Kinetic Molecular Theory

Gases 1.

Kinetic Molecular Theory

A.

Main Points 1. All matter consists of particles: either atoms or molecules. For a gas, if it is monoatomic (like He or Ar), it will consist of atoms. If it consists of I2, H2, N2, Br2, O2, Cl2, F2, it will consist of molecules. A gas in the form of a compound also consists of molecules. 2. All particles are in constant motion. The States of Matter Solids

Solids' particles vibrate. This is the only motion experienced by this state of matter.

Animation of water’s three characteristic vibrations: http://www2.ess.ucla.edu/~schauble/MoleculeHTML/H2O_html/H2O_page.html page: http://www2.ess.ucla.edu/~schauble/molecular_vibrations.htm )

1

(main

Liquids

Liquid molecules also vibrate but they rotate as well, giving them their familiar freedom to assume the shape of whatever container they are poured in.

Gases

Gas molecules move from one point to another; they're said to translate. Of course gas molecules still rotate and vibrate (monoatomic gases do not). Their volume is determined by their container, not by the size of individual molecules.1

This is because the size of the molecules is miniscule compared to the distance between them.

1

LaurenHill Chemistry 534 3. Their collisions are perfectly elastic, meaning that particles do not lose kinetic energy (energy of motion) by colliding with each other. 4.

For ideal gases only, the forces between particles are negligible(can be ignored). This is virtually the case for gases like helium, nitrogen and oxygen, at least under conditions of low pressure or high temperatures, but is not as valid for gases like H2O(water vapour) which have attractions for one another even at room temperature. Example 1

Show the difference between an ideal gas and a "real" gas.

Example 2 Under what conditions does ideal gas behaviour break down for most gases? Show on graph. V

Ideal gas behaviour

2

Gases 5.

The kinetic energy is directly proportional to the temperature of a gas in Kelvin degrees (absolute temperature). At a given temperature the molecules of all gases have the same average kinetic energy. That does not imply that they all move at the same speed. Lightweight (lower molar mass) ones move faster; heavier molecules are slower. But since kinetic energy depends on both mass and speed, it balances out: the avg. kinetic energy of a lightweight molecule will equal that of a slower heavyweight. Even if mass is the same, some molecules will move a little faster, others slower. But on average, at the same temperature, the speed of molecules will be the same for a given molecule of the same size.

Example 1

What happens to the kinetic energy of oxygen gas molecules if they are heated from 20 oC to 50 oC?

Example 2

What moves faster? O2 molecules at 34 oC? Or He at 34 oC?

B.

Heat Versus Temperature Heat is energy and it depends on temperature but also on the total mass of matter at that temperature. The temperature of a substance can only vary with the changing speed of its molecules. Temperature is a measurement directly proportional to the kinetic energy ( Ek) of the molecules. T = (2/3)Ek/R, where R = gas constant = 8.31 L kPa/([K mole Ek K = 273.15 + oC T(K) Example: A 200.0 g cup of coffee at 60.0 oC have fast moving molecules but contains less heat than a cooler 30 oC swimming pool containing 300.0 kg. Not convinced? Use Q = mc(Tf-Ti) from last year, using an initial temperature of 20.0 o C for both.

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LaurenHill Chemistry 534 Exercises 1.

All gases consist of particles. What type of particles?

2.

A gas’ volume is determined by its container, not by the size of individual molecules. Why?

3.

What are the differences between a liquid and a gas from a molecular point of view?

4.

How is the behaviour of suspended dust particles (Think of dislodged dust in a room as sunlight beams through) evidence for the idea that gases are in constant motion?

5.

a. Do all particles at the same temperature necessarily have the same speed? Explain. (Hint: if two class averages are equal, does everyone have the same mark?) b. If a gas particle's kinetic energy increases, what happens to its speed?

6.

If the temperature of a material is high, does it necessarily contain a lot of heat? Explain.

7.

At the same temperature, which of these gases’ particles will move faster, on average? Kr, CO2, C2H6 or O2?

8.

Consider two molecules, A and B, of different mass, mA and mB, each at the same temperature. Use Ek = 0.5mv2, where Ek = kinetic energy; m = mass; v = velocity of the particles to derive a formula revealing that small molecules move faster if they have as much energy as larger molecules.

