The Gas Laws

Dr. Molecular HazlettMandeville High School

Kinetic Molecular Theory

• All gas behavior is based on this theory• Gases are made up of particles (atoms and/or

molecules) that have KE• They move in random, straight paths• They do collide with each other and with the

walls of their container – It is assumed these collisions are elastic in Ideal

Gases

In more detail, Kinetic Molecular Theory of Gases states:

Note: These statements are made only for what is called an ideal gas. They cannot all be rigorously applied (i.e. mathematically) to real gases, but can be used to explain their observed behavior qualitatively.

1. All matter is composed of tiny, discrete particles (molecules or atoms).

2. Ideal gases consist of small particles (molecules or atoms) that are far apart in comparison to their own size. The molecules of a gas are very small compared to the distances between them.

3. These particles are considered to be dimensionless points which occupy zero volume. The volume of real gas molecules is assumed to be negligible for most purposes.

(However this is NOT TRUE. Real gas molecules do occupy volume and it does have an impact on the behavior of the gas. This impact can BE IGNORED when discussing ideal gases).

4. These particles are in rapid, random, constant straight line motion. This motion can be described by well-defined and established laws of motion.

5. There are no attractive forces between gas molecules or between molecules and the sides of the container with which they collide.

In a real gas, there actually is attraction between the molecules of a gas. Once again, this attraction WILL BE IGNORED when discussing ideal gases.

6. Molecules collide with one another and the sides of the container (this is pressure).

7. Energy is conserved in these collisions, although one molecule may gain energy at the expense of the other.

8. Energy can be transferred in collisions among molecules.

9. Energy is distributed among the molecules in a particular fashion known as the Maxwell-Boltzmann Distribution.

10. At any particular instant, the molecules in a given sample of gas do not all possess the same amount of energy. The average kinetic energy of all the molecules is proportional to the absolute temperature (measured in Kelvin).

KE of Gases• KE is proportional to temperature

–If it is monotomic: KE = 3/2 nRT–If diatomic: KE = 5/2 nRT

• Where n represents the number of moles• T is the temperature in Kelvin• R is the gas constant

Collision Rate (z)• Remember – we are in three dimensions (x, y,

and z axes)• As a gas particle moves, it clears a path which

reduces the volume of the gas and sets a frequency of collisions (f)

• Therefore: z = √2 π n2 d2 vavg

2

n = # particles d = distance between particlesv = average velocity z = collision rate

When dealing with Gases, there are four major variables utilized:

Volume

Pressure

Temperature

Amount of Gas (Moles)

Volume:

• All gases must be enclosed in a container that, if there are openings, can be sealed with no leaks. The three-dimensional space enclosed by the container walls is called volume. When the generalized variable of volume is discussed, the symbol V is used.

• Volume is usually measured in liters (symbol = L) or milliliters (symbol = mL). A liter is also called a cubic decimeter (dm3).

• Other units of volume are used such as cubic feet (cu. ft. or ft3) or cubic centimeters (cc or cm3). The main point to remember is: whatever units of volume are used, use them all the way through the problem. If you must convert from one unit to another, make sure you do it correctly.

Pressure• Gas pressure is created by the molecules of gas hitting the walls

of the container. The symbol P is used.• There are different units of pressure used in chemistry. You

must be able to use all of them:

– atmospheres (symbol = atm) – millimeters of mercury (symbol = mm Hg) – Pascals (symbol = Pa) or kiloPascals (kPa) – And, Pounds per Square Inch (psi)

• Standard pressure is defined as one atm.; or 760.0 mm Hg; or 101.325 kPa; or 101325 Pa; or 14.7 psi.

• Standard temperature and pressure is a very common phrase in chemistry, it is abbreviated to STP.

Measuring Pressure• The Manometer

A manometer is an device employed to measure pressure. There are a variety of manometer designs. A simple, common design is to seal a length of glass tubing and bend the glass tube into a U-shape. The glass tube is then filled with a liquid, typically mercury, so that all trapped air is removed from the sealed end of the tube. The glass tube is then positioned with the curved region at the bottom. The mercury settles to the bottom (see the picture at the left).After the mercury settles to the bottom of the manometer, a vacuum is produced in the sealed tube (the left tube in the picture). The open tube is connected to the system whose pressure is being measured. In the sealed tube, there is no gas to exert a force on the mercury (except for some mercury vapor). In the tube connected to the system, the gas in the system exerts a force on the mercury. The net result is that the column of mercury in the right (sealed) tube is higher than that in the left (unsealed) tube. The difference in the heights of the columns of mercury is a measure of the pressure of gas in the system.

