Chapter 15: Q3P (page 765)

The probability density function for the component of the $x$ velocity of a molecule of an ideal gas is proportional torole="math" localid="1665039876656" $e^{\frac{- m v^{2}}{(2 k T)}}$ where $v$ is the $x$ component of the velocity, $m$ is the mass of the molecule, $T$ is the temperature of the gas androle="math" localid="1665040106251" $k$ is the Boltzmann constant. By comparing this with (8.1), find the mean and standard deviation of $v$ , and write the probability density function $f (v)$ .

Short Answer

Expert verified

The probability density function is $\frac{e^{\frac{- m v^{2}}{(2 k T)}}}{\sqrt{\frac{2 πkT}{m}}}$ .

Step by step solution

Given information

The velocity of a molecule of an ideal gas is proportional to $e^{\frac{- m v^{2}}{(2 k T)}}$ .

Definition of variance

A statistical measurement of the dispersion between values in a data collection is known as variance.

Find the average of the function

The function is $f (v) \infty e^{\frac{- m v^{2}}{(2 k T)}}$ …….(1)

role="math" localid="1665042441825" $f (v) = c e^{\frac{- m v^{2}}{(2 k T)}} f (x) = \frac{1}{σ \sqrt{2 π}} e^{\frac{- {(x - μ)}^{2}}{(2 σ^{2})}}$ ……………..(2)

Compare (1) and (2)

$\begin{array}{rcl} - \frac{1}{2 σ^{2}} & = & - \frac{m}{2 k T} \\ σ & = & \sqrt{\frac{k T}{m}} \end{array}$

The value of average is $\begin{array}{rcl} μ & = & 0 \end{array}$ .

Substitute in (1).

$\begin{array}{rcl} f (v) & = & \frac{e^{\frac{- m v^{2}}{(2 k T)}}}{\sqrt{\frac{2 πkT}{m}}} \end{array}$

The variance is $\begin{array}{rcl} σ & = & \sqrt{\frac{k T}{m}} \end{array}$ .

The average is $\begin{array}{rcl} μ & = & 0 \end{array}$ .

The probability density function is $\begin{array}{rcl} f (v) & = & \frac{e^{\frac{- m v^{2}}{(2 k T)}}}{\sqrt{\frac{2 π k T}{m}}} \end{array}$ .

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Short Answer

Step by step solution

Given information

Definition of variance

Find the average of the function

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