Sure! Let's solve each part step by step. --- ### **Step 1: Find the Number Density of Electrons** **Given:** - Density, \(\rho = 1.4 \times 10^6 \, \text{g/cm}^3\) - The white dwarf is **pure carbon-12** (\({}^{12}\)C: 6 protons, 6 neutrons, 6 electrons per atom). **Find:** - Number density of **electrons**, \(n_e\). **Calculation:** 1. **Molar mass of C-12:** \(12\, \text{g/mol}\) 2. **Number of atoms per cm³:** \[ n_\text{C} = \frac{\rho}{12\,\text{g/mol}} \times N_A \] where \(N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}\) (Avogadro's number). \[ n_\text{C} = \frac{1.4 \times 10^6}{12} \times 6.022 \times 10^{23} \] \[ = 1.167 \times 10^5 \times 6.022 \times 10^{23} \] \[ = 7.03 \times 10^{28} \text{ atoms/cm}^3 \] 3. **Number of electrons per atom:** 6 (for C-12) \[ n_e = n_C \times 6 = 7.03 \times 10^{28} \times 6 = 4.22 \times 10^{29} \text{ electrons/cm}^3 \] **End of Step 1 Explanation:** We found the number density of electrons in the white dwarf to be \(n_e \approx 4.2 \times 10^{29} \, \text{cm}^{-3}\). --- ### **Step 2: Electron Spacing and Heisenberg Uncertainty Principle** #### **(b) Spacing between electrons** The mean spacing between electrons, \(d\), is given by: \[ d = n_e^{-1/3} \] \[ d = (4.2 \times 10^{29})^{-1/3} \text{ cm} \] Calculate the cube root: - \(\log_{10}(4.2 \times 10^{29}) = \log_{10}(4.2) + 29 = 0.623 + 29 = 29.623\) - \(\frac{29.623}{3} = 9.874\) - \(10^{9.874} \approx 7.5 \times 10^{9}\) - \(d = 1 / 7.5 \times 10^{-10} \, \text{cm} = 1.33 \times 10^{-10} \,\text{cm}\) So, \[ d \approx 1.3 \times 10^{-10} \text{ cm} = 1.3 \, \text{Å} \] **End of Step 2 Explanation:** The mean spacing between electrons is about \(1.3 \times 10^{-10}\) cm, or \(1.3\) Ångström. --- ### **Step 3: Electron Velocity from Heisenberg Uncertainty Principle** The **uncertainty in position** is on the order of the spacing, \(d\). By the **Heisenberg Uncertainty Principle**: \[ \Delta x \, \Delta p \sim \hbar \] \[ \Delta p \sim \frac{\hbar}{d} \] - \(\hbar = 1.055 \times 10^{-27}\) erg·s - \(d = 1.3 \times 10^{-10}\) cm \[ \Delta p \sim \frac{1.055 \times 10^{-27}}{1.3 \times 10^{-10}} = 8.12 \times 10^{-18} \text{ g cm/s} \] **Electron velocity:** \[ p = mv \implies v = \frac{p}{m} \] Electron mass \(m_e = 9.11 \times 10^{-28}\) g \[ v = \frac{8.12 \times 10^{-18}}{9.11 \times 10^{-28}} = 8.92 \times 10^{9} \, \text{cm/s} \] \[ v = 8.9 \times 10^9 \text{ cm/s} \] **Compare with the speed of light (\(c = 3.0 \times 10^{10}\) cm/s):** \[ \frac{v}{c} = \frac{8.9 \times 10^9}{3.0 \times 10^{10}} \approx 0.30 \] So, the electron velocity is about **30% the speed of light**. #### **Justification: Is non-relativistic approximation valid?** Generally, the non-relativistic approximation is valid if \(v \ll c\). Here, \(v \approx 0.3c\), so relativistic corrections start becoming important, and the non-relativistic formula is not strictly valid. **End of Step 3 Explanation:** The typical electron velocity is about \(0.3c\), so relativistic effects are becoming important, and the non-relativistic approximation is not strictly valid. --- ## **Final Answers** ### (a) **Velocity of electrons from Heisenberg Uncertainty Principle:** \[ v \approx 8.9 \times 10^9 \, \text{cm/s} \approx 0.3c \] **Non-relativistic approximation is not strictly valid** since \(v\) is a significant fraction of \(c\). ### (b) **Spacing between electrons:** \[ d \approx 1.3 \times 10^{-10} \, \text{cm} \] --- **Summary of steps:** 1. Found the number density of electrons (\(n_e \approx 4.2 \times 10^{29}\) cm\(^{-3}\)). 2. Calculated the mean spacing between electrons (\(d \approx 1.3 \times 10^{-10}\) cm). 3. Used the uncertainty principle to estimate electron velocity (\(v \approx 0.3c\)), and concluded non-relativistic formulas are not strictly valid here.

To answer the question, I first calculated the electron number density in the white dwarf using its mass density and the atomic composition of carbon-12, leading to a high electron density. Next, I estimated the average spacing between electrons by taking the inverse cube root of this density, which gives the typical distance between neighboring electrons. Then, applying the Heisenberg Uncertainty Principle, I related the uncertainty in position (the electron spacing) to a momentum uncertainty, which I converted into an electron velocity. Comparing this velocity to the speed of light indicated that electrons move at a significant fraction of \(c\), making relativistic effects important and thus questioning the validity of a purely non-relativistic approach.