Verified Solution • 2025–26 Session moderate

QPages 419-421: Q1: HC Verma: Concepts of Physics - Volume 2 Semiconductors and Semiconductor Devices Pages 419-421: Q1 Solution: Calculate the number of states per cubic metre of sodium in 3s…

Calculate the number of states per cubic metre of sodium in 3s band. The density of sodium is 1013 kg m−310^{13} \text{ kg m}^{-3}. How many of them are empty?

Step-by-Step Verified Solution

**Answer:** Total states in 3s band = 5.3 _ 10__ per m_. Number of empty states = 2.65 _ 10__ per m_. **Step-by-Step Explanation:** Density of sodium _ = 1013 kg/m_. Molar mass = 23 g/mol = 0.023 kg/mol. Number of atoms per m_:
n=ρNAM=1013×6.02×10230.023=2.65×1028 atoms/m3n = \frac{\rho N_A}{M} = \frac{1013 \times 6.02 \times 10^{23}}{0.023} = 2.65 \times 10^{28} \text{ atoms/m}^3
Each atom contributes 2 states to the 3s band (spin up and spin down):
Total states=2n=5.3×1028 per m3\text{Total states} = 2n = 5.3 \times 10^{28} \text{ per m}^3
Sodium has one 3s electron per atom, so the number of filled states = 2.65 _ 10__. Empty states = total - filled = 5.3 _ 10__ - 2.65 _ 10__ = 2.65 _ 10__ per m_ .

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