Question

The wave motion of electron in a Bhor's orbit of hydrogen is as shown in diagram. The potential energy of electron
in the given orbit of hydrogen atom is​

Answer 1

Answer:

Explanation:

In this question we must combine dynamics with quantum physics.

According to the bohr's model electrons orbit the nucleus in a circular path. A body will only move in a circular path when a force constantly pulls it towards the center of a circle in this case that force is the coulumb's force.

Coulumb's force : F=kq₁q₂/r²

Both electron and proton have the same amount of electric charge so : F=kq²/r²

(the charge amount is “e”, in other words q=e)

Centripetal force : F=mv²/r

We will equate these two forces and cancel out one “r” from each side so : mv²=ke²/r

We will call this equation (1)

The electric potential energy : U=kq₁q₂/r

When the electron is infinitly far from proton(further than the seventh layer) the potential is zero and it's also increasing when the electron moves further away(because two opposite charge don't like to be seperated and if they do they'll get angry and their potential energy increases :D) so we must put a negative in the formula above for that to make sense.(the only way for a number to increase and reach zero is to be a negative number so we put a negative there)

The mechanical energy is the sum of kinetic energy and potential energy.(E=k+U)

k=1/2mv²

Substitude equation (1) and solve the E=k+U :

E=-ke²/2r

From here on there will be a somewhat complex calculation it's not hard but it's really confusing it has too many variables so I'll skip that. You can look it up if you want. ( Bohr model - Wikipedia )

r=n²r₀

r₀ is the bohr's radius. So our equation will become like this : E=-ke²/2n²r₀

We will call “ke²/2r₀” the rydberg energy(E-sub-R, my keyboard doesn't support sub letters I'll write ER instead so don't be confused)

Now our equation looks like this: E=-ER/n² much simpler isn't it ;)

Now we have to subtract kinetic energy from it(we wanted the potential energy,remember?)

U=-2ER/n²

Now comes the easy part :D

ER=-13.6ev , n=1 : U=-(2×13.6)÷(1^2)=27.2ev

understood?