Chapter 6 Hines

Gas Physical Characteristics

Gases assume the volume and shape of their containers.

Gases are the most compressible state of matter.

Gases will mix evenly and completely when confined to the same container.

Gases have much lower densities than liquids and solids.

Pressure

Pressure is the force exerted per unit area by gas molecules as they strike the surfaces around them.

Just as a ball exerts a force when it bounces against a wall, a gaseous atom or molecule exerts a force when it collides with a surface.

The pressure of a gas depends on several factors:

• Number of gas particles in a given volume (concentration!)

• Volume of the container

• Average speed of the gas particles

Atmospheric Pressure Effects

Variation in pressure in Earth’s atmosphere creates wind and pressure.

The number of gas particles in a given volume decreases with increasing altitude.

Pressure decreases with increasing altitude.

Gas Pressure and Particle Density

Pressure exerted by a gas is dependent on the number of gas particles in a given volume.

The fewer gas particles, the lower the force per unit area and the lower the pressure.

• A low density of gas particles results in low pressure.

• A high density of gas particles results in high pressure.

Pressure Imbalance in the Ear

If there is a difference in pressure across the eardrum membrane, the membrane will be pushed out causing pain.

Barometer

A barometer is an evacuated glass tube, the tip of which is submerged in a pool of mercury (Hg).

Atmospheric pressure on the liquid mercury’s surface forces the mercury upward into the evacuated tube.

Due to the high density of mercury (13.5 times more dense than water), atmospheric pressure can support a column of Hg that is 760 millimeter tall.

Units

• Pascal (1 Newton per square meter) - Pa - 101,325 Pa

• Pounds per square inch - psi - 14.7 psi

• Torr (1 mmHg) - torr - 760 torr (exact)

• Inches of mercury - in Hg - 29.92 in Hg

• Atmosphere - atm - 1 atm

Manometer

The pressure of a gas trapped in a container can be measured with an instrument called a manometer.

Manometers are U-shaped tubes partially filled with a liquid that is connected to the sample on one side and open to the air on the other.

A competition is established between the pressures of the atmosphere and the gas.

The difference in the liquid levels is a measure of the difference in pressure between the gas and the atmosphere.

Four Basic Properties of a Gas

pressure (P), volume (V), temperature (T), and amount in moles (n).

• These properties are interrelated—when one changes, it affects the others.

• The simple gas laws describe the relationships between pairs of these properties.

Boyle’s Law

an inverse relationship between volume and pressure.

The pressure of a gas is inversely proportional to its volume.

• Constant T and amount of gas.

• Graph P versus V is a curve.

• Graph P versus 1/V is a straight line.

As P increases, V decreases by the same factor.

• P1 V1 = Constant = P2 V2

Boyle’s Law and Diving

For every 10 metre of depth, a diver experiences approximately one additional atmosphere of pressure due to the weight of the surrounding water.

If a diver holds his or her breath and rises to the surface quickly, the outside pressure drops to 1 atm.

• According to Boyle’s law, what should happen to the volume of air in the lungs?

• Because the pressure is decreasing by a factor of 3, the volume will expand by a factor of 3, causing damage to internal organs.

• Always exhale when rising!

Charles’ Law

Volume and Temperature

The volume of a fixed amount of gas at a constant pressure increases linearly with increasing temperature in kelvins:

• The volume of a gas increases with increasing temperature.

• Kelvin T = Celsius T + 273

• V = constant × T (if T is measured in kelvins)

Avogadro’s Law

Volume and Amount (Moles)

At constant P and T , Volume is directly proportional to the number of gas molecules.

• V = constant × n – More gas molecules = larger volume

• Count the number of gas molecules by moles.

• Equal volumes of gases contain equal numbers of molecules:

• The type of gas doesn’t matter.

• V1/N1 = V2/N2

When the amount of gas in a sample increases at constant temperature and pressure, its volume increases in direct proportion because the greater number of gas particles fill more space.

The volume of a gas sample increases linearly with the number of moles of gas in the sample.

This one is literally just you blowing up a balloon with your mouth!

Ideal Gas Law

The relationships that we have discussed so far can be combined into a single law that encompasses all of them.

• V∞(nT/P)

By combining the gas laws, and adding a constant, we can write a general equation.

• PV = nRT

R is called the gas constant, with P in atm and V in liters.

The other gas laws are found in the ideal gas law if two variables are kept constant. The ideal gas law allows us to find one of the variables if we know the other three.

