Class 11 Thermodynamics Physics

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1. Introduction to Kinetic Theory of Gases (KTG) Purpose: Explains the macroscopic properties of gases (pressure, temperature) based on the microscopic behavior of their constituent particles (atoms/molecules). Foundation: Relates the motion and energy of gas molecules to temperature and pressure. 2. Postulates of Kinetic Theory of Gases Gases consist of a large number of identical, tiny particles (atoms or molecules) called molecules. Molecules are in a state of continuous, random motion, moving in all possible directions with all possible velocities. The size of the molecules is negligible compared to the average distance between them and the volume of the container. There are no intermolecular forces of attraction or repulsion between gas molecules, except during collisions. Collisions between molecules and with the walls of the container are perfectly elastic (no loss of kinetic energy). The duration of a collision is negligible compared to the time between successive collisions. The motion of molecules is governed by Newton's laws of motion. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature ($T$) of the gas. 3. Ideal Gas Equation Equation: $PV = nRT$ $P$: Pressure (Pa) $V$: Volume ($\text{m}^3$) $n$: Number of moles $R$: Universal Gas Constant ($8.314 \text{ J mol}^{-1} \text{ K}^{-1}$ or $0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}$) $T$: Absolute Temperature (K) In terms of number of molecules: $PV = NkT$ $N$: Total number of molecules $k$: Boltzmann Constant ($1.38 \times 10^{-23} \text{ J K}^{-1}$) Relationship: $R = N_A k$, where $N_A$ is Avogadro's number ($6.022 \times 10^{23} \text{ mol}^{-1}$). In terms of mass: $PV = \frac{m}{M}RT$, where $m$ is mass of gas, $M$ is molar mass. 4. Pressure Exerted by an Ideal Gas Formula: $P = \frac{1}{3} \frac{Nm}{V} v_{rms}^2 = \frac{1}{3} \rho v_{rms}^2$ $N$: Total number of molecules $m$: Mass of one molecule $V$: Volume of the container $\rho = \frac{Nm}{V}$: Density of the gas $v_{rms}$: Root Mean Square speed of the gas molecules. Derivation Insight: Pressure arises from the elastic collisions of gas molecules with the container walls. 5. Speeds of Gas Molecules Root Mean Square Speed ($v_{rms}$): Definition: Square root of the mean of the squares of the speeds of individual molecules. $v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + ... + v_N^2}{N}}$ Formula: $v_{rms} = \sqrt{\frac{3RT}{M_0}} = \sqrt{\frac{3kT}{m}}$ $R$: Universal Gas Constant $T$: Absolute Temperature (K) $M_0$: Molar mass (in kg/mol for $R$) $k$: Boltzmann Constant $m$: Mass of one molecule (in kg for $k$) Average Speed ($\bar{v}$ or $v_{avg}$): Definition: Arithmetic mean of the speeds of individual molecules. $\bar{v} = \frac{v_1 + v_2 + ... + v_N}{N}$ Formula: $\bar{v} = \sqrt{\frac{8RT}{\pi M_0}} = \sqrt{\frac{8kT}{\pi m}}$ Most Probable Speed ($v_p$): Definition: Speed possessed by the maximum number of molecules in a gas sample. Formula: $v_p = \sqrt{\frac{2RT}{M_0}} = \sqrt{\frac{2kT}{m}}$ Relationship between Speeds: $v_p : \bar{v} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$ Order: $v_p 6. Kinetic Energy of a Gas Average Kinetic Energy per molecule: Per molecule, for any gas (monoatomic, diatomic, polyatomic): $\bar{E} = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} kT$ This is valid only for translational kinetic energy. It implies that average kinetic energy per molecule depends only on absolute temperature, not on the nature of the gas. Total Translational Kinetic Energy of $N$ molecules: $E_{total} = N \left(\frac{3}{2} kT\right) = \frac{3}{2} NkT = \frac{3}{2} nRT$ Internal Energy ($U$) for an Ideal Gas: For a monoatomic gas (only translational KE): $U = \frac{3}{2} nRT$ For diatomic and polyatomic gases, rotational and vibrational energies also contribute to internal energy. 