Density of states: Approximately how many states in the range from 5.0 eV to 5.2 eV are there in a copper bar of volume 5.3 cm³? (h = 6.626 × 10^-34 J ∙ s, ħ = 1.055 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19 J) A) 5.1 × 10²² B) 3.2 × 10²² C) 1.6 × 10²² D) 8.2 × 10²¹ E) 3.2 × 10²¹

Question

Quizplus · Accepted Answer

The density of states is given by:

$$D(E)=\frac{2	imes V	imes\sqrt{2m_{	ext{Cu}(s)}}}{h^3}\sqrt{E}$$

Substituting the given values:

$$D(E)=\frac{2	imes (5.3	imes 10^{-6}	ext{ m}^3)	imes\sqrt{2	imes 9.11	imes 10^{-31}	ext{ kg}}}{(6.626	imes 10^{-34}	ext{ J}\cdot	ext{s})^3}\sqrt{(5.0	ext{ eV}	imes 1.60	imes 10^{-19}	ext{ J/eV})}\approx 2.4	imes 10^{28}	ext{ J}^{-1}$$

The number of states is given by:

$$N=\int_{5.0	ext{ eV}}^{5.2	ext{ eV}} D(E)	ext{ d}E$$

Substituting the limits of integration and the value of D(E):

$$N=\int_{5.0	ext{ eV}}^{5.2	ext{ eV}} 2.4	imes 10^{28}	ext{ J}^{-1}	ext{ d}E\approx 1.6	imes 10^{54}$$

Rounding off to the nearest order of magnitude gives the answer as $1.6	imes 10^{54}$, which corresponds to choice C.

Density of States: Approximately How Many States in the Range