Radius of a certain orbit of hydrogen atom is 8.48 $\mathop A\limits^o$. If energy of electron in this orbit is $E / x$. then $x=$ ________ (Given $\mathrm{a}_0=0.529$ $\mathop A\limits^o$, $E=$ energy of electron in ground state).

<p>Let's approach this problem by understanding the basics and applying the Bohr model to find the energy levels of a hydrogen atom.</p> <p>The energy of an electron in a hydrogen atom for any given orbit can be defined using the formula:</p> <p>$E_n = \frac{E}{n^2}$</p> <p>where:</p> <ul> <li>$E$ is the energy of the electron in the ground state ($n=1$), and</li> <li>$n$ is the principal quantum number (orbit number).</li> </ul> <p>The radius of an orbit in the hydrogen atom, according to the Bohr model, is given by:</p> <p>$r_n = n^2 a_0$</p> <p>where:</p> <ul> <li>$r_n$ is the radius of the nth orbit,</li> <li>$a_0$ is the Bohr radius ($0.529$ angstroms or $\mathop A\limits^o$), and</li> <li>$n$ is the principal quantum number (orbit number).</li> </ul> <p>Given that:</p> <ul> <li>The radius of a certain orbit of the hydrogen atom is $8.48 \mathop A\limits^o$, and</li> <li>The energy of an electron in this orbit is $E/x$,</li> <li>We need to find $x$.</li> </ul> <p>First, let's find $n$, the principal quantum number for the orbit with radius $8.48$ angstroms:</p> <p>$8.48 = n^2 \times 0.529$</p> <p>Solving for $n^2$:</p> <p>$n^2 = \frac{8.48}{0.529}$</p> <p>$n^2 \approx 16.03$</p> <p>For simplicity and practicality in the quantum model, $n^2$ approximately equal to 16 would imply $n = 4$, considering $n$ must be a whole number and $16.03$ is close to $16$, which is a perfect square of $4$.</p> <p>Now, to find $x$, we'll use the energy relationship. Since the energy levels of the hydrogen atom are inversely proportional to the square of the principal quantum number $n$,</p> <p>$E_{orbit} = \frac{E}{n^2} = \frac{E}{4^2} = \frac{E}{16}$</p> <p>According to the given information, $E_{orbit} = \frac{E}{x}$, which means:</p> <p>$\frac{E}{x} = \frac{E}{16}$</p> <p>Hence,</p> <p>$x = 16$</p>