# Gases Chapter 5.

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Gases Chapter 5

Gas Properties Four properties determine the physical behavior of any gas: Amount of gas Gas pressure Gas volume Gas temperature

Gas pressure Gas molecules exert a force on the walls of their container when they collide with it

Gas pressure Gas pressure can support a column of liquid
Pliquid = g•h•d g = acceleration due to the force of gravity (constant) h = height of the liquid column d = density of the liquid

Standard atmospheric pressure (1 atm) is 760 mm Hg
Torricelli barometer In the closed tube, the liquid falls until the pressure exerted by the column of liquid just balances the pressure exerted by the atmosphere. Patmosphere = Pliquid = ghd Patmosphere  liquid height Standard atmospheric pressure (1 atm) is 760 mm Hg

Units for pressure In this course we usually convert to atm

Gas pressure Pliquid = g•h•d
Pressure exerted by a column of liquid is proportional to the height of the column and the density of the liquid Container shape and volume do not affect pressure

Example A barometer filled with perchloroethylene (d = 1.62 g/cm3) has a liquid height of 6.38 m. What is this pressure in mm Hg (d = 13.6 g/cm3)? P = ghd = g hpce dpce = g hHg dHg hpce dpce = hHg dHg hHg = hpce d pce = (6.38 m)(1.62 g/cm3) = m dHg g/cm3 hHg = 760 mm Hg

Gas pressure A manometer compares the pressure of a gas in a container to the atmospheric pressure

Gas Laws: Boyle In 1662, Robert Boyle discovered the first of the simple gas laws PV = constant For a fixed amount of gas at constant temperature, gas pressure and gas volume are inversely proportional

Examples 6-3A & 6-3B A cylinder contains a gas at 5.25 atm pressure. When the gas is allowed to expand to a final volume of 12.5 L, the pressure drops to 1.85 atm. What was the original volume of the gas? 1.50 L of gas at 2.25 atm pressure expands to a final volume of 8.10 L. What is the final gas pressure in mm Hg?

Gas Laws: Charles In 1787, Jacques Charles discovered a relationship between gas volume and gas temperature: • relationship between volume and temperature is always linear • all gases reach V = 0 at same temperature, – °C volume (mL) • this temperature is ABSOLUTE ZERO temperature (°C)

A temperature scale for gases: the Kelvin scale
A new temperature scale was invented: the Kelvin or absolute temperature scale K = °C Zero Kelvins = absolute zero

Gas laws: Charles Using the Kelvin scale, Charles’ results is
For a fixed amount of gas at constant pressure, gas volume and gas temperature are directly proportional A similar relationship was found for pressure and temperature:

Examples 6-4A & 6-4B A gas at 25 °C and atm is heated under a piston. The volume expands from L to 1.65 L. What is the new temperature of the gas, if pressure has remained constant? If an aerosol can contains a gas at 1.82 atm at 22 °C, what will be the gas pressure in the can in an incinerator at 935 °C?

Standard conditions for gases
Certain conditions of pressure and temperature have been chosen as standard conditions for gases Standard temperature is K (0 °C) Standard pressure is exactly 1 atm (760 mm Hg) These conditions are referred to as STP (standard temperature and pressure)

Gas laws: Avogadro In 1811, Avogadro proposed that equal volumes of gases at the same temperature and pressure contain equal numbers of particles. At constant temperature and pressure, gas volume is directly proportional to the number of moles of gas Standard molar volume: at STP, one mole of gas occupies 22.4 L

Examples 6-6A & 6-6B A small tank of propane is opened and releases 30.0 L of gas at STP. What mass of propane was released? 128 g of dry ice sublimes into CO2 gas. What is the volume of this gas at STP?

Putting it all together: Ideal Gas Equation
Combining Boyle’s Law, Charles’ Law, and Avogadro’s Law give one equation that includes all four gas variables: R is the ideal or universal gas constant R = atm L/mol K

Using the Ideal Gas Equation
Ideal gas equation may be expressed two ways: One set of conditions: ideal gas law Two sets of conditions: general gas equation

Examples What is the volume occupied by 20.2 g NH3 gas at –25 °C and 752 mm Hg? How many moles of He gas are in a 5.00 L tank at 10.5 atm pressure and 30.0 °C? A 1.00 mL sample of N2 gas at 36.2 °C and 2.14 atm is heated to 37.8 °C while the pressure is changed to 1.02 atm. What volume does the gas occupy at this temperature and pressure?