• V = RnT/P

Standard Conditions (STP)

We often specify the volume of a mole of gas under conditions known as standard temperature and standard pressure, abbreviated as ST P.

Standard pressure = 1 atm

Standard temperature = 273 Kelvin = 0 °Celsius

The volume occupied by one mole of a substance is its molar volume at STP (T = 273 Kelvin or 0 °Celsius; P = 1 atm).

For any gas:

• [(1.00)(0.08206)(273)] / 1.00

• = 22.4

Density of a Gas at STP

Density is the ratio of mass to volume (just like with other substances)

Density of a gas is generally given in gram/Litre.

The mass of 1 mole = molar mass.

The volume of 1 mole at STP = 22.4 Litre.

• Density = (molar mass / molar volume)

• Molar Volume always 22.4

But note that ALL gases take up the SAME molar volume!!

So, for all gases, density is directly proportional to molar mass

Molar Mass of a Gas

We can determine the molar mass of an unknown substance by measuring its mass and volume under conditions of known pressure and temperature.

Then, we determine the amount of the gas in moles from the ideal gas law.

Finally, we calculate the molar mass by dividing the mass (in grams) by the amount (in moles).

• Molar Mass = Mass (m) / Moles (n)

Mixtures of Gases

Many gas samples are not pure but are mixtures of gases. Dry air, for example, is a mixture of many gases.

• 78% Nitrogen

• 21% Oxygen

• 0.9% Argon

• 0.04% Carbon Dioxide

In certain applications, the mixture can be thought of as one gas.

• Even though air is a mixture, we can measure the pressure, volume, and temperature of air as if it were a pure substance.

• We can calculate the total moles of molecules in an air sample, knowing P, V, and T, even though they are different molecules.

Partial Pressure

The pressure of a single gas in a mixture of gases is called its partial pressure.

We can calculate the partial pressure of a gas if

• we know what fraction of the mixture it composes and the total pressure, or

• we know the number of moles of the gas in a container of known volume and temperature.

Gases behave independently. So the sum of the partial pressures of all the gases in the mixture equals the total pressure:

• Dalton’s law of partial pressures

• P(total) = Pa+Pb+Pc…

We can calculate partial pressure from the ideal gas law by assuming that each gas component acts independently.

• Pn=nn(RT/V)

Dalton’s Law of Partial Pressures

For a gas mixture, we calculate the partial pressure of each component from the ideal gas law and the number of moles of that component (nn):

• Pa = na(RT/V) ; Pb = nb(RT/V) ; Pc = nc(RT/V) …

The sum of the partial pressures of the components in a gas mixture equals the total pressure:

• P(total) = Pa+Pb+Pc…

P total is the total pressure and Pa, Pb, Pc, . . . are the partial pressures of the components. This relationship is known as Dalton’s law of partial pressures.

• P(total) = Pa+Pb+Pc…

• P(total) = na(RT/V) ; nb(RT/V) ; nc(RT/V) …

• P(total) = (na + nb + nc)(RT/V)

• P(total) = (n(total))(RT/V)

Mole Fraction

The ratio of the partial pressure a single gas contributes and total pressure is equal to the mole fraction. The mole fraction is the number of moles of a component in a mixture divided by the total number of moles in the mixture

• Pa/P(total) = na(RT/V) / n(total)(RT/V) = na/n(total)

• Xa = na/n(total)

The partial pressure of a component in a gaseous mixture is its mole fraction multiplied by the total pressure

• Pa = (Xa)(P(total))

For gases, the mole fraction of a component is equivalent to its percent by volume divided by 100%

Collecting Gases

Gases are often collected by having them displace water from a container.

The problem is that, because water evaporates, there is also water vapor in the collected gas.

The partial pressure of the water vapor, called the vapor pressure, depends only on the temperature.

Vapor Pressure of Water

If you collect a gas sample with a total pressure of 758.2 mmHg at 25 °Celsius, the partial pressure of the water vapor will be 23.78 mmH g, so the partial pressure of the dry gas will be 734.4 mmHg.

Gases in Chemical Reactions

In reactions involving gases, we often specify the quantity of a gas in terms of its volume at a given temperature and pressure.

As we have seen, stoichiometry involves relationships between amounts in moles.

For stoichiometric calculations involving gases, we can use the ideal gas law to determine the amounts in moles from the volumes, or to determine the volumes from the amounts in moles.

