Ideal Gas Law Lecturer: Professor Stephen T. Thornton

Ideal Gas Law

Lecturer: Professor Stephen T. Thornton

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

Reading QuizReading 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 toosmall

B) measured lengths will still beaccurate

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.

Reading QuizReading Quiz

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.

Copyright © 2009 Pearson Education, Inc.

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.

Thermal Expansion

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 QuizConceptual Quiz

Coefficient of volume expansion (1/°C )

Glass HgQuartz Air

AlSteel

A) aluminum

B) steel

C) glass

D) aluminum and steel

E) all three

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?

steel ring

Aluminum Aluminum is the only material that has a larger larger value value than the steel ring, so that means that the aluminum rod will aluminum rod will expand more than the steel ringexpand more than the steel ring. Thus, only in that case does the rod have a chance of reaching the top of the steel ring.


Coefficient of volume expansion (1/°C )

Glass HgQuartz Air

AlSteel

A) aluminum

B) steel

C) glass

D) aluminum and steel

E) all three

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?


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/more_stuff/Applets/brownian/brownian.html

Atomic Theory of Matter



Ideal gasesWe 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 findP ~ 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 findP ~ 1/V.

If we put these three experiments together, we findP ~ NT/V or PV ~ NTSo we add a constant and havePV = NkT, where k is the Boltzmann constant and is 1.38 x 10-23 J/K.

The temperature T must be kelvin K.

Sometimes pressure is lower case p.

Ideal gas lawPV NkT

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 g1 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 ismass (grams)(mole) =

molecular mass (g/mol)n

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

PiVi = PfVf

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

Now called Charles’s Law. This is also consistent with the ideal gas law.

fi

i f

VV

T T

If we keep the temperature constant, then changing the pressure does directly affect the volume.


Ideal GasTemperature 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.


If we use the Ideal Gas Temperature Scale, the temperature is

In order to determine temperature using a real gas, the pressure must be as low as possible.

tp(273.16 K) PT

P

tp 0tp

lim(273.16 K)P

PTP

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.]

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


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.


Which weighs more—a mole

of nitrogen (N2) gas or a mole

of oxygen (O2) gas?

A) oxygen

B) nitrogen

C) both the same


Which weighs more—a mole

of nitrogen (N2) gas or a mole

of oxygen (O2) gas?

A) oxygen

B) nitrogen

C) both the same

The oxygen molecules have a molecular mass of 32, and the

nitrogen molecules have a molecular mass of 28.

Conceptual Quiz Conceptual Quiz

Follow-up:Follow-up: Which one will take up more space? Which one will take up more space?

Conceptual Quiz Conceptual Quiz

A) cylinder A

B) cylinder B

C) both the same

D) it depends on the

pressure P

Two identical cylinders at the same

pressure contain the same gas. If A

contains three times as much gas as

B, which cylinder has the higher

temperature?

Ideal gas law: PVPV = = nRTnRT

Solve for temperature:

For constant V and P, the one with less gas

(the smaller value of the smaller value of nn) has the higher

temperature T.


A) cylinder A

B) cylinder B

C) both the same

D) it depends on the

pressure P

Two identical cylinders at the same

pressure contain the same gas. If A

contains three times as much gas as

B, which cylinder has the higher

temperature?

PVT =

nR

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.

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Ideal Gas Law Lecturer: Professor Stephen T. Thornton