Pressure ( P ) is the ratio of the force ( F ) exerted upon a surface to the surface area ( A ).P = F / A

• This method for measuring pressure led to the use of millimeters of mercury (mm Hg) as a unit of pressure. Today 1 mm Hg is called 1 torr. A pressure of 1 torr or 1 mm Hg is literally the pressure that produces a 1 mm difference in the heights of the two columns of mercury in a manometer.

• To understand how the height of a column of mercury can be used as a unit of pressure and how the unit of torr is related to the SI unit of Pascal (1 Pa = 1 N/m2), consider the following mathematical analysis of the behavior of the manometer.

• The force exerted by the column of mercury in a tube arises from the gravitational acceleration of the column of mercury. Newton's Second Law provides an expression for this force:

F = m g

• In this equation, m is the mass of Hg in the column and g is gravitational acceleration (9.80665 m/s2) and the force (F) is distributed over area (A)

Therefore: P = m g A• The mass of Hg is given by the product of Hg density (dHg) and

its volume (V) [dHg = 13.5951 g/cm3]• In a cylinder – the V of Hg is the product of the area and

height of the column (h)Thus:P = m g = dHg V g = dHg A h g = dHg h g

A A A• This means the column height is directly proportional to the

pressure exerted on it

Temperature• All gases have a temperature. When the

generalized variable of temperature is discussed, the symbol T is used.

• All gas law problems will be done with Kelvin (K) temperatures.

• You must convert into Kelvin before doing any calculations. Standard temperature is defined as zero degrees Celsius or 273.16 K.

• The Kelvin temperature of a gas is directly proportional to its kinetic energy. Double the Kelvin temperature, you double the kinetic energy.

Amount of Gas• The amount of gas present is measured in

moles (symbol = mol) or in grams (symbol = g or gm).

• Typically, if grams are used, you will need to convert to moles at some point. When the generalized variable of amount in moles is discussed, the letter "n" is used as the symbol

Boyle’s Law• Discovered by Robert Boyle in 1662. • His law gives the relationship between pressure and volume if temperature

and amount are held constant.• If the volume of a container is increased, the pressure decreases; and if the

volume of a container is decreased, the pressure increases.• Suppose the volume is increased. This means gas molecules have farther to go

and they will impact the container walls less often per unit time. This means the gas pressure will be less because there are less molecule impacts per unit time.

• If the volume is decreased, the gas molecules have a shorter distance to go, thus striking the walls more often per unit time. This results in pressure being increased because there are more molecule impacts per unit time.

• The mathematical form of Boyle's Law is: PV = k• This means that the pressure-volume product will always be the same value if

the temperature and amount remain constant. This relationship was what Boyle discovered.

• This is an inverse mathematical relationship. As one quantity goes up in the value, the other goes down.

Boyle’s Law Continued:

• Suppose P1 and V1 are a pressure-volume pair of data at the start of an experiment. In other words, some container of gas is created and the volume and pressure of that container is measured. Keep in mind that the amount of gas and the temperature DOES NOT CHANGE. When you multiply P and V together, you get a number that is called k. The exact value is irrelevant.

• Now, if the volume is changed to a new value called V2, then the pressure will spontaneously change to P2. It will do so because the PV product must always equal k. The PV product CANNOT just change to any old value, it MUST go to k. (If the temperature and amount remain the same.)

• So we know this: P1V1 = k

• And we know that the second data pair equals the same constant: P2V2 = k

• Since k = k, we can conclude that P1V1 = P2V2.

• Boyle’s Law will be P1V1 = P2V2

Boyle’s Law Demonstration

Charles’ Law• Discovered by Joseph Louis Gay-Lussac in 1802. He made

reference in his paper to unpublished work done by Jacques Charles about 1787. Charles had found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 degree interval.

