One facet of the central field approximation for many-electronatoms is that inner-shell electrons screen the nuclear charge. Tounderstand how this works quantitatively, first note that theprobability distributions for electrons in different shellsgenerally do not overlap much. For instance, the electrons in the Mshell (n=3) are almost always farther from the nucleus than theelectrons of the K (n=1) and L (n=2) shells. Thus, it is a goodapproximation to assume that the inner shells completely screen thenucleus from the outer shells. For example, if there are tenelectrons altogether in the K and L shells of an atom, then theelectrons in the M shell experience force from a charge of roughlyZ-10, where Z is the charge on the nucleus as an integer multipleof e, the magnitude of the charge on an electron. This is calledthe effective nuclear charge Z_eff.
a. In a beryllium atom (Z=4), how many electrons are in the Kshell?
Express your answer as an integer.
b. n xenon (Z=54), what is the effective charge Z_eff experiencedby an electron in the M (n=3) shell?
Express your answer as an integer.
c. How many electrons are there altogether in the K, L, and Mshells of xenon? Recall that for n=3, the orbital quantum number lmust be zero, one, or two and that m_l can take any value betweenpostive and negative l.
Express your answer as an integer.
d.
The idea of simply subtracting the number of inner-shell electronsworks well only for atoms with low values of Z. For atoms withlarger nuclear charge, the different shapes of the probabilitydistributions for different subshells becomes an important factor.For instance, in the N shell, electrons in the s subshell have amuch higher probability than those in the p or d subshells of beingfound closer to the nucleus than some of the K, L, or M electrons.It is said that the s electrons penetrate the inner shells morereadily than do the p or d electrons. Therefore, it is to beexpected that our simple subtraction model may not work well forouter-shell s electrons in large atoms. Keeping this in mind, wesee that the effective nuclear charge Z_eff for the outermostelectron in an atom may be found experimentally by measuring theionization energy of an atom and then calculating the effectivecharge using the equation
E_n=- \frac{Z_{\rm eff}^2}{n^2}(13.6\;\rm eV).
The energy for the 5p valence electron in indium (Z=49) is -5.79electron volts. What is the effective nuclear charge Z_effexperienced by this electron?
Express your answer to three significant figures.
One facet of the central field approximation for many-electronatoms is that inner-shell electrons screen the nuclear charge. Tounderstand how this works quantitatively, first note that theprobability distributions for electrons in different shellsgenerally do not overlap much. For instance, the electrons in the Mshell (n=3) are almost always farther from the nucleus than theelectrons of the K (n=1) and L (n=2) shells. Thus, it is a goodapproximation to assume that the inner shells completely screen thenucleus from the outer shells. For example, if there are tenelectrons altogether in the K and L shells of an atom, then theelectrons in the M shell experience force from a charge of roughlyZ-10, where Z is the charge on the nucleus as an integer multipleof e, the magnitude of the charge on an electron. This is calledthe effective nuclear charge Z_eff.
a. In a beryllium atom (Z=4), how many electrons are in the Kshell?
Express your answer as an integer.
b. n xenon (Z=54), what is the effective charge Z_eff experiencedby an electron in the M (n=3) shell?
Express your answer as an integer.
c. How many electrons are there altogether in the K, L, and Mshells of xenon? Recall that for n=3, the orbital quantum number lmust be zero, one, or two and that m_l can take any value betweenpostive and negative l.
Express your answer as an integer.
d.
The idea of simply subtracting the number of inner-shell electronsworks well only for atoms with low values of Z. For atoms withlarger nuclear charge, the different shapes of the probabilitydistributions for different subshells becomes an important factor.For instance, in the N shell, electrons in the s subshell have amuch higher probability than those in the p or d subshells of beingfound closer to the nucleus than some of the K, L, or M electrons.It is said that the s electrons penetrate the inner shells morereadily than do the p or d electrons. Therefore, it is to beexpected that our simple subtraction model may not work well forouter-shell s electrons in large atoms. Keeping this in mind, wesee that the effective nuclear charge Z_eff for the outermostelectron in an atom may be found experimentally by measuring theionization energy of an atom and then calculating the effectivecharge using the equation
E_n=- \frac{Z_{\rm eff}^2}{n^2}(13.6\;\rm eV).
The energy for the 5p valence electron in indium (Z=49) is -5.79electron volts. What is the effective nuclear charge Z_effexperienced by this electron?
Express your answer to three significant figures.
