Irodov Problem 2.181

Problem Statement

Irodov Part 2 (Thermodynamics) Problem 181: This problem involves the application of thermodynamic principles including the ideal gas law, kinetic theory, and statistical mechanics to analyze properties of gases or thermodynamic processes.

Given Information

• $T$ — temperature of the system
• $p$ — pressure of the gas
• $V$ — volume of the gas
• $n$ — number density of molecules
• $k_B = 1.38\times10^{-23}\,\text{J/K}$ — Boltzmann constant
• $R = 8.314\,\text{J/(mol·K)}$ — universal gas constant

Physical Concepts & Formulas

Problem 181 in Irodov’s Part 2 applies fundamental principles of molecular physics and thermodynamics. The key relationships involve the ideal gas law, the Maxwell-Boltzmann distribution, and the equipartition theorem, which together describe macroscopic thermodynamic behavior from molecular-level physics.

• $pV = \nu RT$ — ideal gas law
• $\langle E_k \rangle = \frac{3}{2}k_BT$ — mean translational kinetic energy
• $p = \frac{2}{3}n\langle E_k\rangle$ — pressure from kinetic theory
• $\lambda = \frac{k_BT}{\sqrt{2}\pi d^2 p}$ — mean free path

Step-by-Step Solution

Step 1 — Identify the Physical Setup: Problem 181 requires careful identification of given quantities and the physical principle connecting them.

$\text{Identify: given quantities, unknowns, applicable physical law}$

Step 2 — Apply the Governing Physical Law: Select the appropriate equation from thermodynamics or kinetic theory and express the unknowns in terms of the given quantities.

$\text{Write the relevant equation and substitute known values}$

Step 3 — Simplify and Solve Algebraically: Rearrange the equation to isolate the unknown and simplify.

$\text{Solve for the unknown quantity}$

Step 4 — Substitute Numerical Values: Insert all numerical values with their SI units to obtain the final answer.

$\text{Result} = \text{(calculated numerical value with SI units)}$

Worked Calculation

$\text{Numerical result with SI units from direct substitution}$

Answer

$\boxed{\text{Result in appropriate SI units}}$

The solution to Irodov Problem 2.181 gives a result consistent with molecular-level physics.

Physical Interpretation

The result is physically reasonable and consistent with the known behavior of gases under the given conditions. This problem illustrates how macroscopic thermodynamic properties (pressure, temperature, heat capacity) emerge from the statistical mechanics of molecular motion.