Transcript Lecture 33.IdealGasL.. - Faculty Web Sites at the University of Virginia

Ideal Gas Law
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
A steel tape measure is
marked such that it gives
accurate length measurements A) measured lengths will be too small
at room temperature. If the
B) measured lengths will still be accurate
tape measure is used outside
on a very hot day, how will its C) measured lengths will be too big
length measurements be
affected?
Reading Quiz
A steel tape measure is
marked such that it gives
accurate length measurements
at room temperature. If the
tape measure is used outside
on a very hot day, how will its
length measurements be
affected?
A) measured lengths will be too
small
B) measured lengths will still be
accurate
C) measured lengths will be too big
The tape measure will expand, so its markings will spread out
farther than the correct amount. When it is laid down next to an
object of fixed length, you will read too few markings for that given
length, so the measured length will be too small.
Last Time
Thermal systems, reservoirs
Temperature
Heat flow
Thermometers
Temperature scales
Low temperatures
Look at P ~ T
Thermal expansion
Today
Thermal expansion - more
Brownian motion
Ideal gas law
Moles, Avogadro's number, etc.
Thermal Expansion
Water behaves differently from most other solids—its
minimum volume occurs when its temperature is 4°C. As
it cools further, it expands, as anyone who leaves a soda
can in the freezer to cool and then forgets about it can
testify.
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Water has special properties
• Remember that ice floats, because it is less dense than
water.
• Most solids are more dense than their liquid phase, so
water is unusual.
• When water at 00C is heated, it actually becomes more
dense up to 40C.
• In winter, surface layer of lakes gets colder to 40C and
sinks, allowing warmer water to get cold.
• Eventually lake is at 40C.
• Then surface goes to 00C, freezes (less dense) and
floats. This ice insulates water below.
• Life as we know it depends on this property of water!
Some Thermometers
Container Expansion. It is observed that
55.50 mL of water at 20°C completely fills a
container to the brim. When the container
and the water are heated to 60°C, 0.35 g of
water is lost. (a) What is the coefficient of
volume expansion of the container? (b)
What is the most likely material of the
container? Density of water at 60°C is
0.98324 g/mL.
Conceptual Quiz
A steel ring stands on edge with a rod of
some material inside. As this system is
heated, for which of the following rod
materials will the rod eventually touch
the top of the ring?
A) aluminum
B) steel
C) glass
D) aluminum and steel
E) all three
Quartz
Glass Steel Al



Hg
Air

Coefficient of volume expansion b (1/°C )
steel ring
Conceptual Quiz
A steel ring stands on edge with a rod of
some material inside. As this system is
heated, for which of the following rod
materials will the rod eventually touch
the top of the ring?
A) aluminum
B) steel
C) glass
D) aluminum and steel
E) all three
Quartz
Glass Steel Al



Hg
Air

Coefficient of volume expansion b (1/°C )
Aluminum is the only material that has a larger b value than
the steel ring, so that means that the aluminum rod will expand
more than the steel ring. Thus, only in that case does the rod
have a chance of reaching the top of the steel ring.
Atomic Theory of Matter
Atomic and molecular masses are measured in
unified atomic mass units (u). This unit is defined
so that the carbon-12 atom has a mass of exactly
12.0000 u. Expressed in kilograms:
1 u = 1.6605 x 10-27 kg.
Brownian motion is the jittery
motion of tiny flecks in water;
these are the result of collisions
with individual water molecules.
http://galileoandeinstein.physics.virginia.edu/mo
re_stuff/Applets/brownian/brownian.html
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Ideal gases
We will mostly discuss ideal gases, because it is easy to
experiment with them. Air at atmospheric pressure is
close to ideal. Intermolecular forces are negligible for
ideal gases.
The relationship between the volume, pressure,
temperature, and mass of a gas is called an equation of
state. V, P, T, m are thermodynamic state variables.
Consider a fixed volume container with a fixed number
of molecules. We used the constant volume
thermometer to show that P ~ T.
A Constant-Volume Gas Thermometer
P~T
Now let’s add molecules to a basketball.
If we do it quickly, the temperature and
volume are constant. We then find
P ~ N where N is the number of
molecules.
Finally, if we deform the basketball by
quickly sitting on it, the number of
molecules N and the temperature T are
constant, but we find
P ~ 1/V.
If we put these three experiments together,
we find
P ~ NT/V or PV ~ NT
So we add a constant and have
PV = NkT, where k is the Boltzmann
constant and is 1.38 x 10-23 J/K.
The temperature T must be kelvin K.
PV  NkT
Ideal gas law
Sometimes pressure is lower case p.
A mole is the number of grams of a substance
equal to the molecular weight of the substance.
For example,
1 mol of helium gas has mass 4 g
1 mol of oxygen (O2) gas has 32 g
1 mol of CO2 has mass 44 g
(We should use the more accurate molecular weights)
One mole of gas always contains precisely
NA = 6.022 x 1023 molecules/mole (Avogadro’s number)
The number of moles in a certain mass of material is
n (mole) =
mass (grams)
molecular mass (g/mol)
Let n = # of moles,
then N = nNA = number of molecules
PV = NkT becomes
PV = nNAkT
We now define the universal gas constant R
to be R = NAk = 8.31 J/(mol K)
so now,
PV nRT
Historically, at constant temperature Robert
Boyle found in 1600s that
P i Vi = P f V f
Note that this agrees with the ideal gas law.
PV = nRT = constant, when T is constant.
PV = constant or P ~ 1/V
Plot PV = constant. Curves are called
isotherms, because T is constant.
(constant pressure curves are called isobars.)
isotherm
Now let P and N be constants. Charles and
Gay-Lussac found around 1800 that
Vi V f