• Lussac did invent the hydrogen-filled balloon and on December 1, 1783, he ascended into the air and became possibly the first man in history to witness a double sunset. On September 16, 1804, he ascended to an altitude of 7016 meters (just over 23,000 feet - about 4.3 miles). This remained the world altitude record for almost 50 years and then was broken by only a few meters.

• Because of Gay-Lussac's reference to Charles' work, many call the law by the name of Charles' Law. This does cause some confusion.

• This law gives the relationship between volume and temperature if pressure and amount are held constant.

• If the volume of a container is increased, the temperature increases; and if the volume of a container is decreased, the temperature decreases.

• The reason: As the temperature is increased the gas molecules will move faster and they will impact the container walls more often. This means the gas pressure inside the container will increase (but only for an instant). The greater pressure on the inside of the container walls will push them outward, thus increasing the volume. When this happens, the gas molecules will now have farther to go, thereby lowering the number of impacts and dropping the pressure back to its constant value.

• It is important to note that this momentary increase in pressure lasts for only a very, very small fraction of a second.

• Consider another case. Suppose the volume is suddenly increased. This will reduce the pressure, since molecules now have farther to go to impact the walls. However, this is not allowed by the law; the pressure must remain constant. Therefore, the temperature must go up, in order to get the molecules to the walls faster, thereby overcoming the longer distance and keeping the pressure constant.

• Charles' Law is a direct mathematical relationship. This means there are two connected values and when one goes up, the other also increases.

• The mathematical form of Charles' Law is: V ÷ T = k• This means that the volume-temperature fraction will always be

the same value if the pressure and amount remain constant.• Let V1 and T1 be a volume-temperature pair of data at the start of

an experiment. If the volume is changed to a new value called V2, then the temperature must change to T2.

• The new volume-temperature data pair will preserve the value of k. The actual value of k is not important, only that two different volume-temperature data pairs equal the same value and that value is called k So we know this: V1 ÷ T1 = k

• And we know this: V2 ÷ T2 = k

• Since k = k, we can conclude that V1 ÷ T1 = V2 ÷ T2.

• This equation is: V1 ÷ T1 = V2 ÷ T2

• EVERY TEMPERATURE USED IN A CALCULATION MUST BE IN KELVINS, NOT DEGREES CELSIUS.

Gay-Lussac’s Law• Discovered by Joseph Louis Gay-Lussac in the early 1800's. • Gives the relationship between pressure and temperature when

volume and amount are held constant.• If the temperature of a container is increased, the pressure

increases; and if the temperature of a container is decreased, the pressure decreases.

• Suppose the temperature is increased. This means gas molecules will move faster and they will impact the container walls more often. This means the gas pressure inside the container will increase, since the container has rigid walls (volume stays constant).

• Gay-Lussac's Law is a direct mathematical relationship. This means there are two connected values and when one goes up, the other also increases.

• The mathematical form of Gay-Lussac's Law is: P ÷ T = k

• This means that the pressure-temperature fraction will always be the same value if the volume and amount remain constant.

• P1 and T1 represent initial pressure-temperature data. If the temperature is changed to a new value called T2, then the pressure will change to P2. Keep in mind that when volume is not discussed (as in this law), it is constant.

• As with the other laws, the exact value of k is unimportant. It is important to know the PT data pairs obey a constant relationship. Besides which, the value of K would shift based on what pressure units (atm, psi, mmHg, or kPa) were being used

• We know this: P1 ÷ T1 = k; and thus we know this: P2 ÷ T2 = k

• And since k = k, we can conclude that P1 ÷ T1 = P2 ÷ T2.

• The equation is: P1 ÷ T1 = P2 ÷ T2

Charles/Lussac Law Demonstration

Diver’s Law• This law refers to diving beneath the water. The deeper you go the greater

the pressure because of the larger amount of water pressing down on you.• This law gives the relationship between pressure and amount with P and V

held constant. Amount is measured in moles. Since volume is constant, this means the container holding the gas is rigid and cannot change in volume.

• If the amount of gas in a container is increased, the pressure increases; and if the amount of gas in a container is decreased, the pressure decreases.

• Suppose the amount is increased. This means there are more gas molecules and this will increase the number of impacts on the container walls. This means the gas pressure inside the container will increase. It will stay at this higher level because the container walls do not move (the volume is constant).

• The mathematical form of Diver's Law is: P ÷ n = k• Let P1 and n1 be a volume-amount pair of data at the start of an experiment.

