Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

CHEM 161

The Hydrogen Spectrum

Skills:  This experiment will help you learn and practice the following essential techniques:

· How to use Excel  to manipulate large data sets

· How to enter formulas in Excel

· How to model the emission spectral lines based on Bohr’s model and the Rydberg equation

Knowledge:  

· Many times, we are asked to repeat the same calculations on a larger set of numbers.  This could be extremely tedious on a calculator! Excel or similar programs give us a great tool to perform these tasks more effortlessly and much more quickly.  If you are scientist, an insurance broker, a CPA, or you are trying to manage your own finances, you will encounter this situation many times throughout your life.  With this lab you will have a better understanding of how Excel lets us perform calculations on larger data sets fairly easily.

· You will also explore how closely the data set generated by Excel and using Bohr’s model of an atom and Rydberg’s equation matches experimental data.

Tasks:  To complete this assignment you should:

· Calculate all theoretical energy levels of the first 10 stationary states of the hydrogen spectrum.

· Calculate all possible wavelengths of light resulting from energy state transitions.

· Observe the visible spectrum of hydrogen emissions and compare your observation with your calculation.

· Observe the visible spectrum of other elements and compare to the hydrogen spectrum.

Criteria for success:

· Successfully calculate the energy levels for the first 10 shells of the hydrogen atom along with the energies associated with the energy transitions.

· Turn in all parts of A-D.  All material should be organized, well labeled and easy to follow.

Introduction

When atoms are excited, either by electricity (what you will observe today) or by heat (what you may have seen in a flame test experiment or also in fireworks) they often give off light. The light emitted is characteristic of the electronic structure of the atom, and so is specific to an atom. These wavelengths emitted constitute what is called atomic spectrum.

According to quantum theory, electrons can only exist in certain states, and each state has a fixed (quantized) amount of energy. When an electron goes from a lower state (closer to the nucleus) to a higher state (farther from the nucleus), it must absorb the energy required (the energy differential between the states). Conversely, when an electron goes from a higher state to a lower state it emits energy, often in the form of light. The energy of this light (photon) emitted must then be equal to the change in energy between the states.

ΔE = E_{higher state} – E_{lower state} = E_photon = (equ. 1)

Where c = 2.998 x 10⁸m/s

h = Planck’s constant =  6.626 x 10^-34 Js

λ = wavelength of light emitted (let’s keep this in nm, and change c constant to nm/sec)

The equation above is of course based on 1 electron changing from a higher energy to a lower energy, but as always we should work with 1 mole of electrons, and thus calculate the transition per mole. To do this simply multiply by Avogadro’s number, N_A = 6.022 x 10²³ mol^-1

N_A ΔE = E_{higher state} – E_{lower state} = N_A E_photon = N_A  (equ. 2)

Substituting the values for N_A hc above we calculate:

N_A hc = 6.022 x 10²³ mol^-1 x 6.626 x 10^-34 Js x 2.998 x 10⁸m/sec x 1kJ/1000J x 10⁹nm/m =

= 1.196 x 10⁵ kJ·nm/mol (equ. 3) 

ΔE = E_{higher state} – E_{lower state} = 1.196 x 10⁵ kJ·nm/mole      (Notice nm cancel and only kJ/mole is left for ΔE)

λ (in nm)

So if you know that wavelength emitted you can calculate the ΔE between the states. Or rearranging,

(equ. 4)

If you know the ΔE between the states you can predict the λ that should be emitted.

The example in Figure 1 uses the sodium atom transitions to show the usefulness of the equation.  The energies of the sodium atoms first three energy levels are known and shown on the graph to the left. State III has an energy of -187.931 kJ/mole. The first calculation on the right shows the predicted wavelength that would be emitted if an electron fell from state III to state II. The prediction shows that this emission is not visible (1140 nm is not in the visible spectrum).

The other calculations in Figure 1 similarly predict the emission wavelengths. The emission from state II to state I is visible yellow/orange light that we commonly observe in vaporized sodium. Those ugly orange/yellow parking lot lights are sodium vapor lights.

Recall that the energies of an atom’s energy levels are all negative numbers, with the most negative being closest to the nucleus. The level closest to the nucleus has the lowest absolute potential energy because the electrons are close to the nucleus’ protons. The electrons farthest way have a higher potential energy (more positive, but still negative numbers.) Notice the 0-point of energy is when the electron completely leaves the atom (called ionization.)

The energies above are calculated by quantum mechanical equations and modeling, and for most atoms (except hydrogen) these energies are difficult to calculate because multi-electron atoms do not follow the Bohr model. Luckily, hydrogen only has 1 electron and the Bohr model works. For hydrogen we can calculate the energy of each state according to Bohr’s model (the constant -1312 kJ/mole below is empirical from Bohr’s model)

   (equ. 5)

Procedure:

A. Calculate the Energy Levels of the Hydrogen Atom

The goal of this part is simply to calculate the energy of levels 1-10 in the hydrogen atom. This is easily done with equation (5) above substituting in the numbers 1-10 for n.  It may seem that performing calculations for 10 energy levels is a lot, but with Excel you can do this really quickly. One of the goals of this experiment is to show you how scientists process huge swaths of data using powerful software.