Ideal Gas Equation and molar mass
Solving for molar mass (M)

Example A glass vessel weighs g when clean, dry, and evacuated. When filled with an unknown gas at 772 mm Hg and 22.4 °C, the vessel weighs g. What is the molar mass of the gas? 1.27 g of an oxide of nitrogen (believed to be either NO or N2O) occupies 1.07 L at 25 °C and 737 mm Hg. Which oxide is it?

Ideal Gas Equation and gas density

Gas density Gas density depends directly on pressure and inversely on temperature Gas density is directly proportional to molar mass

Examples What is the density of helium gas at 298 K and atm? Why can we say He is lighter than air? Hint: what is the average molar mass of air, which is 78.08% N2, 20.95% O2, 0.93% Ar, and 0.036% CO2? At what temperature will the density of O2 gas be 1.00 g/L if the pressure is kept at 745 mm Hg?

Mixtures of Gases Ideal gas law applies to pure gases and to mixtures
In a gas mixture, each gas occupies the entire container volume, at its own pressure The pressure contributed by a gas in a mixture is the partial pressure of that gas Ptotal = PA + PB (Dalton’s Law of Partial Pressures)

Mixtures of Gases When a gas is collected over water, it is always “wet” (mixed with water vapor). Ptotal = Pbarometric = Pgas + Pwater vapor Example: If 35.5 mL of H2 are collected over water at 26 °C and a barometric pressure of 755 mm Hg, what is the pressure of the H2 gas? The water vapor pressure at 26 °C is 25.2 mm Hg.

Gas mixtures The mole fraction represents the contribution of each gas to the total number of moles. XA = mole fraction of A

Gas mixtures The mole fraction represents the contribution of each gas to the total number of moles. XA = mole fraction of A The pressure fraction is equal to the mole fraction

Gas mixtures The volume composition of a gas mixture is
Avogadro’s hypothesis: at constant T & P, gas V is proportional to moles of gas The volume percent gives the mole fraction

Gas Mixtures For gas mixtures, mole fraction equals pressure fraction
Each gas occupies the entire container. The volume fraction describes the % composition by volume. mole fraction equals pressure fraction volume fraction

Examples What is the total gas pressure in a mixture of 1.0 g H2, 5.00 g He, and 12.5 g Ne, in a 5.0 L container at 55 °C? A mixture of mol CO2 and mol H2O gas are held at 30.0 °C and 2.50 atm. What is the partial pressure of each gas in the mixture? Air is 78.08% N2, 20.95% O2, 0.93% Ar, and 0.036% CO2 by volume. What is the partial pressure of each gas at a barometric pressure of 748 mm Hg?

Gases in Chemical Reactions
To convert gas volume into moles for stoichiometry, use the ideal gas equation: If both substances in the problem are gases, at the same T and P, gas volume ratios = mole ratios. P2 = P1 and T2 = T1

Examples How many grams of NaN3 will produce 20.2 L N2 at 30.0 °C and 776 mm Hg? NaN3 (s)  2 Na(l) N2 (g) What volume of O2 is consumed per liter of NO formed, at constant temperature and pressure: 4 NH3 (g) O2 (g)  4 NO (g) + 6 H2O (g)

A Model for Gas Behavior
Gas laws describe what gases do, but not why. Kinetic Molecular Theory of Gases (KMT) is the model that explains gas behavior. developed by Maxwell & Boltzmann in the mid-1800s based on the concept of an ideal or perfect gas

Ideal gas Composed of tiny particles in constant, random, straight-line motion Gas molecules are point masses, so gas volume is just the empty space between the molecules Molecules collide with each other and with the walls of their container The molecules are completely independent of each other, with no attractive or repulsive forces between them. Individual molecules may gain or lose energy during collisions, but the total energy of the gas sample depends only on the absolute temperature.

