1. **Problem statement:** Evaluate the chemical potential $\mu$ of an ideal gas of helium-4 atoms at normal conditions $T=273.15$ K and $p=760$ mmHg using the normalization condition equation: $$N = \sum_k e^{(\varepsilon_k - \mu)/T} \quad \text{(Eq. 3.4)}$$ 2. **Understanding the formula:** - $N$ is the total number of particles. - $\varepsilon_k$ are the energy levels. - $\mu$ is the chemical potential to find. - $T$ is the temperature. For an ideal gas, the chemical potential $\mu$ relates to pressure $p$, temperature $T$, and thermal wavelength $\lambda$ by: $$\mu = k_B T \ln\left(\frac{p \lambda^3}{k_B T}\right)$$ where $k_B$ is Boltzmann constant and thermal wavelength $\lambda$ is: $$\lambda = \sqrt{\frac{2 \pi \hbar^2}{m k_B T}}$$ 3. **Calculate thermal wavelength $\lambda$:** - Mass of helium-4 atom $m = 6.646 \times 10^{-27}$ kg - Planck constant $\hbar = 1.0545718 \times 10^{-34}$ J\cdots - Boltzmann constant $k_B = 1.380649 \times 10^{-23}$ J/K $$\lambda = \sqrt{\frac{2 \pi (1.0545718 \times 10^{-34})^2}{6.646 \times 10^{-27} \times 1.380649 \times 10^{-23} \times 273.15}}$$ Calculate inside the square root: $$= \sqrt{\frac{2 \pi \times 1.112 \times 10^{-68}}{2.512 \times 10^{-48}}} = \sqrt{2.78 \times 10^{-20}} = 1.67 \times 10^{-10} \text{ m}$$ 4. **Convert pressure to SI units:** - $p = 760$ mmHg - $1$ mmHg $= 133.322$ Pa $$p = 760 \times 133.322 = 101325 \text{ Pa}$$ 5. **Calculate chemical potential $\mu$:** $$\mu = k_B T \ln\left(\frac{p \lambda^3}{k_B T}\right)$$ Calculate $\lambda^3$: $$\lambda^3 = (1.67 \times 10^{-10})^3 = 4.65 \times 10^{-30}$$ Calculate the argument of the logarithm: $$\frac{p \lambda^3}{k_B T} = \frac{101325 \times 4.65 \times 10^{-30}}{1.380649 \times 10^{-23} \times 273.15} = \frac{4.71 \times 10^{-25}}{3.77 \times 10^{-21}} = 1.25 \times 10^{-4}$$ 6. **Evaluate logarithm:** $$\ln(1.25 \times 10^{-4}) = \ln(1.25) + \ln(10^{-4}) = 0.2231 - 9.2103 = -8.9872$$ 7. **Calculate $\mu$ in Joules:** $$\mu = 1.380649 \times 10^{-23} \times 273.15 \times (-8.9872) = -3.39 \times 10^{-20} \text{ J}$$ 8. **Interpretation:** The chemical potential is negative and small in magnitude, typical for an ideal gas at normal conditions. **Final answer:** $$\boxed{\mu = -3.39 \times 10^{-20} \text{ J}}$$