AP Chemistry 3.3-3.4: States of Matter and the Ideal Gas Law

Gas pressure

Pressure is the force exerted per unit area: $P=\frac{F}{A}$. For a gas, pressure results from collisions of gas particles with surfaces.

How does changing force or area affect pressure?

Pressure increases when force increases or the area decreases. It decreases when force decreases or the area increases, according to $P=F/A$.

Why does a gas exert pressure on the walls of its container?

Moving gas particles collide with the container walls and transfer momentum, producing a force. The large number of collisions over a given area creates measurable pressure.

Why can a figure skater exert more pressure on ice than a much heavier elephant?

The skater's narrow blades apply the force over a much smaller area. Since pressure is inversely proportional to area, the skater can exert greater pressure despite having less weight.

Atmospheric pressure

Atmospheric pressure is the force per unit area caused primarily by the weight of the column of air above a surface. Near sea level, its typical value is about $1\ \mathrm{atm}$.

What is the SI unit of pressure, and how is it defined?

The SI unit is the pascal (Pa): $1\ \mathrm{Pa}=1\ \mathrm{N/m^2}$. A newton is $1\ \mathrm{kg\cdot m/s^2}$.

Pressure-unit relationships

$1\ \mathrm{kPa}=1000\ \mathrm{Pa}$; $1\ \mathrm{bar}=100{,}000\ \mathrm{Pa}$; $1\ \mathrm{atm}=101{,}325\ \mathrm{Pa}=760\ \mathrm{torr}$; $1\ \mathrm{psi}\approx 6.895\ \mathrm{kPa}$.

What are the relationships among torr, millimeters of mercury, and atmospheres?

$1\ \mathrm{atm}=760\ \mathrm{torr}$, and $1\ \mathrm{mm\ Hg}\approx1\ \mathrm{torr}$. A torr was historically associated with the pressure of a $1$-mm mercury column, but the units are not exactly identical.

What is the relationship between inches of mercury and pascals?

$1\ \mathrm{in.\ Hg}=3386\ \mathrm{Pa}$.

How should pressure-unit conversions be performed?

Use dimensional analysis and conversion factors so that unwanted units cancel. For example, convert inches of mercury to millimeters of mercury, then to torr or atmospheres if needed.

Hydrostatic pressure

The pressure due to a fluid column is $P=h\rho g$, where $h$ is column height, $\rho$ is fluid density, and $g$ is gravitational acceleration.

Barometer

A barometer measures atmospheric pressure using a liquid column. Atmospheric pressure supports the liquid, and the height of the column is proportional to the pressure.

How can the height of a barometer column be used to compare pressures?

For the same liquid and gravitational field, $P=h\rho g$, so pressure is directly proportional to column height. A taller mercury column indicates greater atmospheric pressure.

What pressure does a $760$-mm mercury column represent under standard conditions?

A $760\ \mathrm{mm\ Hg}$ column corresponds to $1\ \mathrm{atm}$, or approximately $101.3\ \mathrm{kPa}$.

Why is mercury commonly used in barometers instead of water?

Mercury has a much greater density than water, so it supports atmospheric pressure with a much shorter column. A water barometer would need to be more than $10\ \mathrm{m}$ tall, whereas a mercury barometer is about $760\ \mathrm{mm}$ at $1\ \mathrm{atm}$.

How tall would a water column need to be to balance standard atmospheric pressure?

Using $h=P/(\rho g)$ with $P=101{,}325\ \mathrm{Pa}$ and $\rho\approx1000\ \mathrm{kg/m^3}$ gives $h\approx10.3\ \mathrm{m}$.

Manometer

A manometer measures the pressure of a gas using the difference in liquid levels in a U-shaped tube. The height difference corresponds to a pressure difference through $\Delta P=h\rho g$.

Closed-end manometer

A closed-end manometer has one arm sealed and evacuated. The gas pressure is directly balanced by the liquid-column pressure, so $P_{\text{gas}}=h\rho g$; for mercury, the height can be read directly in torr or mm Hg.

How is gas pressure determined from a closed-end mercury manometer reading of $26.4\ \mathrm{cm\ Hg}$?

Convert centimeters to millimeters: $26.4\ \mathrm{cm\ Hg}=264\ \mathrm{mm\ Hg}\approx264\ \mathrm{torr}$. This is approximately $35.2\ \mathrm{kPa}$ or $0.352\ \mathrm{bar}$.

Open-end manometer

An open-end manometer compares gas pressure with atmospheric pressure. The gas pressure equals atmospheric pressure plus or minus the hydrostatic pressure represented by the liquid-level difference.

How do you decide whether to add or subtract the height difference in an open-end manometer?

If the liquid level is higher on the gas side, the gas pressure is lower than atmospheric pressure, so subtract the height difference: $P_{\text{gas}}=P_{\text{atm}}-\Delta P$. If the liquid level is higher on the open atmospheric side, the gas pressure is greater, so add: $P_{\text{gas}}=P_{\text{atm}}+\Delta P$.

What does it mean when the liquid levels in an open-end manometer are equal?

The gas pressure equals the atmospheric pressure because there is no hydrostatic pressure difference between the two arms.

An open-end mercury manometer at sea level has the mercury level $13.7\ \mathrm{cm}$ higher on the atmospheric side. What is the gas pressure?

$P_{\text{gas}}=760\ \mathrm{mm\ Hg}+137\ \mathrm{mm\ Hg}=897\ \mathrm{mm\ Hg}$. This equals approximately $1.18\ \mathrm{atm}$ or $120\ \mathrm{kPa}$.