If the amount is changed to a new value called n2, then the pressure will change to P2.

• We know this: P1 ÷ n1 = k; and we know this: P2 ÷ n2 = k

• Since k = k, we can conclude that P1 ÷ n1 = P2 ÷ n2

Boyle’s Law Applies to Diving

• As P increases withdepth, the V of theDiver’s air compresses• When he surfaces,it will expand and could hurt the diverif he surfaces too fast

Pressure Below and Above Sea Level

• For every 33 ft., add 1 atm • Remember the 1 atm above the surface

– Therefore:• At 33 ft., pressure is 2 atm• At 66 ft., pressure is 3 atm• Etc.

• Pressure decreases the higher you go! Height (m) Pressure (atm)

2750 0.755486 0.58576 0.3316132 0.130901 0.0148467 0.00169464 0.000186282 0.00001

No-Name Law• Gives the relationship between amount and temperature when pressure and volume

are held constant. Remember amount is measured in moles. Also, since volume is one of the constants, that means the container holding the gas is rigid-walled and inflexible.

• If the amount of gas in a container is increased, the temperature must decrease; and if the amount of gas in a container is decreased, the temperature must increase.

• Suppose the amount is increased. This means there are more gas molecules and this will increase the number of impacts on the container walls. This means the gas pressure inside the container will increase, since the volume is constant. However, this cannot be allowed since pressure also remains constant. Lowering the temperature will cause the molecules to move slower, thus lessening the impacts on the container walls and reducing the pressure to the original value.

• The mathematical form of the "no-name" Law is: nT = k• Let n1 and T1 be a amount-temperature pair of data at the start of an experiment. If the

amount is changed to a new value called n2, then the temperature will change to T2.

• We know this: n1T1 = k and we know this: n2T2 = k

• Since k = k, we can conclude that n1T1 = n2T2

• The "no-name" Law is an inverse mathematical relationship.

Combined Gas LawTo derive the Combined Gas Law, do the following:• Step 1: Write Boyle's Law: P1V1 = P2V2

• Step 2: Multiply by Charles Law: P1V1

2 / T1 = P2V22 / T2

• Step 3: Multiply by Gay-Lussac's Law: P1

2V12 / T1

2 = P22V2

2 / T22

• Step 4: Take the square root to get the combined gas law: P1V1 / T1 = P2V2 / T2

• If all six gas laws are included (the three above as well as Avogadro, Diver, and "no-name"), we would get the following: P1V1 / n1T1 = P2V2 / n2T2

Ideal Gas Law• The Ideal Gas Law was first written in 1834 by Emil Clapeyron. • To "derive" the Ideal Gas Law, write each of the six gas laws as follows:

PV = k1

V / T = k2

P / T = k3

V / n = k4

P / n = k5

1 / nT = 1 / k6

• Note that the last law is written in reciprocal form. The subscripts on k indicate that six different values would be obtained.

• When you multiply them all together, you get: P3V3 / n3T3 = k1k2k3k4k5 / k6

• Let the cube root of k1k2k3k4k5 / k6 be called R.

• The units work out:k1 = atm-Lk2 = L / Kk3 = atm / Kk4 = L / molk5 = atm / mol1 / k6 = 1 / mol-K

• Each unit occurs three times and the cube root yields L-atm / mol-K, the correct units for R when used in a gas law context.

• Resuming, we have: PV / nT = R; or PV = nRT• R is called the gas constant. Sometimes it is referred to as the universal

gas constant.

The Numerical Value for R• R's value can be determined many ways. • Assuming 1.000 mol of a gas at STP. The volume of this amount of gas at

STP is known precisely. The value is 22.414 L.• This is called molar volume. It is the volume of ANY ideal gas at standard

temperature and pressure.• Put the numbers into the equation:

– (1.000 atm) (22.414 L) = (1.000 mol) (R) (273.15 K)• Solving for R gives 0.08206 L atm / mol K, when rounded to four

significant figures. This is not the only value of R that can exist. It depends on which units you select. It can also be 8.3145 Joules per mole Kelvin.

