Gases

Gases

• All elements that are gases at standard conditions are nonmetals (N₂, O₂)
• All compounds that are gases at standard conditions are molecular (CO₂, CH₄)
• Gases of all elements/compounds have similar physical properties.
• Substances that are liquid and solid at standard condition can exist as gases – they are usually called vapors (water vapor)

Kinetic Molecular Theory

An explanation of the characteristics and properties of gases (and how they differ from liquids and solids)

Postulates (assumptions)

• Gases are composed of a large number of particles (atoms/molecules) that behave like hard, spherical objects in a state of constant, random motion

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• These particles have insignificant volume compared to the total volume of the gas. The particles are much smaller than the average distances between them. Most of the volume of a gas is empty space between the molecules.

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• There is no force or attraction between the gas particles or between the particles and the walls of the container.

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• When particles of a gas collide a small amount of energy may transfer from one particle to another but the average kinetic energy of the gas remains constant. (Energy is conserved)

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• The average kinetic energy of a collection of gas particles depends only on the temperature of the gas.
• Same temperature = Same average kinetic energy
• KE = ½ mv²

On AP equation sheet

Pressure Exerted by Gases

On AP equation sheet

1 in² column of

air (mass = 14.7 lb)

1 atm of pressure =14.7 lb/in²

Measuring the pressure of collected gases

Principle: Pressure on a gas = the pressure of a gas

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As long as the balloon is not inflating/deflating P_A = P_B

Measuring Equipment

• Eudiometer: gas measuring tube

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• Manometer: instrument which allows for the determination of the pressure of a gas sample

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• Barometer: instrument for measuring air pressure

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Manometer - measuring the pressure of collected gases

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(a) P_gas = P_h1 (b) P_gas = P_atm – P_h2 (c) P_gas = P_atm + P_h3

Barometer – measuring air pressure

Hg can move in and out of the tube

Temperature of Gases

Temperature (Kelvin) of a sample of gas is a measure of its average kinetic energy

KE= ½ mv²

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The faster the particles are moving the higher their average kinetic energy and the higher the temperature.

Maxwell-Boltzmann distribution

• It’s hard to calculate something about one particle so we use many and a statistical analysis.

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• As temperature increases, the average speed increases, and thus KE increases.

KE = ½ mv²

Single gas at multiple temps:

• As molar mass increases, speed decreases (KE is constant at a constant temp)

KE = ½ mv²

Multiple gases at the same temp:

Graham's Law

• Related to the rate at which gases:
• diffuse (spread to fill a volume)
• effuse (move through a small opening in their container)

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Lighter particles (low MM) 🡪 move faster!

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most often stated as:

can also use density:

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KE = ½ mv²

• Volume

Since a gas is mostly empty space it can be readily compressed to a smaller volume or can expand to fill any larger volume. (Takes the volume of its container)

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• Density

Very low! The mass of a gas occupies a much greater volume than an equal mass of the same liquid or solid.

Other Properties of Gases

• Mixtures - All gases that do not chemically react with each other can form homogeneous mixtures

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• High entropy

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Ideal Gas

• Conforms exactly to all aspects of the kinetic theory
• Does NOT exist.
• Real gases have
• attractions between particles
• particles have volume.
• Real gases exhibit ideal behavior when
• Temperature is high (particles have enough energy to overcome any attractions)
• Pressure is low (particles are so far apart their individual volume is insignificant).
• Real gases have near ideal behavior at room conditions.

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The most ideal gases have the weakest IMFs

(remember from 1^st year chem!)

Real Gases

most ideal He 🡪 no bonds (weakest IMF)

N₂ 🡪 nonpolar

least ideal H₂O 🡪 polar (strongest IMF)

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Standard Temperature

• a reference temperature which is 0^oC or 273 K

Standard Pressure

• a reference pressure which is 1 atm or its equivalent

Molar Volume of a gas

• The volume of 1 mole of a gas at standard temperature and pressure (STP) is 22.4 L/mole

On AP equation sheet

Boyle’s Law �Gas Pressure vs. Gas Volume

As the container size decreases, the

particles collide with the walls more

frequently thus raising the pressure

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Qualitatively:

P ↑ , V ↓ or P↓ , V ↑

temperature and moles held constant

Gas Pressure vs. Gas Volume

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As volume increases, pressure decreases.

Boyle’s P-V

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P₁V₁ = P₂V₂

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Inverse variation:

Movie

Amonton’s Law �(a.k.a. Gay-Lussac’s Law)�(Not in your book)�Gas Pressure vs. Gas Temperature

Increasing the temperature increases the KE of the molecules. With higher velocities, the molecules hit the walls more often and harder: more pressure (if volume held constant)

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qualitatively: T ↑ , P ↑ or T↓ , P ↓

volume and moles held constant

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Amonton’s: P-T

P₁T₂ = P₂T₁ ONLY if temp is Kelvin

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• Represents a direct variation: graph is a straight line

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Charles’s Law�Gas Volume vs. Gas Temperature

Increasing the temperature increases the KE of the molecules. The faster moving molecules will hit the walls more often and harder. If the pressure is held constant and the volume is not, the volume will increase.

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Qualitatively:

T ↑ , V ↑ or T ↓ , V ↓

pressure and moles held constant

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Charles’s Law�Gas Temperature vs. Gas Volume

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V₁T₂ =V₂Τ₁ (Temp in Kelvin!!!)

