Using the ideal gas law, find an expression for the mean free path ℓ_M that involves pressure and temperature instead of (N/V). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K.

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Start with the formula for the mean free path: ℓ_M = (k_B * T) / (√2 * π * d^2 * (N/V)), where k_B is the Boltzmann constant, T is the temperature, d is the diameter of the gas molecules, and (N/V) is the number density of molecules.

Use the ideal gas law, PV = Nk_B * T, to express (N/V) in terms of pressure (P) and temperature (T). Rearrange the ideal gas law to get (N/V) = P / (k_B * T).

Substitute (N/V) = P / (k_B * T) into the mean free path formula. This gives ℓ_M = (k_B * T) / (√2 * π * d^2 * (P / (k_B * T))).

Simplify the expression by canceling terms. The final expression for the mean free path becomes ℓ_M = (k_B * T^2) / (√2 * π * d^2 * P).

To calculate the mean free path for nitrogen molecules at 7.5 atm and 300 K, substitute the given values into the derived formula. Use the Boltzmann constant (k_B = 1.38 × 10^-23 J/K), the diameter of nitrogen molecules (d ≈ 3.7 × 10^-10 m), and convert pressure to Pascals (1 atm = 1.013 × 10^5 Pa).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. It is expressed as PV = nRT, where R is the ideal gas constant. This law allows us to derive relationships between these variables, which is essential for understanding gas behavior under various conditions.

Mean Free Path

The mean free path (ℓ_M) is the average distance a molecule travels between collisions with other molecules. It is influenced by factors such as the density of the gas and the size of the molecules. Understanding mean free path is crucial for analyzing gas behavior, especially in kinetic theory and applications involving gas dynamics.

Kinetic Theory of Gases

The kinetic theory of gases provides a molecular-level interpretation of gas behavior, explaining properties like pressure and temperature in terms of molecular motion. It posits that gas pressure arises from collisions of molecules with the walls of a container. This theory is foundational for deriving expressions related to mean free path and other gas properties, linking macroscopic observations to microscopic behavior.