Molecular collisions and pressure
Force of molecular collisions depends on collision frequency molecule kinetic energy, ek ek depends on molecule mass m and molecule speed u molecules move at various speeds in all directions

Molecular speed Molecules move at various speeds
Imagine 3 cars going 40 mph, 50 mph, and 60 mph Mean speed = u = ( ) ÷ 3 = 50 mph Mean square speed (average of speeds squared) u2 = ( ) ÷ 3 = m2/hr2 Root mean square speed urms = √2567 m2/hr2 = mph

Distribution of molecule speeds

The basic equation of KMT
Combining collision frequency, molecule kinetic energy, and the distribution of molecule speeds gives the basic equation of KMT P = gas pressure and V = gas volume N = number of molecules m = molecule mass u2 = mean square molecule speed (average of speeds squared)

Combine the Equations of KMT and Ideal Gas
If n = 1, N = NA and PV = RT Avogadro’s number

Combine the Equations of KMT and Ideal Gas
NA x m (Avogadro’s number x mass of one molecule) = mass of one mole of molecules (molar mass M)

Combine the Equations of KMT and Ideal Gas
We can calculate the root mean square speed from temperature and molar mass

Calculating root mean square speed
To calculate root mean square speed from temperature and molar mass: Units must agree! Speed is in m/s, so R must be J/mol K M must be in kg per mole, because Joule = kg m2 / s2 Speed is inversely related to molar mass: light molecules are faster, heavy molecules are slower

Example 6-17A Which has the greater root mean square speed at 25 °C, NH3 gas or HCl gas? Calculate urms for the one with greater speed.

Interpreting temperature
Combine the KMT and ideal gas equations again Again assume n=1, so N = NA and PV = RT

Interpreting temperature
Absolute (Kelvin) temperature is directly proportional to average molecular kinetic energy At T = 0, ek = 0

Diffusion and Effusion
Diffusion (a) is migration or mixing due to random molecular motion Effusion (b) is escape of gas molecules through a tiny hole

Rates of diffusion/effusion
The rate of diffusion or effusion is directly proportional to molecular speed: The rates of diffusion/effusion of two different gases are inversely proportional to the square roots of their molar masses (Graham’s Law)

Using Graham’s Law Graham’s Law applies to relative rates, speeds, amounts of gas effused in a given time, or distances traveled in a given time.

Using Graham’s Law with times
Graham’s law can be confusing when applied to times rate = amount of gas (n) time (t)

Use common sense with Graham’s Law
When you compare two gases, the lighter gas escapes at a greater rate has a greater root mean square speed can effuse a larger amount in a given time can travel farther in a given time needs less time for a given amount to escape or travel Make sure your answer reflects this reality!

Examples 2.2 x 10–4 mol N2 effuses through a tiny hole in 105 seconds. How much O2 effuses through the same hole in the same time? How long would it take for 2.2 x 10–4 mol H2 to effuse through the same hole as the 2.2 x 10–4 mol N2, which effused in 105 seconds?

Examples A sample of Kr gas escapes through a tiny hole in 87.3 sec. Under the same conditions, the same amount of unknown gas effuses in sec. What is the molar mass of the unknown gas? How long would it take for the same amount of ethane gas to effuse, under the same conditions as the Kr gas in problem A?

Reality Check Ideal gas molecules Real gas molecules
constant, random, same straight-line motion point masses are NOT points – molecules have volume; Vreal gas > Videal gas independent of each other are NOT independent – molecules are attracted to each other, so Preal gas < Pideal gas gain / lose energy during same (some energy may be collisions, but total energy absorbed in molecular depends only on T (ek) )

Real gas corrections For a real gas,
a corrects for attractions between gas molecules, which tend to decrease the force and/or frequency of collisions (so Preal < Pideal) b corrects for the actual volume of each gas molecule, which increases the amount of space the gas occupies (so Vreal > Videal) The values of a and b depend on the type of gas

An equation for real gases: the van der Waals equation
Add correction to Preal to make it equal to Pideal, because intermolecular attractions decrease real pressure Subtract correction to Vreal to make it equal to Videal, because molecular volume increases real volume

When do I need the van der Waals equation?
Deviations from ideality become significant when molecules are close together (high pressure) molecules are slow (low temperature) At low pressure and high temperature, real gases tend to behave ideally At high pressure and low temperature, real gases do not tend to behave ideally } non-ideal conditions

Example 6-20A Calculate the pressure exerted by 1.00 mol CO2 when confined to a volume of 2.00 L at 273 K. aCO2 = 3.59 L2atm/mol2 and bCO2 = L/mol. Which shows a greater departure from ideality, CO2 or Cl2 (aCl2 = 6.49 and bCl2 =0.0562, same units)?

