Physics307L:People/Allen/Balmer/Balmer Series Lab Summary
Balmer Series Lab Summary
^{SJK 18:57, 4 October 2009 (EDT)}The Balmer Series is specific to Hydrogen spectral line emissions, and supports the concept of quantized energy as light emission occurs only at specific wavelengths that are determined by the energy differences between two electron states. Spectroscopy and especially the Balmer Series finds application in astronomy to analyze the composition of stars and calculate their surface temperatures from the relative strength of spectral lines. Wiki Article on Balmer Series
This lab was taken on to determine the Rhydberg Constant for Hydrogen and Deuterium by applying the Rhydberg Formula to the spectral lines of the Balmer Series measured in the lab. We used a constant deviation spectrometer which we calibrated with the known wavelength measurements of the spectral lines for Mercury vapor and then measured the wavelengths emitted from the Hydrogen and Deuterium vapor tubes.
We found an average value for the Rhydberg Constant for Hydrogen of 1.0965640Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
and for Deuterium of 1.0950235Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
.
Data
- Hydrogen
Red: 657.88+/-0.43nm
Orange: 603.88+/-0.2nm
Yellow: 582.3+/-0.31nm
Blue-Green: 486.05+/-0.07nm
Violet: 434.05nm+/-.07nm
- Deuterium
Red: 660.6+/-3.06nm
Orange: 618.8nm
Yellow: 584.1nm
Blue-Green: 485.45+/-.18nm
Violet: 434.3nm
Analysis
Putting these into the formula for the Rhydberg Constant for Hydrogen:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\frac{1}{\lambda }}{(\frac{1}{2^2}-\frac{1}{n^2})}=R_{H}}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{H}} is the Rydberg Constant for Hydrogen, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda} is the observed wavelength, and n is the excited energy state that the electron was in.
We calculated the constants for the various spectral lines as follows:
Rydberg Constant for Hydrogen(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{H}} )
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{H}} for the red spectral line = 1.0944245+/-.00071Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{H}} for the blue-green spectral line = 1.0972808+/-.000158005Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{H}} for the violet spectral line = 1.0979866+/-.0001769Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
Rydberg Constant for Deuterium(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{D}}
)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{D}} for the red spectral line = 1.0899183+/-.00507Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{D}} for the blue-green spectral line = 0.910980157+/-.00028CNote: Due to a typo, this was miscalculated. Using the proper averaged measurement for the wavelength, 485.54nm, the Rhydberg constant is found to be 1.0986370Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{D}} for the violet spectral line = 1.09645516Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}} (no +/- due to only seeing the two spectral lines and having only one data point)
This gives an average value for the Rhydberg Constant for Hydrogen of 1.0965640Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
and for Deuterium of 1.0950235Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
.
The accepted value for the Rhydberg Constant for Hydrogen is 1.0973732Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle *10^{7}m^{-1}}
, so our finding has an error of 0.7%.
Comments
This was a good lab to begin with as the equipment gave precise readings and our percent error was very low, which was very encouraging. It seems that there would be good reason to not include the data from the first day in the analysis for the Hydrogen spectral lines as three of the five data were outside the standard error for the mean. Another time, or in future labs, I would take more multiple readings to more accurately determine an average reading, and would consider the cause of an outlier at the moment of taking the reading to allow for an immediate retake.
In addition, I would take more care in the set up of the equipment, taking time to clean the scale markings and the prism for clearer sighting of the spectral lines and the measurements. It may also have helped to work more with adjusting the vapor tube position to improve the clarity and intensity of the spectral lines.
When I first learned of spectral lines in the past, I was fascinated by the concept. It would be interesting to be able to use the spectrometer for analysis of the spectral lines of other vapors or light sources as well.
Concerns and Sources of Error
- Systematic Error: When we first calibrated the spectrometer, we noticed that we had not fully tightened the set screw on the prism, so there was some slippage of the prism as we moved through the range of our measurements. As a result we repeated the first calibration with the prism set properly. However, this may still be a concern on a small scale.
In addition, the scale on the dial to the wavelengths was difficult to read and did not have as much accuracy for the longer wavelengths. The dial had a significant amount of play in the gears as well, which we tried to minimize by always taking the readings from the same direction.
- Random Error: This could come from reading the markings or aligning the spectral lines slightly differently for each repitition of a data point, even though they were all taken from the same direction.
- The one outlier reading for the Deuterium red spectral line on day two may have been due to inaccurate reading of the scale markings or due to approaching the spectral line from the opposite direction, leading to error from the play in the adjustment gears. In addition, I did not know how to include the error of our readings for Mercury from its standard spectral line values into our data collected for Hydrogen and Deuterium It seems we should have adjusted our readings due to the amount that the Mercury readings varied across the scale. In doing future labs, I would work on developing this correction to obtain more accurate results with other calibrations.
- Another concern comes from the fact that we observed five spectral lines for Hydrogen, two of them yellow, whereas in the lab notes and other information thaat I've read there are only four listed, two of them violet. So it may be that our vapor tube was contaminated with another element. Although we still found the Rhydberg Constant within a small degree of error, this seems a major issue of uncertainty.