1. Open an Excel spreadsheet and in column A, line 1, type the title of the column (maybe you would call it, energy level or stationary state or ‘n’…)

2. Skip down to line 3 (still column A) and type the number 1

3. Go to line 4 (cell A4) and type an equals sign (=). The equal sign tells Excel you want to perform a function.  

· What you want to do is keep adding 1 to the number in Cell A3 until you get to the number 10. Now, obviously you could just type the numbers 1-10 down the column, but what if you had 50000 numbers to type?!

4. After the equals sign click on the cell with the number 1 in it (should be cell A3), the symbol ‘A3’ should now appear in cell. Now type +1. This tells Excel to add 1 to the number in cell A3. Press Enter.

5. Next, copy and paste the contents of cell A4 into the celles below.  Excel will add 1 to every cell directly above and now you should have the numbers 1-10 in the column.  Click Cntrl C on A4, then outline (click and drag) the cells below A4 (so that you have n=1 to 10),  then Cntrl V to paste, then Enter.

Neat, huh!

6. Go to column B, line 1 and type the name of the next column (Energy). Using equation 5, input this equation into cell B3, using cell A3 as the n value. Then you have to square it (n²).  Then cut and paste it down the column.

· Don’t forget to type = first. The / is a division sign and the ^ is the power symbol. Use parenthesis where necessary). Try to figure it out with your partner or lab group next door, but if you need help ask. It’s easy once you get it. J

7. You should now have an excel spreadsheet with two filled columns. Column A is the energy levels (n = 1 through 10) and column B is the energy of those levels as calculated by equation (5).

B. Calculation of the wavelengths in the Hydrogen Spectrum

Look back at equation (4). This equation allows you to calculate the λ of emission if you know the ΔE between the two states. In part A above (column B on your spreadsheet) you calculated the E of the energy states from 1-10.  So if you calculate the ΔE between any two states, you can use equation (4) to calculate the λ of emission.  Holy Cow!  That’s a lot of ΔE values (2 → 1, 3→1….10→1, 3→2, 4→2…10→2 etc etc etc.)  Excel to the rescue.

8. In column C, you will calculate the difference of energy between any state and state 1. (So that’s 2 → 1, 3→1….10→1). To make sure you don’t forget, click on C1 and type a title for this column. Perhaps it would say something like ΔE (x →1), meaning ‘this column is the energy change from state x to state 1.

· You can use Insert – Symbol to find things like Δ or Greek letters.  Go to the “Insert” tab and then select “Symbol” at the very right of the menu bar.

9. Click on cell C4. In this cell you will input the formula to calculate the energy difference from 2 → 1. Start by typing = then click on cell B4 then type minus – then click on cell B3.  Ok  now you are going to learn a new Excel function.  

· Later when you cut and paste this down the column if you just use the formula as written excel will subtract B4 – B3 (in cell C4) and B5 – B4 (on cell C5) etc etc etc.   But you want to keep the B3 cell constant. What you want is B4-B3, B5-B3, B6-B3 etc etc etc, (this is ΔE 2→1, then ΔE 3→1, then ΔE 4→1 etc etc etc). So to keep cell B3 always as the subtracted cell, type a $ sign on both sides of the B in B3. It should look like this:  =B4-$B$3

10. Press Enter, then copy and paste down the column.  Now you have calculate all the ΔE values between every energy state and state 1!! (2 → 1, 3→1….10→1). The numbers get bigger, right? It’s more energy from 10→1 than from 2→1.

11. In column D you would like to calculate the λ for each ΔE. This is using Equation 4. So go to cell D1 and type a title for the column (probably just λ is good enough), and then go to cell D4. Tell Excel to calculate λ by using equation 4 on cell C4. You should be able to do this on your own. Don’t forget to type =.  Ask for help if you need it! Copy and Paste down the column.

12. Now that you know how to do this, you are going to calculate the ΔE for every stationary state 1-10. Go to column E, type a title (probably ΔE (x →2)). Then go to E5 and type the formula to calculate the Energy difference of 3→2. [=B5-B4, but then add the $ sign….B5-$B$4]Copy and Paste Down.

13. Column F—this should be the calculated λ for the ΔE in column E. (Refer to Column D if you need a refresher.)

14. Getting it? Okay finish the spreadsheet now with all of the rest of the columns. (Yes, you will have a lot of data…that’s the point!)

C. Assignment of wavelengths

Listed below are experimental transition wavelengths for the hydrogen atom. Using the data (meaning look at the spreadsheet) you just generated (remember those are theoretical values), assign each wavelength a transition. For example: 3039 nm represents a transition from n = 10 → n = 5

Note that some of the values will be slightly different from your calculations.  The listed values above are experimental values, what you calculated on the spreadsheet are theoretical values.

Organized and tabulate the answers neatly on your print-out.

D. The Balmer Series

The part of the Hydrogen emission spectrum that is in the visible range is called the Balmer series.  Answer the following questions.

1. Using your calculated, theoretical data, predict how many spectral lines you should see in the Balmer series.  How many emissions are in the visible wavelength range (400-800nm)?

2. What are the state transitions for the Balmer series?  (From n =? → n=?)

3. Now observe the hydrogen emission spectrum using the gas discharge tubes that are set up in the lab. Verify that your predictions for spectral lines match with your observations.

4. Comment on how many spectral lines you predicted you should see and how many you actually saw.  Why do you think you don’t see all of them?  Come up with a reasonable explanation.

5. Also observe the emission spectrum for Neon. Note any differences from the hydrogen spectrum.  How many spectral lines can you see?