9.

a) Which of these is more likely to be behave like an ideal gas: Ne or NH3? b) Even the gas that shows more ideal behavior will experience attractions in certain conditions. What conditions are we referring to, and what effect will attractions have on volume at constant pressure? c) Under what conditions will real gas behavior disappear and look like that of an ideal gas?

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Gases C.

Changes of State

What happens at the molecular level during changes of state? (melting, evaporation, condensation etc) Example 1

Freezing

When water molecules are above zero, they vibrate and also rotate. As the temperature cools, the molecules begin to rotate more slowly. The attractions between molecules start to get stronger, and they begin to move into a geometrical pattern (hexagon). At O oC, many molecules stop rotating; the attractions are strong enough so that the molecules become locked into that pattern. Only vibrations exist. We have a solid; ice has formed from liquid water.

Example 2

Explain sublimation at the molecular level. GUIDELINE

STEPS

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LaurenHill Chemistry 534 D. Pressure Pressure is force acting per unit area. Increasing pressure means that molecules will collide with both themselves and their container more frequently. So in chemistry pressure is also defined as "collision frequency". Note that in the diagram below, the arrows on each individual molecule, have the same total length, symbolizing that they are at the same temperature. (Notice that the volume is changing)

The second diagram reveals why there is greater air pressure at sea level compared to higher altitudes.

Example 1

What would happen if you filled a paint-can with steam and waited for it to cool?

Example 2

Why is it easier to inflate a long thin bag by leaving a small distance between your mouth and the opening of the bag?

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Gases Exercises 1.

What happens during evaporation at the molecular level?

2.

Explain melting at the molecular level.

3.

A student at a fashion show forgot to bring a camera. Since he was unable to take pictures, he decided to do some science. He applied the limewater test to the "smoke" coming from the stage and found it to be CO2. a.

Explain how solid carbon dioxide turns into gas.

b.

Why does a smoke machine, which uses dry ice, create "smoke"? Isn't carbon dioxide gas invisible? Hint: What other gas can be in a room that is filled with people?

4.

Why does a balloon burst if too much air is blown into it? Avoid three-word answers.

5.

Explain why water rises into a jar soon after the candle's flame is extinguished.

6.

Which of the following is the slowest molecule? Assume constant temperature.

a.

Ne

7.

The average velocity of four gases was measured at the same temperature and pressure.

b.

F2

c.

Ar

d.

CO2

The four gases are: He, Ne, CH4, and CO2. In which of the following are the gases arranged in increasing order of their average velocity? A)

He, Ne, CH4, CO2

B)

CO2, Ne, CH4, He

C)

CO2, CH4, Ne , He

D)

Ne, He, CH4, CO2

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LaurenHill Chemistry 534 2.

The Basic Gas Laws

A.

Charles' Law: applies to ideal gases under constant pressure.

If you heat a gas, its volume, V, will expand at a rate that is directly proportional to its absolute temperature measured in Kelvin. Jacques Charles (1746-1822)

V(L)_

V(L)_

T (K) T (oC)

-273

Notice that a plot of volume versus temperature in degrees Celsius (oC) leads to an xintercept of about -273. To obtain an x-intercept of 0, we add 273 to the Celsius temperature to obtain the absolute temperature measured in Kelvin. This leads to a simpler relationship known as Charles' Law:

V1 V2  T1 T2 at constant pressure with T in Kelvin

Example 1

Derive the above formula.

Example 2

A 250 mL balloon is warmed under constant atmospheric pressure from 0.00 oC to 20.0 oC. Find its final volume. (Assume that the rubber does not offer more resistance as it stretches; in reality it does and pressure would not be constant.)

Example 3

What happens to the volume of a gas if its absolute temperature is tripled?

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Gases B.

Avogadro's Law: Under the same conditions of pressure and temperature, equal volumes of different gases have the same number of molecules, or in other words the same number of moles.2 Ar

Kr

O2

If the above are all, for example, at room temperature and at atmospheric pressure, and if their containers are all of the same volume, then they all have the same number of moles. The size of the individual molecules is irrelevant. Conversely, if you were told that under certain conditions 10 moles of Ar occupied 224 L, then 10 moles of oxygen would also take up 224 L. At OoC and under 101.3 kPa of pressure (STP conditions) , one mole of any ideal gas occupies 22.4 L.

