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Chapter 5

Gases

Shaun Williams, PhD

Why Study Gases

A Gas

Pressure

$$ \text{Pressure}=\frac{\text{force}}{\text{area}} $$

Barometer

  • Device used to measure atmospheric pressure.
  • Mercury flows out of the tube until the pressure of the column of mercury standing on the surface of the mercury in the dish is equal to the pressure of the air on the rest of the surface of the mercury in the dish.
A graphical representation of a barometer.

Manometer

  • Device used for measuring the pressure of a gas in a container.
A photograph of a woman usinga manometer to measure the air pressure inside her car's tire.

Collapsing Can

A open container being heated with steam escaping from the top. The cap is placed on the contain and when it cools it collapses under the atmospheric pressure.

Pressure Conversions: An Example

The pressure of a gas is measured as 2.5 atm. Represent this pressure in both torr and pascals.

Pressure Conversions: An Example - Answer

The pressure of a gas is measured as 2.5 atm. Represent this pressure in both torr and pascals.

$$ \left( 2.5\,\chem{atm} \right) \times \left( \frac{760\,\chem{torr}}{1\,\chem{atm}} \right) = 1.9\times 10^3\,\chem{torr} $$ $$ \left( 2.5\,\chem{atm} \right) \times \left( \frac{101325\,\chem{Pa}}{1\,\chem{atm}} \right) = 2.5\times 10^5\,\chem{Pa} $$

The Gas Laws of Boyle, Charles, and Avogadro

Liquid Nitrogen and a Balloon

When an inflated balloon is placed in liquid nitrogen, the balloon shrinks (collapses).

Gas Laws

Boyle's Law

Exercise 1

A sample of helium gas occupies 12.4 L at 23°C and 0.956 atm. What volume will it occupy at 1.20 atm assuming that the temperature stays constant?

Exercise 1 - Answer

A sample of helium gas occupies 12.4 L at 23°C and 0.956 atm. What volume will it occupy at 1.20 atm assuming that the temperature stays constant?

$$ 9.88\,\chem{L} $$

Charles' Law

Exercise 2

Suppose a balloon containing 1.30 L of air at 24.7°C is placed into a beaker containing liquid nitrogen at –78.5°C. What will the volume of the sample of air become (at constant pressure)?

Exercise 2 - Answer

Suppose a balloon containing 1.30 L of air at 24.7°C is placed into a beaker containing liquid nitrogen at –78.5°C. What will the volume of the sample of air become (at constant pressure)?

$$ 0.849\,\chem{L} $$

Avogadro's Law

Exercise 3

If 2.45 mol of argon gas occupies a volume of 89.0 L, what volume will 2.10 mol of argon occupy under the same conditions of temperature and pressure?

Exercise 3 - Answer

If 2.45 mol of argon gas occupies a volume of 89.0 L, what volume will 2.10 mol of argon occupy under the same conditions of temperature and pressure?

$$ 76.3\,\chem{L} $$

The Ideal Gas Law

Bringing the Law Together

$$ PV=nRT $$ where \(R=0.08206\,\bfrac{\chem{L\cdot atm}}{\chem{mol\cdot L}}\), the universal gas constant.

Exercise 4

An automobile tire at 23°C with an internal volume of 25.0 L is filled with air to a total pressure of 3.18 atm. Determine the number of moles of air in the tire.

Exercise 4 - Answer

An automobile tire at 23°C with an internal volume of 25.0 L is filled with air to a total pressure of 3.18 atm. Determine the number of moles of air in the tire.

$$ 3.27\,\chem{mol} $$

Exercise 5

What is the pressure in a 304.0 L tank that contains 5.670 kg of helium at 25°C?

Exercise 5 - Answer

What is the pressure in a 304.0 L tank that contains 5.670 kg of helium at 25°C?

$$ 114\,\chem{atm} $$

Exercise 6

At what temperature (in °C) does 121 mL of \(\chem{CO_2}\) at 27°C and 1.05 atm occupy a volume of 293 mL at a pressure of 1.40 atm?

Exercise 6 - Answer

At what temperature (in °C) does 121 mL of \(\chem{CO_2}\) at 27°C and 1.05 atm occupy a volume of 293 mL at a pressure of 1.40 atm?

$$ 696^\circ\,\chem{C} $$

Gas Stoichiometry

Molar Volume of an Ideal Gas

Exercise 7

A sample of oxygen gas has a volume of \(2.50\,\chem{L}\) at STP. How many grams of \(\chem{O_2}\) are present?

Exercise 7 - Answer

A sample of oxygen gas has a volume of \(2.50\,\chem{L}\) at STP. How many grams of \(\chem{O_2}\) are present?

$$ 3.57\,\chem{g} $$

Molar Mass of a Gas

$$ \text{Molar Mass}=\frac{dRT}{P} = \frac{\left( \frac{\chem{g}}{\cancel{\chem{L}}} \right)\left( \frac{\chem{\cancel{L}\cdot \cancel{atm}}}{\chem{mol\cdot \cancel{K}}} \right)\left( \chem{\cancel{K}} \right)}{\left( \chem{\cancel{atm}} \right)} $$

Exercise 8

What is the density of \(\chem{F_2}\) at STP (in \(\bfrac{\chem{g}}{\chem{L}}\))?

Exercise 8 - Answer

What is the density of \(\chem{F_2}\) at STP (in \(\bfrac{\chem{g}}{\chem{L}}\))?

$$ 1.70\,\bfrac{\chem{g}}{\chem{L}} $$

Dalton's Law of Partial Pressures

If we consider the pressure of two seperated gases and then mix the two gases, the pressure of the mixture is the sum of the pressures of the serparated gases.

