Chapter 6: Q. 6.52 (page 256)

Consider an ideal gas of highly relativistic particles ( such as photons or fast-moving electrons) whose energy-momentum relation is $E = p c$ instead of $E = \frac{p^{2}}{2 m}$ . Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single particle partition function, $Z_{1}$ , for one particle in the gas.

Short Answer

Expert verified

The partition function is given by $Z = \frac{2 L k T}{h c}$ .

Step by step solution

Step 1. Given information

The energy of the particle is given by

$E = p c . . . . . . . . . . . . (1)$

Step 2. Calculation

The allowed values of the energy for the gas molecule is given by

$E_{n} = \frac{h c n}{2 L} . . . . . . . . . . . . . . . . . (2)$

The formula to calculate the single particle partition function is given by

$Z_{1} = \sum_{n} e^{- \frac{E_{n}}{k T}} . . . . . . . . . . . . . . . (3)$

Substitute the value of the allowed energy from equation (2) into equation (3) and transferring the summation by integral solve to calculate the required partition function of the gas.

$\begin{array}{rcl} Z_{1} & = & \int_{0}^{\infty} e^{- \frac{h c n}{2 L k T}} d n \\ = & - \frac{2 L k T}{h c} {[e^{- \frac{h c n}{2 L k T}}]}_{0}^{\infty} \\ = & \frac{2 L k T}{h c} \end{array}$

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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