Example 1:

If a jar filled with 56 g nitrogen gas was emptied and replaced with oxygen gas under identical conditions of pressure and temperature, what would be the mass of the oxygen gas?

Example 2:

If a sample containing 2.5 moles of He gas has a volume of Vx, what is the volume of 5.0 moles of N2 gas?

Example 3:

What is the STP volume of 3.0 grams of hydrogen gas?

2

Avogadro's own words: "Volumi ugali di gas nelle stesse condizioni di temperaturaa e di pressione contegono lo stesso numero di molecule."

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LaurenHill Chemistry 534 C.

Boyle's Law: applies to ideal gases at constant temperature. The product of a gas' pressure and volume are constant, implying that P1V1 = P2V2. at constant temperature. The relation is an inverse one: if for instance you double pressure, the volume becomes half of the original.

V (L)_

P (kPa)

Robert Boyle (1627 – 1691)

The curve is known as a hyperbola, and it reveals how volume decreases as more pressure is applied. Example 1

Experimentally, how could you derive the above formula?

Example 2

If the same gas exerting 500 kPa is moved from a 300 mL container to a 100 mL container, what will the pressure become?

Example 3:

At constant temperature, the pressure of a gas is increased threefold. The final volume of the gas is 36 L. What was the original volume?

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Gases D.

Gay-Lussac's Law: At constant volume, the pressure of an ideal gas is directly proportional to its absolute temperature (K).

P1 P2 under constant volume. Temp in K.  T1 T2

P(kPa)_

Joseph Louis Gay-Lussac (1778 - 1850)

T (K)

The graph reveals how increasing the temperature will cause the molecules to move faster, and since they are not permitted extra room for maneuvering, they collide more often, resulting in more pressure. Example 1

At 300K, what pressure would be exerted by hydrogen gas if it exerts 200 kPa at 100 K?

Example 2

Show algebraically that halving the absolute temperature of a gas at constant volume will cause pressure to halve. Also show how the relationship does not apply to Celsius temperature.

Exercises 1.

At constant pressure, 35.0 L are heated from 268 to 287 K. Find the new volume.

2.

What happens to the volume of a gas if its absolute temperature changes by a factor of 0.88 at constant pressure?

3.

The volume of a gas increased by a factor of 1.05. If the final temperature was 40.0 oC, what was its original temperature in Celsius degrees? (Assume constant pressure.)

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LaurenHill Chemistry 534 4.

A gas was cooled to half of its absolute temperature at constant pressure. If the volume became 10.0 L, what was its original volume?

5.

A student measured the volume of a gas under constant pressure while she gradually increased the temperature of the gas. Find the experimental equivalent in Celsius degrees of absolute zero by obtaining the x-intercept, either graphically, or with the statistical mode of your calculator. Temperature (oC) 10.0 20.0 30.0 40.0 50.0

6.

Volume (L) 19.88 20.59 21.29 21.99 22.69

Examine the information given about the following two gas samples. 1.0 L of NH3 gas at 25°C and 101.3 kPa 1.0 L of N2 gas at 25°C and 101.3 kPa TRUE? Or FALSE? A. B. C. D.

NH3 and N2 have the same mass. NH3 and N2 molecules have the same average velocity. NH3 and N2 have the same number of molecules. NH3 and N2 molecules have the same average kinetic energy.

7.

If two astronauts exposed to same conditions of pressure and temperature are known to have the same number of molecules within their lungs, what do you conclude about their lungs?

8.

Two identical tanks filled with gas are under the same conditions of temperature and pressure. One is filled with hydrogen H2 and the other with nitrogen N2. According to Avogadro's law, which of the following statements is true? A)

Nitrogen molecules are more numerous than hydrogen molecules.

B)

Nitrogen molecules are as numerous as hydrogen molecules.

C)

The two tanks contain equal masses of gases.

D)

Nitrogen molecules are less numerous than hydrogen molecules.

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Gases

9.

