Fundamental Concepts in Physical Chemistry and Introductory Statistics

Atomic and Molecular Masses

Atomic Weight and Atomic Mass

Atomic weight is defined as the ratio of the average mass of an element's atom to 1/12 the mass of the isotope 12C. The term relative atomic mass is often used interchangeably with atomic weight.

• Atomic mass: The average mass of an atom, typically expressed in atomic mass units (amu).

• Example: Atomic weight of 12C = 12

Molecular Weight and Molecular Mass

Molecular weight is the ratio of the average mass of molecules to 1/12 the mass of 12C. It is also referred to as relative molecular mass.

• Molecular mass: The average mass of a molecule, usually in amu.

• Example: Molecular weight of H2O = 18.015

Average Mass and Formula Units

The term "average mass" is used to account for the existence of naturally occurring isotopes. For ionic compounds, the molecular weight is based on the mass of the formula unit.

• Example: Molecular weight of NaCl = 58.443

• No individual NaCl molecules exist in a crystal; the formula unit is used instead.

Dimensionless Numbers

Atomic and molecular weights are relative masses and are dimensionless.

Avogadro's Number and the Mole

Avogadro's Number

Avogadro's number is the number of 12C atoms in 12 grams of 12C.

• Avogadro's number: $6.02214076 \times 10^{23}$

The Mole

The mole is the SI unit for the amount of a substance, defined as containing Avogadro's number of elementary entities.

• Atomic mass: Average mass of an atom

• Molecular mass: Average mass of a molecule

• Atomic masses and molecular masses use atomic mass units (amu)

• 1 amu = 1/12 mass of a 12C atom

• Some biologists prefer "dalton" to amu units

• Example: Atomic mass of C = 12.011 amu

• Example: Molecular mass of H2O = 18.015 amu

Molar Mass and Mole Fractions

Molar Mass

Molar mass is the mass per mole of a pure substance.

• Formula: $M_i = m_i / n_i$

• $m_i$ = mass of substance i in sample

• $n_i$ = number of moles of i in sample

• Example: Molar mass of H2O: $M_{H_2O} = 18.015$ g/mol

• In most calculations, g/mol and amu are interchangeable

Mole Fraction

The mole fraction of a species is the ratio of the number of moles of that species to the total number of moles in the mixture.

• Formula: $x_i = n_i / n_{tot}$

• $n_i$ = moles of chemical species i

• $n_{tot}$ = total number of moles of all species

• Sum of mole fractions of all species: $x_1 + x_2 + x_3 + ... = 1$

Example Calculation

Given a solution of HCl in water that is 12.0% HCl by mass, the mole fractions of HCl and H2O can be calculated using the masses and molar masses of each component.

Gas Laws and Properties

Boyle's Law

Boyle's Law describes the relationship between pressure and volume for a fixed amount of gas at a constant temperature.

• Formula: $P V = k$ (where $k$ is a constant for a given amount of gas)

• Pressure and volume are inversely proportional

• Boyle's Law is approximate for real gases; deviations approach zero at the limit of zero pressure

Molecular Picture of Gases

Gases are composed of many noninteracting particles. Pressure is exerted by gas molecules colliding with container walls.

• If molecules interact, the gas is not ideal

• Decrease in volume increases collision frequency, raising pressure

Pressure and Units

Pressure is defined as force divided by area.

• Formula: $P = F / A$

• SI unit: Pascal (Pa), where $1 \text{ Pa} = 1 \text{ N} / \text{m}^2$

• Other units: kilopascal (kPa), megapascal (MPa), torr, bar

• Conversions:

Unit	Equivalent
1 atm	760 torr = 101325 Pa
1 bar	105 Pa = 0.986923 atm = 750.062 torr

Common Volume Units

• Cubic centimeter: cm3

• Cubic decimeter: dm3

• Cubic meter: m3

• Liter: 1 L = 1000 cm3

Charles's Law and Temperature Scales

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature at constant pressure and fixed amount of gas.

• Formula: $V = a_1 + a_2 \theta$ (where $a_1$ and $a_2$ are constants)

• In the limit of zero pressure, all gases show the same $V$ vs. $\theta$ behavior

Temperature Measurement and Scales

Absolute temperature scales are based on the properties of gases at the zero pressure limit.

• Kelvin scale: $T_K = t_{\text{Celsius}} + 273.15$

• Triple point of water used as reference: $T_{\text{TP}} = 273.16$ K

Ideal Gas Law and Related Equations

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and amount of gas.

• Formula: $P V = n R T$

• $R$ = universal gas constant

• Values of R:

Unit	Value
J / mol K	8.3145
cm3 atm / mol K	82.057

• Obeyed by real gases only in the limit of zero density

Other Forms of the Ideal Gas Law

• Using molar mass $M = m / n$:

• $P V = m R T / M$

• $M = m R T / P V$

• Using density $\rho = m / V$:

• $P V = m R T / M$

• $P M R T = m V$

• $P = P M R T$

Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure of a mixture of gases equals the sum of the partial pressures each gas would exert if it were alone in the container.

• Formula: $P = n_1 R T / V + n_2 R T / V + ...$

• Partial pressure of gas $i$: $P_i = x_i P$

• $x_i$ = mole fraction of gas $i$

Functions, Limits, and Slope

Functions and Limits

In mathematics, a function $y = f(x)$ expresses $y$ as a function of $x$. The limit of a function as $x$ approaches a value $c$ is the value the function approaches.

• Limit notation: $\lim_{x \to c} f(x)$

• As $x$ approaches $0$,$\sin(x)/x$approaches $1$

Slope

The slope of a straight line is defined as "rise over run" and is given by the equation of a straight line.

• Equation: $y = mx + b$

• $m$ = slope, $b$ = y-intercept

• The slope of a curve at point $P$ equals the slope of the tangent at $P$

Unit Conversions and Measurement

Unit Conversions

Conversions between units are essential in physical chemistry and statistics.

• 1.0 GPa = 109 Pa

• 1.0 bar = 105 Pa

• 1.0 atm = 101325 Pa

• 1.0 kg = 1000 g

• 1.0 cm3 = 1.0 mL

Measurement Units

• Know the measurements: kg, g, cm, m

Summary Table: Key Constants and Units

Constant/Unit	Value
Avogadro's Number	$6.02214076 \times 10^{23}$
Universal Gas Constant (J / mol K)	8.3145
Universal Gas Constant (cm3 atm / mol K)	82.057
1 atm	101325 Pa = 760 torr
1 bar	105 Pa = 0.986923 atm
1 L	1000 cm3

Additional info: Some content inferred and expanded for clarity, including definitions, formulas, and context for calculations and physical chemistry/statistics applications.