B.3 Gas laws

Solution outline

1. $F = mg = 24\times9.8 = 235\ \mathrm{N}$
2. $P = \dfrac{F}{A} = \dfrac{235}{0.32} \approx 7.3\times10^{2}\ \mathrm{Pa}$
3. $\text{This value represents the force per unit area on the floor, which is the definition of pressure.}$

Solution outline

1. $n = \dfrac{N}{N_A}$
2. $n = \dfrac{4.0\times10^{24}}{6.02\times10^{23}} \approx 6.6\ \text{mol}$
3. $\text{Here } N_A \text{ is Avogadro’s constant, the number of particles in 1 mol.}$

Solution outline

1. $\text{Convert to SI units: } V = 2.5\times10^{-3}\ \mathrm{m^{3}},\ P = 1.50\times10^{5}\ \mathrm{Pa}.$
2. $PV = nRT \Rightarrow n = \dfrac{PV}{RT}$
3. $n = \dfrac{1.50\times10^{5}\times2.5\times10^{-3}}{8.31\times300} \approx 0.15\ \text{mol}$

Solution outline

1. $\text{In kinetic theory, temperature is proportional to the mean kinetic energy of the molecules.}$
2. $\text{Higher temperature } \Rightarrow \text{higher average speed } c \Rightarrow \text{larger } \langle c^{2} \rangle.$
3. $P = \dfrac{1}{3}\,\rho \langle c^{2} \rangle \Rightarrow \text{greater } \langle c^{2} \rangle \text{ gives greater pressure at the same density.}$

Solution outline

1. $U = \tfrac{3}{2}nRT$
2. $U = \tfrac{3}{2}\times0.80\times8.31\times450 \approx 4.5\times10^{3}\ \mathrm{J}$
3. $\text{For a monatomic ideal gas, internal energy is purely translational kinetic energy, which depends only on temperature, not on volume or pressure.}$

Solution outline

1. $\text{A real gas behaves approximately ideally at relatively high temperature, low pressure and low density.}$
2. $\text{At low density and low pressure, molecules are far apart so intermolecular forces are negligible except during brief collisions.}$
3. $\text{This matches the kinetic theory assumption that there are no significant forces between molecules and that collisions are effectively elastic.}$