7. Degrees of Freedom ($f$) Definition: The number of independent ways in which a molecule can possess energy (translational, rotational, vibrational). Monoatomic Gas (e.g., He, Ne, Ar): $f=3$ (3 translational: along x, y, z axes). No practical rotation or vibration. Diatomic Gas (e.g., O$_2$, N$_2$, H$_2$): At room temperature: $f=5$ (3 translational, 2 rotational). At very high temperatures, vibrational modes become active, $f=7$ (3 translational, 2 rotational, 2 vibrational). Polyatomic Gas (e.g., CO$_2$, CH$_4$): Non-linear: $f=6$ (3 translational, 3 rotational). Linear (like CO$_2$): $f=5$ (3 translational, 2 rotational) similar to diatomic, but often considered $f=6$ for general polyatomic. Vibrational modes also contribute at higher temperatures. 8. Law of Equipartition of Energy Statement: For a system in thermal equilibrium, the total energy is equally distributed among its degrees of freedom, and the average energy associated with each degree of freedom is $\frac{1}{2} kT$ per molecule (or $\frac{1}{2} RT$ per mole). Total Internal Energy of $n$ moles: $U = n \left(\frac{f}{2} RT\right)$ Total Internal Energy of $N$ molecules: $U = N \left(\frac{f}{2} kT\right)$ 9. Molar Specific Heats from KTG Molar Specific Heat at Constant Volume ($C_V$): Using First Law: $dU = dQ - dW$. At constant volume, $dW=0$, so $dU = dQ = nC_VdT$. Thus, $C_V = \frac{1}{n} \left(\frac{dU}{dT}\right)_V$. From Equipartition: $U = n \frac{f}{2} RT \implies C_V = \frac{f}{2} R$. Molar Specific Heat at Constant Pressure ($C_P$): Using Mayer's Relation: $C_P = C_V + R$. So, $C_P = \frac{f}{2} R + R = \left(\frac{f}{2} + 1\right) R$. Ratio of Specific Heats ($\gamma$): $\gamma = \frac{C_P}{C_V} = \frac{(f/2 + 1)R}{(f/2)R} = 1 + \frac{2}{f}$. Summary Table for Ideal Gases: Gas Type $f$ $C_V$ $C_P$ $\gamma = C_P/C_V$ Monoatomic 3 $\frac{3}{2} R$ $\frac{5}{2} R$ $\frac{5}{3} \approx 1.67$ Diatomic (rigid rotator, room T) 5 $\frac{5}{2} R$ $\frac{7}{2} R$ $\frac{7}{5} = 1.4$ Polyatomic (non-linear, room T) 6 $3 R$ $4 R$ $\frac{4}{3} \approx 1.33$ 10. Mean Free Path ($\lambda$) Definition: The average distance traveled by a molecule between two successive collisions. Formula: $\lambda = \frac{1}{\sqrt{2} \pi d^2 n_V}$ $d$: Diameter of the gas molecule. $n_V = N/V$: Number density of molecules (number of molecules per unit volume). Alternative Formula (using $P, T$): $\lambda = \frac{kT}{\sqrt{2} \pi d^2 P}$ Dependencies: $\lambda \propto T$ (at constant pressure) $\lambda \propto \frac{1}{P}$ (at constant temperature) $\lambda \propto \frac{1}{d^2}$ 11. Important Concepts for Exams (School & Competitive) Postulates of KTG: Be able to list and explain them, especially the assumptions about molecular size, forces, and collisions. Ideal Gas Equation: Master its application in various problems, including changes in $P, V, T$. Pressure Formula: Understand the derivation conceptually and its dependence on density and $v_{rms}$. All Three Speeds ($v_{rms}, \bar{v}, v_p$): Memorize all three formulas and their inter-relationships. Understand that they all depend on $\sqrt{T/M_0}$ (or $\sqrt{T/m}$). Be able to compare their magnitudes. Average Kinetic Energy per Molecule: $\frac{3}{2} kT$. This is a fundamental result and frequently tested. Emphasize its independence from gas type and dependence only on $T$. Degrees of Freedom: Correctly identify 'f' for monoatomic, diatomic, and polyatomic gases at different temperature ranges (especially room temperature). This is crucial for specific heat calculations. Law of Equipartition of Energy: Understand its statement and how it leads to the internal energy formula $U = n \frac{f}{2} RT$. Molar Specific Heats ($C_V, C_P$) and $\gamma$: Know the formulas for $C_V, C_P, \gamma$ in terms of $f$. Be able to calculate these values for different types of gases. Mayer's relation ($C_P - C_V = R$) is essential. Mean Free Path: Understand its physical meaning and how it depends on temperature, pressure, and molecular diameter. Maxwell-Boltzmann Distribution: While the exact curve might not be in Class 11, understand its qualitative features: As $T$ increases, the peak shifts to higher speeds, and the curve broadens and flattens. It shows the distribution of speeds, not just a single value. Numerical Problems: Practice numericals involving all the formulas, especially those combining KTG concepts with Ideal Gas Law and Thermodynamics (e.g., calculating $v_{rms}$ and then $\Delta U$ for an adiabatic process).