Grade 11 : Physics - Kinetic Theory (Set 1)

Explanation: The average translational kinetic energy per molecule of an ideal gas is given by $E_k = \frac{3}{2} k_B T$, where $k_B$ is Boltzmann's constant and $T$ is the absolute temperature. Thus, it depends solely on absolute temperature.

Explanation: The root mean square speed of ideal gas molecules is defined as $v_{rms} = \sqrt{\frac{3RT}{M}}$, where $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass.

Explanation: Absolute zero is 0 K, which corresponds to $-273.15^\circ\text{C}$ on the Celsius scale ($T(\text{K}) = t(^\circ\text{C}) + 273.15$).

Explanation: A monoatomic gas molecule can move freely in three independent spatial directions (x, y, z), so it has 3 translational degrees of freedom.

Explanation: A rigid diatomic molecule has 3 translational degrees of freedom and 2 rotational degrees of freedom (rotation about two axes perpendicular to the interatomic axis), giving total degrees of freedom $f = 3 + 2 = 5$.

Explanation: The ideal gas state equation combines Boyle's law, Charles's law, and Avogadro's law into $PV = nRT$.

Explanation: According to kinetic theory, pressure exerted by gas molecules on the container walls is $P = \frac{1}{3} \rho v_{rms}^2$.

Explanation: Since $v_{rms} \propto \sqrt{T}$, quadrupling temperature increases speed by $\sqrt{4} = 2$ times (it doubles).

Explanation: For 1 mole of monoatomic gas with $f = 3$ degrees of freedom, internal energy $U = \frac{f}{2} RT = \frac{3}{2} RT$.

Explanation: Kinetic theory assumes collisions are perfectly elastic, meaning total kinetic energy and momentum are conserved during collisions.

Explanation: Boltzmann constant $k_B = R / N_A$ has energy per unit temperature units, which is Joules per Kelvin ($\text{J/K}$).

Explanation: The law of equipartition states that total energy is equally distributed among all degrees of freedom, each contributing $\frac{1}{2} k_B T$ per molecule.

Explanation: For monoatomic gas, $C_v = \frac{3}{2} R$ and $C_p = \frac{5}{2} R$. Thus $\gamma = \frac{C_p}{C_v} = \frac{5/2}{3/2} = \frac{5}{3} \approx 1.67$.

Explanation: Mayer's relation states $C_p - C_v = R$ for an ideal gas, showing specific heat at constant pressure exceeds that at constant volume by gas constant $R$.

Explanation: Mean free path is $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$, where $n$ is number density (molecules per unit volume) and $d$ is molecular diameter. Thus $\lambda \propto \frac{1}{n}$.

Explanation: Total kinetic energy $E = \frac{3}{2} n R T = \frac{3}{2} \times 1 \times 8.314 \times 300 = 1.5 \times 2494.2 = 3741.3\text{ J}$.

Explanation: At constant volume, Gay-Lussac's Law states $P \propto T$. Doubling absolute temperature $T$ doubles pressure $P$.

Explanation: Dalton's law states total pressure is the sum of partial pressures only when gases do not react chemically with each other.

Explanation: From $P = \frac{\rho R T}{M}$, pressure $P$ is directly proportional to density $\rho$ at constant temperature. Doubling density doubles pressure.

Explanation: Mean free path $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$ depends only on molecular density $n = N/V$. At constant volume, $n$ is constant, so $\lambda$ remains unchanged when temperature changes.