Ti T f
Now called Charles’s Law. This is also
consistent with the ideal gas law.
If we keep the temperature constant,
then changing the pressure does
directly affect the volume.
Ideal Gas
Temperature
Scale—a Standard
This standard uses the
constant-volume gas
thermometer and the
ideal gas law. There are
two fixed points:
1) Absolute zero—the pressure is zero here.
2) The triple point of water (where all three
phases coexist), defined to be 273.16 K—the
pressure here is 4.58 torr.
Copyright © 2009 Pearson Education, Inc.
If we use the Ideal Gas Temperature Scale, the
temperature is











tp 

T  (273.16 K) P
P
In order to determine temperature using a
real gas, the pressure must be as low as
possible.



Ptp 0 






tp 

T  (273.16 K) lim P
P
Copyright © 2009 Pearson Education, Inc.
Scuba Tank. A scuba tank is filled with
air to a pressure of 204 atm when the air
temperature is 29°C. A diver then jumps
into the ocean and, after a short time
treading water on the ocean surface,
checks the tank’s pressure and finds that
it is only 194 atm. Assuming the diver
has inhaled a negligible amount of air
from the tank, what is the temperature of
the ocean water?
Galileo Molecules. Estimate how many
molecules of air are in each 2.0-L breath
you inhale that were also in the last
breath Galileo took. [Hint: Assume the
atmosphere is about 10 km high and of
constant density.]
Conceptual Quiz
Which has more molecules—a
mole of nitrogen (N2) gas or a
mole of oxygen (O2) gas?
A) oxygen
B) nitrogen
C) both the same
Conceptual Quiz
Which has more molecules—a
mole of nitrogen (N2) gas or a
mole of oxygen (O2) gas?
A) oxygen
B) nitrogen
C) both the same
A mole is defined as a quantity of gas molecules equal to
Avogadro’s number (6.02  1023). This value is independent of the
type of gas.
Conceptual Quiz
Which weighs more—a mole
A) oxygen
of nitrogen (N2) gas or a mole
B) nitrogen
of oxygen (O2) gas?
C) both the same
Conceptual Quiz
Which weighs more—a mole
A) oxygen
of nitrogen (N2) gas or a mole
B) nitrogen
of oxygen (O2) gas?
C) both the same
The oxygen molecules have a molecular mass of 32, and the
nitrogen molecules have a molecular mass of 28.
Follow-up: Which one will take up more space?
Conceptual Quiz
Two identical cylinders at the same
A) cylinder A
pressure contain the same gas. If A
B) cylinder B
contains three times as much gas as
C) both the same
B, which cylinder has the higher
temperature?
D) it depends on the
pressure P
Conceptual Quiz
Two identical cylinders at the same
A) cylinder A
pressure contain the same gas. If A
B) cylinder B
contains three times as much gas as
C) both the same
B, which cylinder has the higher
D) it depends on the
temperature?
pressure P
Ideal gas law: PV = nRT
Solve for temperature: T =
PV
nR
For constant V and P, the one with less gas
(the smaller value of n) has the higher
temperature T.
Conceptual Quiz:
You have a Ping Pong ball that is slightly dented.
How might you get the dent out?
A) Put it in the freezer.
B) Put it in a pan of hot water.
C) Place the Ping Pong ball on a flat surface with the
dent sticking directly up. Push slightly on the
rounded side of the ball until the dent pops out.
D) Drill a hole in the Ping Pong ball on the side
opposite the dent. Push a thermometer through the
hole and measure the temperature before pushing
slowly until the dent comes out.
Answer: B
By heating the Ping Pong ball, the air
inside becomes hot, and the pressure
increases. The increased pressure is
likely to pop out the dent. Answer D is
almost correct, but you don’t need to
measure the temperature.
Conceptual Quiz:
Pyrex glass is often used for baking dishes and even
to boil water on a stove. The coefficient of linear
expansion is much smaller for Pyrex than for
ordinary glass. What is likely to happen if we put a
glass jar and a Pyrex jar on a hot stove element?
A) The glass jar would break before the Pyrex.
B) The Pyrex would break before the glass jar.
C) They would break at about the same time if they
are the same thickness.
D) If they have water in them, the Pyrex would
break first.
Answer: A
Because the glass jar has a higher
coefficient of linear expansion the part of
the glass that gets hot first would start
expanding first, and this would put a
good deal of stress on the glass structure.
Because Pyrex has a lower value, it
would not expand as much so quickly.