Gas Laws and Kinetic Molecular Theory (4B) - MCAT Chemical and Physical Foundations of Biological Systems

$R = 8.314\ \text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$. This value of $R$ is employed in the ideal gas law when energy units (joules) are required.

$T(\text{K}) = T(^\circ\text{C}) + 273.15$. Absolute temperature in Kelvin is obtained by adding 273.15 to the Celsius temperature to align with the ideal gas law scale.

$T = 273.15\ \text{K},\ P = 1\ \text{atm}$. STP conditions provide a reference point for gas properties, with temperature at freezing point of water in Kelvin and pressure at sea level.

$P_i = x_i P_{\text{total}}$. Partial pressure of a gas in a mixture is its mole fraction times the total pressure, assuming ideal behavior.

$22.4\ \text{L mol}^{-1}$. At STP, one mole of ideal gas occupies this volume, derived from the ideal gas law with $P=1$ atm and $T=273$ K.

$P_{\text{total}} = \sum_i P_i$. Dalton's law states that in a mixture of non-reacting gases, total pressure equals the sum of each gas's partial pressure.

$P_{\text{He}} = 1\ \text{atm}$. Partial pressure equals mole fraction times total pressure in an ideal gas mixture per Dalton's law.

$P_1V_1 = P_2V_2$. Boyle's law describes the inverse relationship between pressure and volume for a fixed amount of gas at constant temperature.

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$. Charles's law indicates that volume is directly proportional to absolute temperature for a gas at constant pressure and moles.

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$. Gay-Lussac's law shows that pressure is directly proportional to absolute temperature for a gas at constant volume and moles.

$\frac{V_1}{n_1} = \frac{V_2}{n_2}$. Avogadro's law states that volume is directly proportional to the number of moles for a gas at constant pressure and temperature.

$PV = nRT$. The ideal gas law combines relationships among pressure, volume, moles, and temperature for an ideal gas using the gas constant $R$.

$R = 0.08206\ \text{L}\cdot\text{atm}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$. This value of $R$ is used when pressure is in atm, volume in L, and temperature in K for the ideal gas law.

$n \approx 0.41\ \text{mol}$. Solving the ideal gas law for $n$ gives moles as pressure times volume over $R$ times temperature.

$P_2 = 1.0\ \text{atm}$. By Gay-Lussac's law, pressure decreases with the ratio of temperatures at constant volume.

$V_2 = 6\ \text{L}$. Using Charles's law, volume increases proportionally with the ratio of temperatures at constant pressure.

$V_2 = 6\ \text{L}$. Applying Boyle's law, volume doubles when pressure halves at constant temperature.

High $P$ and low $T$. Real gases deviate from ideality when intermolecular forces and particle volume become significant under high pressure and low temperature.

Negligible particle volume; no intermolecular attractions or repulsions. Ideal gas assumptions simplify behavior by treating particles as point masses with no volume and no interactions except elastic collisions.

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$. Graham's law indicates that diffusion or effusion rates are inversely proportional to the square root of molar masses.

$u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$. Root-mean-square speed measures the square root of the average of squared speeds, depending on temperature and molar mass $M$.

$\langle KE \rangle_{\text{mol}} = \frac{3}{2}RT$. For one mole, average kinetic energy relates to temperature via the gas constant $R$, derived from per-molecule kinetic energy.

$\langle KE \rangle = \frac{3}{2}k_BT$. Kinetic molecular theory posits that average kinetic energy per molecule is directly proportional to absolute temperature, with $k_B$ as Boltzmann's constant.

$x_i = \frac{n_i}{n_{\text{total}}}$. Mole fraction represents the ratio of moles of one component to total moles in the mixture.