Chapter 6

Chapter 6: Properties and Behavior of Gases

6.2 Pressure

Pressure is a fundamental property of gases, defined as the force exerted per unit area. In gases, pressure results from collisions of gas particles with the walls of their container.

• Definition: Pressure ($P$) = force/area

• Origin: Gas particles collide with the sides of the container, creating pressure.

States of Matter and Volume:

• Solid: Fixed volume and shape.

• Liquid: Fixed volume, takes the shape of the container.

• Gas: Volume and shape depend on the container; particles are evenly distributed.

Pressure Units:

• 1 mm Hg = 1 torr

• 1 atm = 760 mm Hg = 760 torr

• Other units: psi (pounds per square inch), Pa (pascal)

Example: Boyle's Law

• When holding your breath while scuba diving, the volume and pressure of air in your lungs change with depth.

• Boyle's Law: $P_1V_1 = P_2V_2$ (at constant temperature)

• Application: If $P_1 = 1$ atm, $V_1 = 6.0$ L, $P_2 = 2$ atm, then $V_2 = 3.0$ L

Charles's Law:

• Relates volume and temperature at constant pressure.

• $V_1/T_1 = V_2/T_2$

• Temperature must be in Kelvin: $K = ^ ext{o}C + 273.15$

• Example: $V_1 = 1.25$ L at $T_1 = 91.3^ ext{o}C$ cooled to $T_2 = 0.0^ ext{o}C$. Find $V_2$.

• Convert temperatures to Kelvin: $T_1 = 364.45$ K, $T_2 = 273.15$ K

• Calculate: $V_2 = V_1 imes (T_2/T_1) = 1.25 imes (273.15/364.45) = 0.937$ L

Avogadro's Law:

• Relates volume and number of moles at constant temperature and pressure.

• $V_1/n_1 = V_2/n_2$

• Example: $V_1 = 334$ mL, $n_1 = 0.87$ moles. What is the volume after adding 0.22 moles?

• $n_2 = 0.87 + 0.22 = 1.09$ moles $V_2 = V_1 imes (n_2/n_1) = 334 imes (1.09/0.87) = 418$ mL

6.4 The Ideal Gas Law

The ideal gas law combines several gas laws into one equation, relating pressure, volume, temperature, and number of moles.

• Equation: $PV = nRT$

• P: Pressure (atm)

• V: Volume (L)

• n: Moles

• R: Universal gas constant ($0.08206$ L·atm/mol·K)

• T: Temperature (K)

Example:

• Given: $V = 1.50$ L, $n = 0.448$ moles, $T = 325$ K

• Find $P$:

$P = rac{nRT}{V}$

• Plug in values: $P = rac{(0.448)(0.08206)(325)}{1.50} = 0.797$ atm

Gas Properties Table

This table compares the properties of three common gases: hydrogen, nitrogen, and methane.

	H2(g)	N2(g)	CH4(g)
n (moles)	1 mol	1 mol	1 mol
V (volume)	22.4 L	22.4 L	22.4 L
Mass	2.0 g	28.0 g	16.01 g
# of gas molecules	$6.022 imes 10^{23}$	$6.022 imes 10^{23}$	$6.022 imes 10^{23}$
Density	0.089 g/L	1.25 g/L	0.715 g/L

Density and the Ideal Gas Law

The ideal gas law can be rearranged to relate density to molar mass:

$PV = nRT$ $n = rac{m}{M}$ $P = rac{mRT}{MV}$ $D = rac{m}{V}$ $P = rac{DRT}{M}$ Example:

• Given: 243 mL of gas, mass = 0.433 g, $P = 745$ mm Hg, $T = 25^ ext{o}C$

• Convert units: $P = 745/760 = 0.980$ atm, $V = 243/1000 = 0.243$ L, $T = 25 + 273.15 = 298$ K

• Find molar mass:

$M = rac{mRT}{PV}$ $M = rac{0.433 imes 0.08206 imes 298}{0.980 imes 0.243} = 46.2$ g/mol

6.6 Mixture of Gases and Partial Pressure

In a mixture, each gas exerts its own pressure independently of the others. The total pressure is the sum of the partial pressures.

