Lines resolved by the instrument

[utkarshakash]

Homework Statement

A spectroscopic instrument can resolve two nearby wavelengths λ and λ+Δλ if λ/Δλ is smaller than 8000. This is used to study the spectral lines of the Balmer series of hydrogen. Approximately how many lines will be resolved by the instrument.

The Attempt at a Solution

For Balmer series

$\dfrac{1}{\lambda} = R_H \left( \dfrac{1}{2^2} - \dfrac{1}{n^2} \right)$

Homework Statement

But what should I substitute for n?

 

[ehild]

Homework Statement

A spectroscopic instrument can resolve two nearby wavelengths λ and λ+Δλ if λ/Δλ is smaller than 8000. This is used to study the spectral lines of the Balmer series of hydrogen. Approximately how many lines will be resolved by the instrument.

The Attempt at a Solution

For Balmer series

$\dfrac{1}{\lambda} = R_H \left( \dfrac{1}{2^2} - \dfrac{1}{n^2} \right)$

Homework Statement

But what should I substitute for n?

n is 3, 4, 5, 6, 7, 8, ...any positive integer >2. You get a wavelength for each n. Find that N so that λ(N)/(λ(N-1)-λ(N))<8000, but λ(N)/(λ(N)-λ(N+1))>8000

ehild

 

[utkarshakash]

n is 3, 4, 5, 6, 7, 8, ...any positive integer >2. You get a wavelength for each n. Find that N so that λ(N)/(λ(N-1)-λ(N))<8000, but λ(N)/(λ(N)-λ(N+1))>8000

ehild

I'm facing difficulty finding the n. I tried forming an equation but it is not easy to solve. Hit and trial method does not work for me.

 

[ehild]

You know that the change of a function can be estimated as Δf(x)=(df/dx) Δx.
Your function is λ(n). Take the derivative, and write Δλ/λ as function of n, with Δn=1.

What is the equation you arrive at? It is not easy to solve, you need to estimate. How big can be n? 5? 10? 50? 1000?

Estimation and trial is a method, frequently applied.


ehild

 

[HalfLostAndSearching]

In this case, n represents the energy level of the electron in the hydrogen atom. The Balmer series includes transitions from higher energy levels (n=3,4,5,...) to the second energy level (n=2). So, you can substitute n=3 and n=4 to calculate the wavelengths of the lines that will be resolved by the instrument.