´
= 737 m L
1 ba r
(44 + 273) K
760 m m Hg
Check: Pressure down, volume up, pressure fraction > 1, OK.
Temperature down, volume down, temperature fraction < 1, OK.
The Combined Gas Law
Applying the Combined Gas Law
• What is the STP volume of a sample of argon gas which had
an initial volume of 3.62 L when at 16°C and 0.843 bar?
Homework
Homework for chapter 4:
19, 21, 31, 37, 41, 51, 56
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