Suppose we want to compute the exact standard molar electronic entropy of helium at a finite but not very high temperature, say $T = 298.15 \textrm{ K}$. By "standard" I mean the standard state, i.e. the helium is an ideal gas at 1 bar, where all interactions between different helium atoms are neglected, or equivalently speaking, the helium is at an infinitesimal pressure and the resulting entropy is converted to 1 bar by applying the ideal gas formula:
$$\tag{1}S_1 - S_0 = R \ln \frac{p_0}{p_1}$$.
The molar electronic entropy is given as follows: \begin{align}\tag{2} S_e = -R \sum_i p_i \ln p_i \\ = -R \sum_i \frac{1}{Q} e^{-\frac{E_i-E_0}{kT}} \ln \frac{1}{Q} e^{-\frac{E_i-E_0}{kT}} \tag{3}\\ = R \sum_i \frac{1}{Q} e^{-\frac{E_i-E_0}{kT}} (\frac{E_i-E_0}{kT} + \ln Q),\tag{4} \end{align} where $E_i$ is energy of the electronic state $i$, and $Q$ is the partition function: $$\tag{5} Q = \sum_i e^{-\frac{E_i-E_0}{kT}} $$ Now, for the helium atom, there are an infinite number of electronic states below the first ionization potential of helium, $I_1$. Thus, $$ Q > \sum_i^{+\infty} e^{-\frac{I_1-E_0}{kT}} = +\infty\tag{6} $$ Consequently, we have \begin{align} S_e > R \sum_i \frac{1}{Q} e^{-\frac{E_i-E_0}{kT}} (\frac{E_0-E_0}{kT} + \ln Q) \tag{7}\\ = R \sum_i \frac{1}{Q} e^{-\frac{E_i-E_0}{kT}} \ln Q \tag{8}\\ = R \ln Q = +\infty \end{align} However, this would mean that the total standard molar entropy of helium is also infinite. Actually we may substitute helium by practically any gaseous molecule, and conclude that the standard molar entropies of all molecular gasses are infinite! Of course the entropy diverges very slowly, because $e^{-\frac{I_1-E_0}{kT}}$ is extremely small at room temperature and the entropy is the logarithm of the sum, but since I apparently didn't make any approximation, this infinity does pose a theoretic problem.
My intuition is that as long as the interactions between the helium atoms are taken into account, the summation becomes finite, however small the interactions are (but I cannot prove this at the moment). But an ideal gas reference state is, by definition, a hypothetical state where the atoms cannot see each other at all, as if the gas is infinitely thin.
My questions are thus:
- Are the above derivations correct?
- If yes, does this imply that it is inherently flawed to define the standard molar entropy of a gas using an ideal gas reference state?
- Are there any alternative definitions of the standard molar entropy of a gas, that can get rid of this divergence? One may think of choosing the reference state as a state with very small but non-zero pressure, small enough so that the effect of intermolecular interaction on translational entropy can be neglected but large enough so that the divergence of the electronic partition function is effectively suppressed. But then, how should one choose the value of the pressure, without introducing arbitrariness into the definition of the standard molar entropy?