If a sample containing 2.5 moles of H2 gas has a volume of 20.00 L, what is the volume of 7.5 moles of N2 gas? (Assume identical conditions of temperature and pressure.)

10.

Boyle's Law describes the relationship between the pressure and the volume of a fixed mass of gas at constant temperature. a) b)

What is the mathematical relationship? Sketch the graph.

11.

When a H2-filled balloon rises into the atmosphere, its volume increases. Why?

12.

If the pressure of a gas decreases from 900.0 kPa to 100.0 kPa, what will happen to its volume, assuming that temperature remains constant?

13.

The diagram below represents two closed containers of different volumes connected by a tube with a valve. The 1.0 L bulb on the left contains hydrogen gas, H2, at a pressure of 75 kPa. The 2.0 L bulb on the right is a vacuum. If the valve is opened, what will be the total pressure of hydrogen throughout the system? Assume that temperature is constant.

H2

1.0 L

2.0 L Vacuum

14.

The pressure of a gas is suddenly 300 kPa after the volume decreased from 30.9 L to 20.4 L. What was the original pressure if the compression took place gradually at constant temperature?

15.

Graph the relationship between pressure and temperature at constant volume.

16.

What is the final pressure of a 500.0 mL container as its temperature increases from 20.0 to 75.0 oC, if its original pressure was 100.0 kPa?

17.

If the volume of a gas remains constant, by what factor does pressure increase if absolute temperature increases from 273.0 K to 300.0 K?

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LaurenHill Chemistry 534

18.

A certain container will explode if its internal pressure increases from 150.0 kPa to beyond 500.0 kPa. It is meant to be kept at room temperature (20.0 oC), but what is the highest temperature it can withstand before it explodes?

19.

Draw two containers of equal volume and show that the number of molecular collisions (hence the pressure) increases as temperature increases.

20.

Galileo Galilei discovered that if there’s and area of very low pressure at the top of a sealed tube, a tube whose bottom was immersed in water, the water rose in the tube. But the water never rose higher than 10.34 m, no matter howthe vessel was shaped. His student Torricelli found out why.

a)

Explain.

b)

How high would liquid mercury rise if its density is13.6 g/ml?

http://www.eoht.info/page/Nature+abhors+a+vacuum

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Gases

3.

A Closer Look at Molar Volume and a Review of Stoichiometry In 1808, Gay Lussac published experiments on combining gas volumes. Three years later Avogadro proposed an explanation for Gay-Lussac's results by proposing his famous hypothesis. That in turn paved the way for atomic masses, which were tediously worked out by Stanislao Cannizzaro in 1860. Example

Amadeo Avogadro(1776 – 1856)

How did Avogadro's hypothesis make sense of the following results from Gay Lussac's experiments? 

22L of HBr(g) reacted with 11 L of NH3(g) to produce 11L of NH4Br(g) and leftover HBr



11L of HBr(g) reacted with 11 L of NH3(g) to produce 11L of NH4Br(g)



11L of H2(g) reacted with 11 L of Cl2(g) to produce 22 L of HCl(g)

Once Avogadro's Hypothesis was accepted, scientists such as Cannizzaro had a way of getting the relative weights of different gas molecules. Why?

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LaurenHill Chemistry 534

Because of Avogadro's Law, we can extend our stoichiometry chart from last year to the following: Volume of a gas (liters)

Mass (grams)

Molecules

 22.4 L / mole, if STP

 6.02  10 23 molecules / mole

 molar mass

MOLES

Ratio from balanced chemical equation

MOLES  6.02  10 23 molecules / mole  22.4 L / mole, if STP

 molar mass

Volume of a gas (liters)

Molecules

Mass (grams)

Example 1 At STP, how many litres of hydrogen gas would you collect by reacting 0.10 g of Na? 2 Na(s) +

2 HCl(aq)



2 NaCl(aq) +

H2(g)

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Gases

Example 2

a. b.

Find the density of Ne at S.T.P. Repeat for -20 oC and 101.3 kPa.

Example 3

If carbon dioxide gas was allowed to reach STP before we measured its volume, which turned out to be 60.0 L, how many molecules of propane were burned to create that quantity of gas?

