Animated Solution for Chemistry - Atomic Structure: The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to $\frac{h^2}{xm{a}_0^2}$. The value of $10x$ is ……… . ($a_0$ is radius of Bohr's orbit) (Nearest integer) [Given, $\pi =3.14$]

Imagine you are looking at a hydrogen atom. At the center, we have the positively charged nucleus. Revolving around it in the second orbit, where $n=2$, is our electron. This is the classic Bohr model setup, and it provides a beautiful, semi-classical way to understand atomic energy levels.

According to Bohr's famous postulate, the angular momentum of this revolving electron is not arbitrary; it is strictly quantized. It must be an integral multiple of $\frac{h}{2\pi }$. This gives us our first master equation:

From this equation, we can easily isolate the velocity $v$. We just move the mass $m$ and the radius $r$ to the denominator on the right side. This gives us the velocity of the electron purely in terms of its orbit radius:

Now, the question asks for the kinetic energy of this electron. We know the classic formula for kinetic energy is:

Let's substitute our expression for velocity into the kinetic energy formula. Squaring the terms inside the bracket gives us $\frac{n^2{h}^2}{4{\pi }^2{m}^2{r}^2}$. Multiplying by $\frac{1}{2}m$ simplifies it beautifully:

We need our final answer in terms of the Bohr radius, $a_0$. Remember, the radius of the $n$-th orbit is given by the formula:

For a hydrogen atom, the atomic number $Z$ is $1$, and here we are dealing with the second orbit, so $n$ is $2$. Therefore, the radius $r$ is simply:

Now, let's bring it all together. We substitute $n=2$ and $r=4a_0$ back into our kinetic energy equation.

Let's simplify this carefully. Squaring $4a_0$ gives $16a_0^2$. The $4$ in the numerator cancels partially with the $8$ in the denominator, leaving a $2$. Two times sixteen is thirty-two. So we get:

Comparing our result with the given expression $KE=\frac{h^2}{xm{a}_0^2}$, we can clearly see that $x$ is exactly $32{\pi }^2$.

The question asks for the value of $10x$. Multiplying by ten and substituting $\pi =3.14$, we get:

Rounding to the nearest integer, our final answer is 3155.