Ideal gas

An ideal gas is a model whose particles are assumed to have negligible volume and no intermolecular attractions, with collisions that conserve kinetic energy. Real gases most closely approximate ideal behavior at low pressure and moderate or high temperature.

What four macroscopic variables are related by the ideal gas law?

The ideal gas law relates pressure ($P$), volume ($V$), amount in moles ($n$), and absolute temperature ($T$).

Ideal gas law

The ideal gas law is $PV=nRT$, where $R$ is the gas constant. Common choices are $R=0.08206\ \mathrm{L\cdot atm\,mol^{-1}\,K^{-1}}$ or $R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}$.

What unit requirements must be satisfied when using $PV=nRT$?

Temperature must be in kelvins, and the pressure-volume units must match the chosen value of $R$. For example, use liters and atmospheres with $R=0.08206\ \mathrm{L\cdot atm\,mol^{-1}\,K^{-1}}$.

Why must gas-law temperatures be expressed in kelvins rather than degrees Celsius?

Gas-law proportionalities are based on absolute temperature, whose zero corresponds to absolute zero. Convert using $T(\mathrm{K})=T(^{\circ}\mathrm{C})+273.15$.

How does pressure change when a fixed amount of gas is compressed at constant temperature?

Pressure increases as volume decreases because $PV=\text{constant}$ at constant temperature and amount. This inverse relationship is Boyle's law.

How does the amount of gas affect volume when pressure and temperature are constant?

Volume is directly proportional to the number of moles: $V\propto n$. Adding gas particles increases the volume of a flexible container maintained at constant pressure and temperature.

How does increasing the number of gas particles affect pressure in a rigid container at constant temperature?

Pressure increases because $P\propto n$ when volume and temperature are constant. More particles produce more frequent collisions with the container walls.

Amontons's law (Gay-Lussac's law)

For a fixed amount of gas at constant volume, pressure is directly proportional to kelvin temperature: $P\propto T$. The two-state form is $\frac{P_1}{T_1}=\frac{P_2}{T_2}$.

A sealed rigid container of gas is heated from $300\ \mathrm{K}$ to $450\ \mathrm{K}$. How does its pressure change?

At constant volume and amount, $P_2=P_1(T_2/T_1)=1.50P_1$. The pressure increases by $50\%$.

Why can heating a sealed aerosol can be dangerous?

The can has nearly constant volume and amount of gas, so heating increases pressure according to $P/T=\text{constant}$. Sufficiently high pressure can rupture the container, and combustible contents add an additional hazard.

Charles's law

For a fixed amount of gas at constant pressure, volume is directly proportional to kelvin temperature: $V\propto T$. The two-state form is $\frac{V_1}{T_1}=\frac{V_2}{T_2}$.

A flexible balloon is cooled while its pressure and amount of gas remain approximately constant. What happens to its volume?

Its volume decreases in proportion to its absolute temperature, according to Charles's law. Warming the balloon causes it to expand.

Combined gas law

For a fixed amount of gas, pressure, volume, and temperature are related by $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$. This combines Boyle's, Charles's, and Amontons's relationships when the amount of gas is constant.

What conditions make the ideal gas law most accurate for a real gas?

The ideal-gas approximation is most accurate at low pressure and moderate or high temperature, where particles are relatively far apart and intermolecular attractions are less significant.

Why does extrapolating gas volume or pressure to zero lead to absolute zero?

For an ideal gas under the appropriate constant conditions, volume and pressure are linear functions of kelvin temperature. Extrapolating these relationships predicts zero volume or pressure at $0\ \mathrm{K}$, the theoretical absolute-zero limit.

Boyle's law

For a fixed amount of gas at constant temperature, pressure is inversely proportional to volume: $P\propto\frac{1}{V}$. Thus, $P_1V_1=P_2V_2$.

Avogadro's law

At constant temperature and pressure, the volume of a gas is directly proportional to the amount of gas: $V\propto n$. In two states, $\frac{V_1}{n_1}=\frac{V_2}{n_2}$.

Dalton's law of partial pressures

The total pressure of a mixture of nonreacting gases equals the sum of the partial pressures of the individual gases: $P_{\mathrm{total}}=P_1+P_2+P_3+\cdots$.

What is a gas's partial pressure?

A gas's partial pressure is the pressure it would exert if it alone occupied the container at the same temperature and volume. For an ideal-gas mixture, $P_i=X_iP_{\mathrm{total}}$, where $X_i$ is the mole fraction.

How is the mole fraction of a gas related to its partial pressure?

The mole fraction is $X_i=\frac{n_i}{n_{\mathrm{total}}}$. For an ideal-gas mixture, it equals the ratio of the gas's partial pressure to the total pressure: $X_i=\frac{P_i}{P_{\mathrm{total}}}$.

What are the key kinetic-molecular characteristics of gases?

Gas particles are far apart relative to their size, move continuously and randomly, undergo elastic collisions, and have negligible intermolecular attractions in the idealized model. Their average kinetic energy increases with kelvin temperature.

How do solids, liquids, and gases differ in particle motion and spacing?

Solid particles are closely packed and vibrate about fixed positions. Liquid particles remain close together but can flow past one another. Gas particles are widely separated and move freely throughout their container.

How do the states of matter differ in shape, volume, and compressibility?

Solids have definite shape and volume and are nearly incompressible. Liquids have definite volume but take the shape of their container. Gases have neither definite shape nor volume and are highly compressible.

How does temperature affect the average kinetic energy of particles?

Temperature measures the average kinetic energy of particles. Increasing temperature increases their average kinetic energy, while decreasing temperature lowers it.

Why are gases much more compressible than liquids and solids?

Gas particles are separated by large amounts of empty space, so their volume can be reduced substantially. Particles in liquids and solids are already packed closely together.