Given:

C3H8(g)

+

5 O2(g) 

3 CO2(g)

+ 4 H2O(g)

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LaurenHill Chemistry 534

Exercises 1.

Consider the first step in the industrial production of nitric acid: 4 NH3(g)

+

5 O2(g)  4 NO(g)

+

6 H2O(g)

a.

What is the most oxygen, in liters at STP, that could react with 34.0 g of NH3(g)?

b.

In burning ammonia, 224 L of oxygen were consumed at STP. How many litres of NO must be contended with at STP?

c.

How many molecules of oxygen gas are needed to completely consume 2.0 moles of ammonia?

2.

A student reacts an unknown mass of iron (Fe) in an excess of sulfuric acid (H2SO4). The balanced equation is: 2 Fe(s) + 3 H2SO4(aq) 

Fe2(SO4)3(aq) +

She records the following data: Temperature: Pressure: Volume of H2 collected:

3 H2(g)

0o C 101.3 kPa 47.3 mL

Find the mass of iron used in the experiment. 3.

Given:

2 H2(g) +

O2(g)



2H2O(l)

At 40 oC, how many liters of O2 will react with 12 g of hydrogen? 4.

There are two gas cylinders in the room. Pressure and temperature are the same for both cylinders. Container A is full with 4 moles of argon. Container B is full with 1.204 X 1024 molecules of neon. How big is container A compared to container B? Why?

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Gases 5.

a.

What is argon's STP density?

b.

What is the density of argon at –50oC and 101.3 kPa?

4.

The Ideal and Combined Gas Laws

A.

Ideal Gas Law

Recall that at constant volume, the relationship between pressure and temperature (K) is a direct variation: P

T (K) Since the relationship is linear, we can say that P/T = m1. The relationship between volume, V, and the number of moles, n, is also linear, assuming constant pressure and temperature. (Avogadro's Hypothesis). If you double the number of particles or moles of particles, you will double the volume. V

n Since the relationship is linear, we can say that V/n = m2.

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LaurenHill Chemistry 534

Multiply the above fractions: cross multiply:

Let m1m2 = a new constant, R, and

PV = nRT.

Example 1

Ideal Gas Law

To get the value of R, we can substitute familiar STP conditions into the expression for R. What do we obtain?

Note, however, that the gas constant R applies to all conditions. If we increase P, then volume will decrease accordingly and still produce 8.31 L*kPa/(K mole) Example 2

What is the volume occupied by 6.0 g of hydrogen (H2) at 30.0 oC if it was subjected to a pressure of 150.0 kPa?

Example 3

List three ways of increasing the pressure of a gas.

Since

Example 4

, we can

Calculate the molar mass of an unknown gas using the data below: Volume of syringe: Mass of syringe + unknown gas Mass of empty syringe

0.100 L 90.99 g 90.67 g

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Gases 30.0 oC 100 kPa

Temperature Pressure

B.

The Combined Gas Law Since PV/nT is a constant, there is another version of the ideal gas law known as the Combined Gas Law: P1V1 P2V2  n1T1 n2T2

where n1 = original number of moles n2 = final number of moles What's nice about this law is that it simplifies into familiar relationships under special conditions. For instance under conditions of constant pressure and the same number of moles, P1 = P2 and n1 = n2, so those terms would cancel and leave us with Charles Law, V1/T1 = V2/T2. If volume and temperature are constant, and we want to know how the number of moles affects pressure, the formula simplifies to P1/n1 = P2/n2. But let's get back to an example where we can use the entire formula: Example 1

A student injects 2.0 moles of cold helium into a cylinder originally containing 2.5 moles of helium. As he does this, the piston rises and the volume increases from 0.505 L to 0.539 L. The pressure increases from 80 kPa to 90 kPa. If the gas was originally at 20.0 oC, what is the final temperature of the mixture? P1V1 P2V2  n1T1 n2T2

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LaurenHill Chemistry 534 Example 2

A cylinder contains a certain amount of krypton at room temperature. The same gas is then transferred to a smaller container only 1/3 as big as the original, and it's placed in a storage room with only 80% of the original absolute temperature. What happens to the pressure of the gas?

Exercises 1.