• n=PV/RT : V = nRT/P

Note: the pressures above could also be partial pressures! The general conceptual plan for these kinds of calculations is as follows:

• Shortcut: when gases are at STP, use 1 mole = 22.4 L

Kinetic Molecular Theory

The simplest model for the behavior of gases is the kinetic molecular theory.

In this theory, a gas is modeled as a collection of particles in constant motion.

Basic Postulates of Kinetic Molecular Theory (and therefore assumptions about ideal gases)

1. The attraction between particles is negligible.

2. The particles of the gas (either atoms or molecules) are constantly moving and collisions are elastic. When one particle collides with another (or with the walls of its container) they do not stick, but they bounce off and continue moving in another direction with no loss of energy.

3. There is a lot of empty space between the gas particles compared to the size of the particles. Therefore the size of the gas molecules is negligibly small. (The gas molecules themselves take up no space.)

4. The average kinetic energy of a particle is proportional to the temperature in kelvins. As you raise the temperature of the gas, the average speed of the particles increases. But not all the gas particles are moving at the same speed!

Temperature and Molecular Velocities

Average kinetic energy of the gas molecules depends on the average mass (m) and velocity (u).

• KE(avg) = (1/2)(N(A)mu²) line on top of the square 2

Gases in the same container have the same temperature; therefore, they have the same average kinetic energy.

If they have different masses, the only way for them to have the same kinetic energy is to have different average velocities.

Lighter particles will have a faster average velocity than more massive particles.

• u(rms) = root(3RT / M)

• u(rms) is root mean square velocity (a type of average velocity)

• urms is root mean square velocity (a type of average velocity)

Temperature and Molecular Velocities

As the temperature of a gas sample increases, the velocity distribution of the molecules shifts toward higher velocity.

• The distribution function “spreads out,” resulting in more molecules with faster speeds.

• u(rms) = root(3RT← / M)

Mean Free Path

Molecules in a gas travel in straight lines until they collide with another molecule or the container.

The average distance a molecule travels between collisions is called the mean free path.

Mean free path decreases as the pressure increases.

Diffusion and Effusion

The process of a collection of molecules spreading out from high concentration to low concentration is called diffusion.

The process by which a collection of molecules escapes through a small hole into a vacuum is called effusion.

The rates of diffusion and effusion are both related to the root mean square velocity.

For gases at the same temperature, this means that the rate of gas movement is inversely proportional to the square root of its molar mass.

• Graham’s Law of Effusion (rates): for two gases at the same temperature:

• rate(A) / rate(B) = root(M(A) / M(B))

Real Gases

Real gases are not actually ideal at any temperature! But simplifying assumptions are nice when small errors are permissible.

For example look at the most common one: 1mol gas at STP = 22.4L

Real gases really deviate from ideal behavior at high pressure or low temperature.

Ideal gas laws assume

1. no attractions between gas molecules.

2. gas molecules do not take up space.

At low temperatures and/or high pressures these assumptions are not valid.

The Effect of the Finite Volume of Gas Particles

At low pressures, the molar volume of argon is nearly identical to that of an ideal gas but as the pressure increases, the molar volume of argon becomes greater than that of an ideal gas.

• At the higher pressures, the argon atoms themselves occupy a significant portion of the gas volume, making the actual volume greater than that predicted by the ideal gas law.

Because real atoms/molecules take up space, the molar volume of a real gas is larger than predicted by the ideal gas law at high pressures.

Real Gases: The Effect of Intermolecular Attractions

At high temperature, the pressures of the gases are nearly identical to that of an ideal gas but at lower temperatures, the pressures of the gases are less than that of an ideal gas.

• At the lower temperatures, the gas atoms spend more time interacting with each other and less time colliding with the walls, making the actual pressure less than that predicted by the ideal gas law.

Modification of the Ideal Gas Equation

In 1873, Johannes van der Waals (1837–1923) modified the ideal gas equation to fit the behavior of real gases.

Van der Waals’s Equation:

• [P + a(n/V)²] * [V - nb] = nRT

• [P + a(n/V)²] - Correction for intermolecular forces

• [V - nb] - Correction for particle volume

a and b are both called Van der Waals constants and are correction factors for intermolecular forces and particle volume, respectively.

Van der Waals equation - nonideal gas

(P + (an²/V²))(V-nb) = nRT

• (P + (an²/V²)) - corrected pressure

• (V-nb) - corrected volume

Constants are empirically determined to help us match "real world" gas behavior.