Examples 6-18 & 6-19A 2.2 x 10–4 mole of N2 gas effuses through a pinhole in 105 s. How much O2 effuses through the same hole in that amount of time? How long would it take for 2.2 x 10–4 mol H2 to effuse through the same hole? A sample of Kr gas escapes through a pinhole in 87.3 s. Gas X requires s for the same amount to escape. What is the molar mass of X?

Exercise 28 A gas is collected over water when the barometric pressure is mm Hg, but the water level inside the container of gas is 4.5 cm higher than outside. What is the total pressure of the gas inside the container, in mm Hg?

Exercise 29 A 35.8-L cylinder of Ar gas is connected to an evacuated 1875-L tank. If the temperature is held constant and the final pressure is 721 mm Hg, what must have been the original gas pressure in the cylinder, in atm?

Exercise 31 A fixed amount of gas held at a constant volume of 275 mL exerts a pressure of 798 mm Hg at 23.4 °C. At what temperature in °C will the pressure of the gas become exactly 1.00 atm?

Exercise 33 A 27.6 mL sample of PH3 gas is obtained at STP.
a. What is the mass of this gas, in mg? b. How many molecules of PH3 are present?

Exercise 37 A 12.8 L cylinder contains 35.8 g O2 at 46 °C. What is the pressure of this gas, in atm?

Exercise 42 A 10.0 g-sample of a gas has a volume of 5.25 L at 25 °C and 762 mm Hg. If to this constant volume is added 2.5 g of the same gas, and the temperature raised to 62 °C, what is the new gas pressure?

Exercise 44 A g sample of a gaseous compound occupies 428 mL at 24.3 °C and 742 mm Hg. The compound consists of 15.5% C, 23.0% Cl , and 61.5% F, by mass. What is its molecular formula?

Exercise 49 In order for a gas-filled balloon to rise in air, the density of the gas in the balloon must be less than that of air. a. Consider air to have a molar mass of g/mol. Determine the density of air at 25 °C and 1 atm, in g/L. b. Show by calculation that a balloon filled with carbon dioxide at 25 °C and 1 atm would not be expected to rise in air at 25 °C.

Exercise 56 A 3.57-g sample of a KCl-KClO3 mixture is decomposed by heating and produces 119 mL O2 measured at 22.4 °C and 738 mm Hg. What is the mass percent of KClO3 in the mixture? KClO3 (s)  KCl (s) + O2 (g) Hint: the KCl is unchanged

Exercise 61 A mixture of 4.0 g H2 and 10.0 g He gases in a 4.3 L flask is maintained at 0 °C. a. What is the total pressure in the container? b. What is the partial pressure of each gas?

Exercise 64 A typical producer gas has this composition by volume: 8.0% CO2, 23.2% CO, 17.7% H2, 1.1% CH4, and 50.5% N2. a. What is the density of this gas mixture at 23 °C and 763 mm Hg, in g/L? b. What is the partial pressure of CO in this mixture at STP?

Exercise 68 A g sample of He gas is found to occupy a volume of L when collected over hexane at 25.0 °C and mm Hg barometric pressure. Determine the vapor pressure of hexane at 25 °C from these data.

Exercise 75 If mol N2O gas effuses through an orifice in a certain period of time, how much NO2 gas would effuse in the same time under the same conditions?

Exercise 76 A sample of N2 gas effuses through a tiny hole in 38 s. What must be the molar mass of a gas that requires 64 s to effuse under identical conditions?

Exercise 79 Use the van der Waals equation to calculate the pressure exerted by 1.50 mol SO2 gas when it is confined to a volume of 5.00 L at 298 K. For SO2, a = 6.71 L2 atm/mol2 and b = L/mol. Compare this real pressure to what you would predict assuming SO2 to be an ideal gas.

Exercise 87 A 3.05-g sample of solid NH4NO3 is placed in an evacuated 2.18-L flask and heated to 250 °C. What is the total gas pressure in atm in the flask at 250 °C after the solid has completely decomposed: NH4NO3 (s)  N2O (g) H2O (g)

Exercise 92 What is the apparent molar mass of air, given that it is 78.08% N2, 20.95% O2, 0.93% Ar, and 0.036% CO2 by volume?