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• Direct variation: graph is a straight line

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Movie

Charles’s Law

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As temperature increases, volume increases

Absolute zero can be determined by determining T when volume is zero.

COMBINED GAS LAW

• This law combines Boyle’s, Amonton’s and Charles’s Laws into one law.
• It allows you to do calculations for situations in which only the amount of gas is constant

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P₁V₁ = P₂V₂

P₁T₂ = P₂T₁ P₁V₁T₂ = P₂V₂T₁

V₁T₂ = V₂T₁

• If you remember only this one equation – you should be able to derive all 3 of the gas laws!

Avogadro’s Hypothesis

• Avogadro used Gay-Lussac’s work and realized:

Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. (it doesn’t matter what gas it is – H₂, H₂O, CO₂, etc)

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Ex: 22.4L of any gas at 273K and 1atm contains 6.02x10²³ particles (1 mole).

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Avogadro’s Law�Gas Volume vs. Amount of Gas

• Increasing number of molecules will increase collisions and will increase volume if pressure is held constant   

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•   qualitatively: n ↑ , V ↑ or n ↓ , V ↓

P and T held constant

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•    quantitatively: V/n = k

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V₁n₂ = V₂n₁

Avogadro’s Law�Gas Volume vs. Amount of Gas

Ideal Gas Law

• combines all of the above into one equation or relationship
• PV = nRT

o      P is pressure

o      V is volume

o      n is the number of moles of gas

o      T is the temperature in KELVIN

o      R is the universal gas constant

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On AP equation sheet

Value of the Gas Constant (R)

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• other values of R

8.314 J/mol K

62.36 L torr/mol K

1.987 cal/mol K

 8.314 m³ Pa/mol K

Units must cancel when using this equation!

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On AP equation sheet

 �Other Applications of the Ideal Gas Law

• The ideal gas equation can be stated in other ways incorporating other variables while still keeping the same general relationship

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g = grams

D = density

mm = molar mass

Example: A sample of gas has a mass of 0.311 g. Its volume is 0.255 L at a temperature of 55°C and a pressure of 886 mm Hg. Find its molar mass.

MM = 28.1 g/mol

328K

mmHg = torr so use the 62.36 value of R

van der Waals Equation �(Ideal vs. Real Gases)

Graham's Law

• Related to the rate at which gases:
• diffuse (spread to fill a volume)
• effuse (move through a small opening in their container)

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Lighter particles (low MM) 🡪 move faster!

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most often stated as:

can also use density:

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KE = ½ mv²

Dalton's Law of Partial Pressure

• Total number of collisions is based on total number of molecules. Collisions from one kind of gas molecule are based only on that kind of molecule.

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• The total pressure of a mixture of gases is the sum of the pressures of each individual gas (each gas is said to have a partial pressure)

P_tot = P_A + P_B + P_C + …..

On AP equation sheet

Example: A 2.0 L flask contains a mixture of nitrogen gas and oxygen gas at 25ºC. The total pressure of the gaseous mixture is 0.91 atm, and the mixture is known to contain 0.050 mol of N₂. Calculate the partial pressure of oxygen and the moles of oxygen present.

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P_totV = n_totRT

(0.91 atm) (2.0 L) = n(.0821 L atm/mol K) (298K)

n_tot= 0.074 mol (total)

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n_tot = n_O2 + n_N2

0.074 mol total = n_O2 + 0.050 mol N₂

n_O2 = 0.024 mol O₂

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P_O2V = n_O2RT

P_O2(2.0 L) =(0.024 mol) (.0821 L atm/mol K) (298K)

P_O2 = 0.29 atm

Dalton’s Law Application #1

• Dalton's Law can be stated in a slightly different way emphasizing one component of the gas mixture

       P₁n₂ = P₂n₁

• the ratio is called the "mole fraction" of the gas and is symbolized by X

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• substituting in the above equation we get: P_A = (X_A) ( P_total )

this works because the total pressure depends on

the total moles of all the gases

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On AP equation sheet

Dalton’s Law Application #2

• Dalton’s Law is especially useful when collecting a gas by water displacement

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1. A gas collected by water displacement will have some water vapor mixed in with the gas

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2. Since we want only the pressure of the gas:

P_tot = P_gas + P_{H2O 🡪} P_gas = P_tot – P_H2O

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3. Values for water vapor pressure are in your text- page 219

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Play at 5 min

Gay-Lussac’s Law (of Combining Volumes)

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Gas volumes during a chemical reaction are proportional to the coefficients of the balanced equation. (just like mole ratios!)

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2 H₂(g) + O₂(g) 🡪 2H₂O(g)

2 mol 1 mol 2 mol

2L 1L 2 L

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Gas Stoichiometry

(Reactions with Gases)

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AT STP ONLY!

Methanol (CH₃OH) can be synthesized by the reaction below. What volume (in L) of hydrogen gas, at a temperature of 355 K and a pressure of 738 mm Hg, is needed to synthesize 35.7 g of methanol?

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CO(g) + 2H₂(g) 🡪CH₃OH(g)

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PV = nRT

(738)(V) = (2.23)(62.36)(355)

66.9L

35.7g

?L

Root-mean-square (rms) speed

• The speed (velocity) of molecules with exactly the average kinetic energy
• KE_average= ½ mv_rms²
• Some molecules in a gas sample move faster
• Some molecules in a gas move slower
• rms speed is close to the average speed

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Speed

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(Most probable)

Root mean square (rms) speed (symbolized by μ)

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As molar mass increases, rms speed decreases

(heavy particles are slower!)