Premed · Premed · General Chemistry 1
Lecture 14: Gases: Ideal Gas Law
General Chemistry I
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the general properties of gases and how they differ from solids and liquids
- Define and interconvert units of pressure
- State and apply Boyle's law, Charles's law, Avogadro's law, and the combined gas law
- Derive and use the ideal gas law (PV = nRT)
- Calculate gas density and molar mass using the ideal gas law
- Apply Dalton's law of partial pressures and the concept of mole fraction
Lecture Content
I. General Properties of Gases
Gases differ dramatically from solids and liquids in several ways. They expand to fill their container completely, having neither fixed shape nor fixed volume. They are highly compressible because their molecules are separated by large distances. Their densities are low compared to condensed phases, and they mix completely and homogeneously in all proportions. A remarkable feature of gases is that their behavior is largely independent of chemical identity -- all gases obey the same laws to a good approximation. Gas behavior is described by four interrelated variables: pressure (P), volume (V), temperature (T), and amount (n, in moles).
II. Pressure
Pressure is defined as force per unit area (P = F/A). Atmospheric pressure arises from the weight of the column of air above a given point. Several units are used to express pressure: the pascal (Pa = N/m^2), which is the SI unit; the atmosphere (1 atm = 101,325 Pa = 101.325 kPa); millimeters of mercury (1 atm = 760 mmHg = 760 torr); the bar (1 bar = 100,000 Pa; 1 atm = 1.01325 bar); and psi (1 atm = 14.696 psi).
A barometer measures atmospheric pressure by the height of a mercury column; at sea level and 0 C, the standard atmosphere supports a column of 760 mm Hg. A manometer measures the pressure of a gas sample relative to atmospheric pressure. In an open-end manometer, the gas pressure equals atmospheric pressure plus or minus the height difference of the mercury columns, depending on which side is higher.
III. The Simple Gas Laws
A. Boyle's Law (P-V Relationship at Constant T and n)
Boyle's law states that at constant temperature and amount, pressure and volume are inversely proportional: P1 V1 = P2 V2. As volume decreases, pressure increases (compression), and vice versa. Graphically, P versus V traces a hyperbola, while P versus 1/V yields a straight line through the origin.
B. Charles's Law (V-T Relationship at Constant P and n)
Charles's law states that at constant pressure and amount, volume is directly proportional to absolute temperature: V1/T1 = V2/T2, where temperature must be expressed in Kelvin. As temperature increases, volume increases. Extrapolating the V-T relationship to zero volume defines absolute zero (0 K = -273.15 C). Gas law calculations must always use Kelvin.
C. Avogadro's Law (V-n Relationship at Constant T and P)
Avogadro's law states that at constant temperature and pressure, volume is directly proportional to the number of moles: V1/n1 = V2/n2. This means that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. At standard temperature and pressure (STP: 0 C = 273.15 K, 1 atm), one mole of any ideal gas occupies 22.4 L, a quantity known as the molar volume.
IV. The Ideal Gas Law
Combining Boyle's, Charles's, and Avogadro's laws yields the ideal gas law: PV = nRT, where P is pressure in atm, V is volume in liters, n is moles, T is temperature in Kelvin, and R is the ideal gas constant (0.08206 Latm/(molK) or 8.314 J/(mol*K)). This equation describes the behavior of an ideal gas -- a hypothetical gas that perfectly obeys the law under all conditions. Real gases approximate ideal behavior at high temperatures and low pressures.
<image>A four-panel graphic summarizing the gas laws. Panel A (Boyle's Law): a graph of P vs. V showing a hyperbolic curve at constant T, with two states (P1,V1) and (P2,V2) marked; beside it, a piston diagram showing compression. Panel B (Charles's Law): a graph of V vs. T(K) showing a straight line through the origin at constant P, with T extrapolated to 0 K; beside it, a balloon expanding with heat. Panel C (Avogadro's Law): a graph of V vs. n showing a straight line through the origin; beside it, two containers of equal volume at same T and P containing equal numbers of molecules of different gases. Panel D (Combined -- Ideal Gas Law): the equation PV = nRT displayed prominently in a box, with R = 0.08206 Latm/(molK) and the derivation path from the three simple laws shown with arrows.</image>
V. Combined Gas Law
For a fixed amount of gas (constant n), the ideal gas law reduces to the combined gas law: (P1 V1) / T1 = (P2 V2) / T2. This form is useful when pressure, volume, and temperature all change simultaneously. When any one variable is held constant, the combined law reduces to the appropriate simple gas law.
VI. Gas Density and Molar Mass
By rearranging PV = nRT and substituting n = m/M (where m is mass and M is molar mass), one obtains d = PM / (RT), where d is the gas density in g/L. Gas density is directly proportional to molar mass and pressure, and inversely proportional to temperature. Rearranging to solve for molar mass gives M = dRT / P. At STP, the density of any ideal gas simplifies to d = M / 22.4 (in g/L).
VII. Stoichiometry with Gases
Gas stoichiometry uses PV = nRT to convert between volume and moles, after which standard stoichiometric ratios apply. At STP, the molar volume (22.4 L/mol) provides a convenient shortcut. For example, to find how many liters of O2 at STP are needed to burn 16.0 g of CH4 in the reaction CH4 + 2 O2 -> CO2 + 2 H2O, first convert mass to moles: 16.0 g / 16.04 g/mol = 0.998 mol CH4. The stoichiometric ratio gives 0.998 x 2 = 1.996 mol O2, which occupies 1.996 x 22.4 = 44.7 L at STP.
VIII. Dalton's Law of Partial Pressures
Dalton's law states that the total pressure of a gas mixture equals the sum of the partial pressures of each component: P_total = P_1 + P_2 + P_3 + ... The partial pressure of a gas is the pressure it would exert if it alone occupied the container, calculated from the ideal gas law applied to that component: P_i = n_i RT / V. The mole fraction (chi_i) is defined as chi_i = n_i / n_total = P_i / P_total, so that P_i = chi_i P_total. The sum of all mole fractions in a mixture always equals 1.
IX. Collecting Gases over Water
When a gas is collected by displacing water in an inverted container, the collected gas is inevitably mixed with water vapor. The total pressure therefore equals the sum of the gas pressure and the vapor pressure of water: P_total = P_gas + P_water. To find the pressure of the dry gas, subtract the vapor pressure of water: P_gas = P_total - P_water. The vapor pressure of water depends on temperature and must be looked up in a reference table. This correction is essential for accurately calculating the moles of collected gas using the ideal gas law.
<image>A diagram of gas collection over water. A reaction flask on the left produces gas that travels through tubing into an inverted bottle filled with water in a trough. The gas displaces water and collects at the top of the inverted bottle. Labels indicate: reaction flask, gas delivery tube, water trough, inverted collection bottle, collected gas (mixture of product gas + water vapor), and water level. A callout box shows the equation P_gas = P_atm - P_H2O, with a note that water levels inside and outside the bottle must be equalized for P_total = P_atm. A small table of P_H2O values at different temperatures (20 C = 17.5 torr, 25 C = 23.8 torr, 30 C = 31.8 torr) is shown.</image>