The Gas Constant (R)

• R = 8.314 L kPa / mol Kor

• R = 0.0821 L atm / mol K or• R = 62.4 L mmHg / mol K

– Form depends on unit of Pressure used

Remember: 1 atm = 14.696 psi = 101 325 Pa = 760 mmHg = 760 torrs

Ideal Gas law

Dalton’s Law of Partial Pressures• Discovered by John Dalton in 1801.• For any pure gas (let's use helium), PV = nRT holds true. Therefore,

P is directly proportional to n if V and T remain constant. As n goes up, so would P and versa vice.– Suppose you were to double the moles of helium gas present. What

would happen?– Answer: the gas pressure doubles.

• However, suppose the new quantity of gas added was a DIFFERENT gas. Suppose that, instead of helium, you added neon.– What would happen to the pressure?– Answer: the pressure doubles, same as before.

• Dalton's Law immediately follows from this example since each gas is causing 50% of the pressure. Summing their two pressures gives the total pressure.

• Written as an equation, it looks like this: PHe + PNe = Ptotal

• Dalton's Law of Partial Pressures: each gas in a mixture creates pressure as if the other gases were not present. The total pressure is the sum of the pressures created by the gases in the mixture.

Therefore: Ptotal = P1 + P2 + P3 + .... + Pn

• The only necessity is that the two gases do not interact in some chemical fashion, such as reacting with each other.

• The pressure each gas exerts in mixture is called its partial pressure.

Dalton’s Law of Partial Pressures

Graham’s Law of Diffusion• Discovered by Thomas Graham of Scotland in the 1830s .• For example: Samples of two different gases at the same

Kelvin temperature.• Since temperature is proportional to the kinetic energy of the

gas molecules, the KE of the two gases is also the same.• In equation form, we can write: KE1 = KE2

• Since KE = (1/2) mv2, (m = mass and v = velocity) we can write the following equation: m1v1

2 = m2v22

(Note that the value of one-half cancels out).• The equation above can be rearranged algebraically into the

following:The square root of (m1 / m2) = v2 / v1

Non-Ideal Gases / Real Gases• Gives more accurate measure of P and V• Takes into account V of particles and IMF

interactions• Uses van der Waals Equation

• “a” is the measure of attraction; and “b” is the measure of molecular volume• These will be found in reference sources

– Vary by gas type– For example:

Molar Volume• Molar volume is the volume occupied by one mole of

ideal gas at STP. • Its value is 22.414 L/mol for any gas!

– 1 mole = 6.022 x 1023 particles– 1 mole of gas at STP will fit into 11 inch3 volume

• This value has been known for about 200 years and it is not a constant of nature like, say, the charge on the electron. If we had picked a different standard temperature, then the molar volume would be different.

• Using PV = nRT, you can calculate the value for molar volume. V is the unknown and n = 1.00 mol. Set P and T to their standard values and use R = 0.08206.

Gas Density• Discussing gas density is slightly more complex than discussing solid/liquid density. Since

gas volume is VERY responsive to temperature and pressure, these two factors must be included in EVERY gas density discussion.

• By the way, solid and liquid volumes are responsive to temperature and pressure, but the response is so little that it can usually be ignored in most cases.

• For gases there is "standard gas density." This is the density of the gas (expressed in grams per liter) at STP. If you discuss gas density at any other set of conditions, drop the word standard and specify the pressure and temperature. When you say "standard gas density," you do not need to add "at STP."

• You can calculate the ideal standard gas density fairly easily. Just take the mass of one mole of the gas and divide by the molar volume.

For nitrogen, we would have: 28.014 g mol¯1 / 22.414 L mol¯1 = 1.250 g / L For water, we have: 18.015 g mol¯1 / 22.414 L mol¯1 = 0.8037 g / L

• The behavior of "real" gases diverges from predictions based on ideal conditions. Small gases like H2 at high temperatures approach ideal behavior almost exactly while larger gas molecules (NH3) at low temperatures diverge the greatest amount. These "real" gas differences are small enough to ignore right now.

• Density = mass/volume• Densitygas = PM M is molar mass

RT• Density is in units of grams per liter (g/L)

Vapor Pressure• Vapor pressure is the pressure of the vapor over a liquid (and some solids) at equilibrium. • Now, what does that definition mean?

(1) Imagine a closed box of several liters in size. It has rigid walls and is totally empty of all substances.