• Partial Pressure: $P_A = X_A P_{total}$, where $X_A$ is the mole fraction of gas A.

• Dalton's Law of Partial Pressures: $P_{total} = P_A + P_B + P_C + ...$

Example:

• Find the total pressure: $P_{O_2} = 125$ mm Hg, $P_{N_2} = 214$ mm Hg, $P_{H_2} = 187$ mm Hg

• $P_{total} = 125 + 214 + 187 = 526$ mm Hg

Percent Composition by Pressure:

• Percent of $O_2$ in total pressure: $100 imes (125/526) = 23.8 ext{ extpercent}$

Percent Composition by Moles:

• Percent of $O_2$ in total moles: $n_{O_2}/n_{total} imes 100$

Example:

• Gas mixture contains 1.25 g $O_2$ and 0.85 g $N_2$ at 15°C, $V = 1.55$ L

• Calculate moles: $n_{O_2} = 1.25/32.00 = 0.039$ mol, $n_{N_2} = 0.85/28.02 = 0.030$ mol

• Total moles: $n_{total} = 0.039 + 0.030 = 0.069$ mol

• Calculate total pressure using ideal gas law:

$P = rac{nRT}{V}$ $P = rac{0.069 imes 0.08206 imes 288}{1.55} = 1.06$ atm

6.7 Gas Stoichiometry

Gas stoichiometry involves using balanced chemical equations to relate volumes, masses, and moles of gases in reactions.

Example:

• Reaction: $2H_2O(l) ightarrow 2H_2(g) + O_2(g)$

• Find mass of $H_2O$ needed to form 1.4 L $O_2$ at 315 K and 0.957 atm.

• Use ideal gas law to find moles $O_2$:

$n = rac{PV}{RT}$ $n = rac{0.957 imes 1.4}{0.08206 imes 315} = 0.0518$ moles $O_2$

• Convert moles $O_2$ to moles $H_2O$ using stoichiometry: $2$moles$H_2O$per $1$ mole $O_2$

• $0.0518 imes 2 = 0.104$ moles $H_2O$

• Convert to grams: $0.104 imes 18.02 = 1.87$ g $H_2O$

Limiting Reactant Example:

• Mix 250 mL of 1.00 M $Na_2CO_3$ with 45.0 mL of 6.00 M HCl and capture $CO_2$ in a 1.00 L flask at 20°C. Find pressure of $CO_2$.

• Calculate moles of each reactant:

• $Na_2CO_3$: $0.250 imes 1.00 = 0.25$ mol

• $HCl$: $0.045 imes 6.00 = 0.27$ mol

• Stoichiometry: $Na_2CO_3 + 2HCl ightarrow 2NaCl + CO_2 + H_2O$

• Limiting reactant: $Na_2CO_3$ (produces 0.25 mol $CO_2$)

• Use ideal gas law to find pressure:

$P = rac{nRT}{V}$ $P = rac{0.25 imes 0.08206 imes 293}{1.00} = 0.601$ atm

6.8 Kinetic Molecular Theory of Gases

The kinetic molecular theory explains the behavior of gases in terms of the motion of their particles.

• Gases consist of tiny particles in constant, random motion.

• Size of gas particles is very small compared to the distance between them.

• Average kinetic energy is proportional to temperature: $KE ext{ (avg)} ightarrow T$

• Collisions are elastic (no energy loss); the rate of energy transfer is proportional to the number of collisions with the container walls.

Key Assumptions:

1. Gas particles are in constant, straight-line motion until they collide.

2. Most of the volume of a gas is empty space.

3. There are no attractive or repulsive forces between particles.

4. Temperature is directly proportional to average kinetic energy.

Example:

• Increasing temperature increases the speed and kinetic energy of gas particles.

Additional info:

• Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and finite particle volume.