Exercise 9

Consider the following apparatus containing helium in both sides at 45°C. Initially the valve is closed. After the valve is opened, what is the pressure of the helium gas?

The situation described in the text above. The left chamber has an intial pressure of 2.00 atm and a volume of 9.00 L. The right chamber has an initial pressure of 3.00 atm and a volume of 3.00 L.

Exercise 9 - Answer

Consider the following apparatus containing helium in both sides at 45°C. Initially the valve is closed. After the valve is opened, what is the pressure of the helium gas?

The situation described in the text above. The left chamber has an intial pressure of 2.00 atm and a volume of 9.00 L. The right chamber has an initial pressure of 3.00 atm and a volume of 3.00 L.

$$ 2.25\,\chem{atm} $$

The Kinetic Molecular Theory of Gases

Postulates of the Kinetic Molecular Theory

  1. The particles are so small compared with the distances between them that the volume of the individual particles can be assumed to be negligible (zero). A graphical representation of the large distance between molecules compared to the size of the molecules.
  2. The particles are in constant motion. The collisions of the particles with the walls of the container are the cause of the pressure exerted by the gas.
  3. The particles are assumed to exert no forces on each other; they are assumed neither to attract nor to repel each other.
  4. The average kinetic energy of a collection of gas particles is assumed to be directly proportional to the Kelvin temperature of the gas.

Root Mean Square Velocity

$$ u_{rms} = \sqrt{\frac{3RT}{M}} $$

Effusion and Diffusion

Effusion of gas molecules through a small hole that separates one chamber (which contains the gas) and a second chamber containing a vaccuum. The wall between the chambers contains a pinhole.

Graham's Law of Effusion

$$ \frac{\text{Rate of effusion for gas 1}}{\text{Rate of effusion for gas 2}} = \frac{\sqrt{M_2}}{\sqrt{M_1}} $$

Real Gases

Plots of \(\bfrac{PV}{nRT}\) Versus \(P\) for Several Gases (\(200\,\chem{K}\))

A graph showing the deviation in behavior of real gases from ideal gases at high pressure.

Plots of \(\bfrac{PV}{nRT}\) Versus \(P\) for Nitrogen Gas at Three Temperatures

A graph showing the deviation in behavior of real gases from ideal gases at high pressure. The higher the temperature, the more ideal nitrogen gas behaves.

Real Gases (van der Waals Equation)

$$ \underbrace{\left[ P_\chem{obs} + a \left(\frac{n}{V}\right)^2 \right]}_{\text{corrected pressure, }P_\chem{ideal}} \times \underbrace{\left( V-nb \right)}_{\text{corrected volume, }V_\chem{ideal}} = nRT $$

Characteristics of Several Real Gases

Values of the van der Waals Constants for Some Gases

  • The value of \(a\) reflects how much of a correction must be made to adjust the observed pressure up to the expected ideal pressure.
  • A low value for \(a\) reflects weak intermolecular forces among the gas molecules.
Gas \(a\,\left(\frac{\chem{atm\cdot L^2}}{\chem{mol^2}}\right)\) \(b\,\left(\frac{\chem{L}}{\chem{mol}}\right)\)
\(\chem{He}\) 0.0341 0.0237
\(\chem{Ne}\) 0.211 0.0171
\(\chem{Ar}\) 1.35 0.0322
\(\chem{Kr}\) 2.32 0.0398
\(\chem{Xe}\) 4.19 0.0511
\(\chem{H_2}\) 0.244 0.0266
\(\chem{N_2}\) 1.39 0.0391
\(\chem{O_2}\) 1.36 0.0318
\(\chem{Cl_2}\) 6.49 0.0562
\(\chem{CO_2}\) 3.59 0.0427
\(\chem{CH_4}\) 2.25 0.0428
\(\chem{NH_3}\) 4.17 0.0371
\(\chem{H_2O}\) 5.46 0.0305

Chemistry in the Atmosphere

Air Pollution

Nitrogen Oxides (Due to Cars and Trucks)

$$ \begin{align} \chem{NO_2(g)} & \xrightarrow{\text{h$\nu$}} \chem{NO(g) + O(g)} \\ \chem{O(g) + O_2(g)} & \rightarrow \chem{O_3(g)} \end{align} $$

Concentration for Some Smog Components vs. Time of Day

Early in the morning molecules of unburned fuel builds up as do NO and NO2 molecules. After sunrise, all of these begin to fall as the concentration of ozone and other pollutants begin to build.

Sulfur Oxides (Due to Burning Coal for Electricity

$$\begin{align} \chem{S}\text{(in coal)}\chem{+O_2(g)} & \rightarrow \chem{SO_2(g)} \\ \chem{2SO_2(g)+O_2(g)} & \rightarrow \chem{2SO_3(g)} \\ \chem{SO_3(g)+H_2O(l)} & \rightarrow \chem{H_2SO_4(aq)} \\ \end{align}$$

A Schematic Diagram of a Scrubber

calcium carbonate, coal, and air enter the combustion chamber. The calcium carbonate is converted into carbon dioxide and calcium oxide by the heat while the sulfur is being converted into sulfur dioxide. The calcium oxide reacts with the sulfur dioxide to form calcium sulfite solid. Any remaining sulfur dioxide is washed from the gases by spraying a water calcium oxide mist through the gas. The calcium sulfite settles to the bottom of a collection chamber as a slurry.

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