Foundations of Modern Physics

Textbook: Foundations of Modern Physics by Steven Weinberg

Ch1 Early atomic theory

1. Gas properties:
   1. Boyle(1627-1691)‘s law: For constant temperature the volume of a gas of fixed mass and composition is inversely proportional to the pressure; Charles’ law(1780s): At constant pressure and mass the volume of gas is proportional to $T-T_0$(absolute temperature).
   2. Theoretical explanations: Bernoulli(1700-1782) stated that in a gas with $n$ particles per unit volume moving isotropically(because of collisions) with a velocity $v$, the pressure is proportional to $n$ and $v^2$. It explains (1) Boyle’s law, if $v^2$ depends only on the temperature; (2) Charles’ law, if $v^2$ is further proportional to $T-T_0$.
      1. Clausius’ proof(1857): $F=\frac{2m{v}_{\perp }}{\delta T}$. Then the total force exerted on the wall(in time $T$) is $pS=FN\frac{\delta T}{T}=2nm{v}_{\perp }^{2}S$, the pressure being $p=2nm{v}_{\perp }^2$. Consider each of the two walls in one direction, $p=2\left(\frac{n}{2}\right)m\overline{v_i^2}$. Since $v^2=v_1^2+v_2^2+v_3^2$, $p=\frac{1}{3}nm\overline{v^2}$. (“but the theoretical explanation for Boyle’s law and Charles’ law had to wait for the development of kinetic theory and statistical mechanics”)
2. Chemistry:
   1. Elements.
   2. Law of combining weights: Dalton(1766-1844) discovered that when hydrogen burns in oxygen, $1$g of hydrogen combines with $5.5$g of oxygen, giving $6.5$g of water. If we know the compound’s formula, e.g. $\text{ce}{H}_{2}O$, we can calculate the atom weight (relative to hydrogen).
   3. Law of combining volumes: People observed chemical reaction of gases(constant T&P) proceeds in definite proportion of volumes. Avogadro(1776-1856) then stated that equal volumes of gases at the same temperature and pressure always contain equal numbers of the gas particles(“molecules”). e.g. The data shows that $2$L of hydrogen combined with $1$L of oxygen give $2$L of water vapour, implying the reaction and molecular formulas to be $\text{ce}2H2+O2->2H2O$.
   4. The gas constant: Clausius’ equation along with Charles’ law yields $p=n\frac{1}{3}m\overline{v^2}=nkT$. Let the number density $n=\frac{N}{V}$. We have $pV=NkT$(ideal gas law). Avogadro stated that the number of molecules in a gas with a given $p,V,T$ is the same for any gas(Avogadro’s principle). Then $k$ must be a constant(Boltzmann’s constant $k_B$). Besides, we can define the molecular weight $\mu =\frac{m}{m_1}$, where $m_1=\frac{1}{12}m(\text{ce}{}^{12}C)$. The law can be written as $pV=\frac{M}{m_1\mu }kT=\frac{M}{\mu }RT$, where the gas constant $R=\frac{k}{m_1}$.
   5. Amount of substance and Avogadro’s number: A mole of any element or compound of molecular weight $\mu$ is defined as $\mu$ grams. Then $N=\frac{M}{m_1\mu }=\frac{1}{m_1}n=:N_{\mathrm{A}}n$. Avogadro’s number $N_{\mathrm{A}}=6.02214076\times 10^{23}$.
3. Electrolysis:
   1. Early developments: Types of charges. Coulomb’s law $F=\frac{k_e{q}_1{q}_2}{r^2}$. Two poles of a lodestone(and the Earth). Volta’s electric batteries. “Turning on an electric currents deflect a nearby compass; Magnets exert force on wires carrying electric currents. “(Oersted) “Wires carrying electric current exert force on each other, $F=\frac{k_m{I}_1{I}_{2}L}{r}$. “(Ampère) “Changing magnetic fields generate electric forces that can drive currents in conducting wires. “(Faraday) Maxwell’s equations.
      1. Fun fact: $1$ ampere is defined as the magnitude of electric current which two parallel wires carry so that $k_m=10^{-7}\text{ce}N/A^2$. Then $1$ coulomb is the electric charge carried in $1\text{ce}s$ by a current of $1\text{ce}A$.
   2. Discovery of electrolysis: