Question

In a dilute gas, the number of molecules with free path length ≥ x is given by N(x) = N₀e^-x/λ, where N₀ is the total number of molecules and A is the mean free path. The fraction of molecules with free path lengths between λ and 2λ is

• A. \(\frac{1}{e}\)
• B. \(\frac{e}{e-1}\)
• C. \(\frac{e^2}{e-1}\)
• D. \(\frac{e-1}{e^2}\)

Answer 1

The Correct Option is D

To find the fraction of molecules with free path lengths between \(\lambda\) and \(2\lambda\), we need to evaluate the expression for \(N(x)\) given by:

\(N(x) = N_0 e^{-x/\lambda}\)

where \(N_0\) is the total number of molecules, and \(\lambda\) is the mean free path.

The number of molecules with free path lengths between \(\lambda\) and \(2\lambda\) can be found by determining the difference \(N(\lambda) - N(2\lambda)\).

1. Calculate \(N(\lambda)\):

\(N(\lambda) = N_0 e^{-\lambda/\lambda} = N_0 e^{-1}\)

1. Calculate \(N(2\lambda)\):

\(N(2\lambda) = N_0 e^{-2\lambda/\lambda} = N_0 e^{-2}\)

1. The number of molecules with path lengths between \(\lambda\) and \(2\lambda\) is:

\(N(\lambda) - N(2\lambda) = N_0 e^{-1} - N_0 e^{-2}\)

\(= N_0 (e^{-1} - e^{-2})\)

1. The fraction of molecules with path lengths between \(\lambda\) and \(2\lambda\) is:

\(\frac{N_0 (e^{-1} - e^{-2})}{N_0} = e^{-1} - e^{-2}\)

Simplify: \(e^{-1} - e^{-2} = \frac{1}{e} - \frac{1}{e^2} = \frac{e - 1}{e^2}\)

Thus, the fraction of molecules with free path lengths between \(\lambda\) and \(2\lambda\) is \(\frac{e-1}{e^2}\).