One winter morning, a driver checked the tires of his car and found the pressure to be 302 kPa at a temperature of -21.0°C. In the middle of the afternoon he rechecked the tires and found that they were over inflated and that the pressure was now 320 kPa. Disregarding any change in the volume of the tire, but assuming that the number of moles of air in the tires decreased from 1.0 to 0.98 moles what was the temperature when he rechecked the tires? (Ans : -0.53 oC)

2.

A 100 mL syringe filled with an unknown gas has a mass of 32.4 g. When empty, its mass is 32.0 g. If the temperature during the experiment is 20.0 oC and the pressure, 221.3 kPa, could the unknown gas have been CO2? Offer evidence!

3.

The valve of an oxygen (O2) cylinder is defective and the cylinder is leaking. The volume of the cylinder is 3400 L. On Monday morning the technician had recorded a pressure of 452 kPa and a temperature of 23°C. Four days later the pressure was 402 kPa and the temperature was 18°C. What was the mass of oxygen lost from the cylinder?

4.

Given:

4 NH3(g)

+

5 O2(g)  4 NO(g)

+

6 H2O(g)

What volume of steam will form for every 1.00 g of NH3 that reacts if the steam is collected at 100 oC and at 180 kPa? 5.

Given:

2 H2(g) +

O2(g)

 2 H2O(g)

If 20.0 L of hydrogen at 100.0 kPa and 10.0 oC burn, how many grams of water will form?

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Gases 6.

What mass of N2 exists in 100.0 L of air at 20.3 oC and 99 kPa? By volume, air is 78% nitrogen.

7.

What is fluorine's (F2) density in g/ml at 28 oC and 200.0 kPa?

8.

71 g of diatomic chlorine were in a 3.0 L flask at 25.0 oC, when suddenly every Cl2 molecule was dissociated by ultraviolet light. What was the new pressure shortly after the UV incident?

9.

A cylinder contains a certain amount of krypton at room temperature. The same gas is then transferred to a smaller container only 20% as big as the original, and it's placed in a storage room with only 5/6ths of the original absolute temperature. In the transfer, 10% of the gas was lost. What happens to the pressure of the gas? Additional examplestextbook:p241 #27, 28, p243 #60

5.

Dalton’s Law of Partial Pressures

If there are two or more gases in a container, the total pressure is the sum of the individual pressures of the various gases: PT = P1 + P2 +… Consider two gases A and B in a container at the same temperature. PT = PA + PB where PA is the partial pressure of gas A(partial pressure is the gas’ individual contribution to the total pressure) If we know nA and nB, is there a way we could calculate PA from PT, nA and nB? PA = (nA/nT)(PT) Use PV = nRT to derive this formula.

Example 1 following:

While kept at a constant temperature, a gas mixture contains the n

gas 0.34 5.55 2.10

H2 He Ar

The manometer attached to the container containing the mixture reads 233 kPa. Find the partial pressure of each gas.

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LaurenHill Chemistry 534

Example 2

Since equal volumes of ideal gases contain the same number of moles under the same conditions of P and T, % volumes of gases are directly proportional to mole fractions. With this in mind, find the partial pressure of oxygen in air at STP. Air is 21% O2 by volume.

Exercises 1. A balloon contains 0.100 moles of oxygen and 0.400 moles of nitrogen. If the balloon is at standard temperature and pressure, what is the partial pressure of the nitrogen? 2. The pressure of a mixture of nitrogen, carbon dioxide, and oxygen is 150.0 kPa. What is the partial pressure of oxygen if the partial pressures of the nitrogen and carbon dioxide are 100.0 kPa and 24.0 kPa, respectively? 3. A gaseous mixture made from 10.0 g of oxygen and 15.0 g of argon is placed in a 8.00 L vessel at 25.2oC. What is the partial pressure of each gas, and what is the total pressure in the vessel?

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Gases 4. a) 0. 888 L of “wet” oxygen (this implies that it’s mixed with water vapour) are collected at a temperature of 25.0 oC. The total pressure of the gases is 99.8 kPa. What is the partial pressure of the dry O2? The partial pressure of water at 25.0o C is 3.17 kPa. b) How many grams of water are in the 0.888 L mixture?

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