(2) Now, inject some liquid into the box, but the box is not full of liquid. What will happen to the liquid? (3) Some, maybe all, of the liquid will evaporate into gas, filling the empty space. Now if all the liquid evaporates, we just have a box of gas. Suppose that only someof the liquid evaporated and that there is both the liquid and gas state present in the box.

• The gas above the liquid is called vapor and creates a pressure called vapor pressure.

• If you put a gas-pressure measuring device into the gas and never touched the liquid it would record a pressure.

• The vapor must be in contact with the liquid at all times. Remove the liquid and you just have a box of gas, you do not have vapor or vapor pressure.

• How is vapor pressure created? Another way to put it - how do molecules of the liquid become molecules of gas?

• Each molecule in the liquid has energy, but not the same amount. The energy is distributed according to the Maxwell-Boltzmann Distribution. Even if you don't know what that is, the point is that some molecules have a fairly large amount of energy compared to the average. Those are the ones we are interested in.

• We are ESPECIALLY interested if one of these high energy molecules happens to be sitting right at the surface of the water. Now, all the molecules are in motion because of their energy, but none have sufficient energy to break the mutual attractive force molecules have for each other. If the surface molecule was moving up away from the surface AND had enough energy to break away from the attractive forces of the molecules around it. . . . .

• Where would that molecule go? It would continue to move away from the liquid surface AND IT BECOMES A MOLECULE OF GAS. This is great because we are now making some vapor pressure. It happens to another molecule and another and another.

• But wait! The vapor pressure stops going up and winds up staying at some fixed value. What's going on?

• As more and more molecules LEAVE the surface, what do some start to do? Some RETURN to the surface and resume their former life as a liquid molecule. Soon the number of molecules in the vapor phase is constant because the rate of returning equals the rate of leaving and so the pressure stays constant. This is equilibrium.

• Here's a REAL IMPORTANT point about vapor pressure: the vapor pressure depends ONLY on the temperature

• That is true because as the temperature goes up, there are more and more molecules with the right combination of energy and direction to break free of the liquid's surface.

• Finally, if you were to put more liquid in, the vapor pressure would not go up. – There are two opposing processes at work: (1) molecules leaving the liquid's

surface and (2) molecules returning to the liquid at its surface. Process #1 depends only on temperature and process #2 depends only on how many molecules are in the vapor phase. Adding more liquid affects neither process.

The Maxwell Distribution• Properties of the Maxwell Distribution The Maxwell distribution describes the distribution of particle speeds in an ideal gas. The distribution may be characterized in a variety of ways.• Average Speed

The average speed is the sum of the speeds of all of the particles divided by the number of particles.• Most Probable Speed

The most probable speed is the speed associated with the highest point inthe Maxwell distribution. Only a small fraction of particles might have this speed, but it is more likely than any other speed.

• Width of the Distribution The width of the distribution characterizes the most likely range of speeds for the

particles. One measure of the width is the Full Width at Half Maximum (FWHM). To determine this value, find the height of the distribution at the most probable speed (this is the maximum height of the distribution). Divide the maximum height by two to obtain the half height, and locate the two speeds in the distribution that have this half-height value. On speed will be greater than the most probably speed and the other speed will be smaller. The full width is the difference between the two speeds at the half-maximum value.

Particle Velocity• Velocity determined by using Root Mean Square• Speeds distributed over a range – called Maxwell

Speed Distribution• Vrms = √3RT

M

R = gas constantT = temperature (K)M = molar mass

Equation Summary• Boyle’s Law P1V1 = P2V2

• Charles’s Law V1 = V2

T1 T2

• Gay-Lussac’s Law P1 = P2

T1 T2

• Combined Gas Law P1V1 = P2V2

T1 T2

• Ideal Gas Law P V = n R T

• Ideal Gas / Mass Law P V = m R T M

(M is molar mass, m is mass in grams)• Dalton’s Law Partial Pressures

Ptotal = P1 + P2 + P3 + . . .

Pgas = Ptotal - Pwater

• Graham’s law of Diffusionrategas 1 / rategas 2 = √mass2 / mass1

m1 v12 = m2 v2

2

• Gas Density dgas = PM

R T• Van der Waals Equation

[ P + a (n/V)2] (V/n – b) = RTValues for (a) and (b) from

references• n (# mols) = m (mass in grams) M (molar mass)