Answer:
The answer is very close to [tex]N_e=2.989\times10^{26}[/tex], where [tex]N_e[/tex] is the number of electrons.
Explanation:
First we take into account that the block weighs 1Kg, and that the number of protons and electrons is the same. As the electron mass is tiny even compared to that of the proton and neutrons we can neglect it in our considerations.
Let's start by equating the mass of all protons and neutrons to the mass of the of the object:
[tex]N_p m_p+N_n m_n=1[/tex]
Where [tex]N_p[/tex] and [tex]N_n[/tex] is the number of protons and neutrons respectively. [tex]m_p[/tex] and [tex]m_n[/tex] is the mass of an proton and a neutron respectively. Because the number of protons and neutrons is equal we can say the following [tex]N_p=N_n=N[/tex], thus we have:
[tex]N_p m_p+N_n m_n=N(m_e+m_p)=1Kg \implies N=\frac{1}{m_p+m_n} [/tex]
On the other hand we have that the sum of all charges is less than the absolute value of 1C, we can express this by the following:
[tex]-1<N_p\cdot e-N_e\cdot e<1[/tex]
[tex]\implies -1<N\cdot e-N_e\cdot e<1[/tex]
Where [tex]e[/tex] is the proton charge (same as for the electron). We continue with the inequality:
[tex]-N\cdot e-1<N_p\cdot e-N_e\cdot e<-N\cdot e+1[/tex]
[tex]\implies \frac{N\cdot e+1}{e}>N_e>\frac{N\cdot e-1}{e}[/tex]
[tex]\implies \frac{(m_e+m_p)^{-1}\cdot e+1}{e}>N_e>\frac{(m_e+m_p)^{-1}\cdot e-1}{e}[/tex]
We have the estimated number of electrons bound. Because
[tex](m_e+m_p)^{-1}\cdot e>>1[/tex] We can neglect the ones on the rightmost and leftmost parts of the inequality. We then have
[tex]N_e\approx\frac{(m_p+m_n)^{-1}\cdot e}{e}[/tex]
Using the table values of the mass of the proton, mass of the neutron and the electron charge e we get
[tex]N_e\approx\frac{(1.672\times 10^{-27}+1.674\times 10^{-27})^{-1}\cdot e}{e}=(1.672\times 10^{-27}+1.674\times 10^{-27})^{-1}[/tex]
[tex]\, =2.989\times 10^{26}[/tex] electrons
