How Can We Calculate the Mass of an Electron Using Rydberg Constant Data?

[rayman123]

b]1. Homework Statement
[/b]
Table 8.6 shows the relative masses of the electron and a number of light atoms is derived from the values of the Rydberg constant (I have uploaded the table)
http://img833.imageshack.us/img833/645/namnlssm.jpg
Turn the problem around and use the data inte last column $$\lambda_{12}$$ (means) to find the mass of the electron given that the mass of the atoms are exact multiples of the unit mass $$1.66\cdot10^{-27}kg$$

Homework Equations

I have started with calculating the Rydberg constant and used the formula

$$\frac{1}{\lambda}=R \cdotZ^2(\frac{1}{(n_{1})^2}-\frac{1}{(n_{2})^2})$$
where Z=1

The Attempt at a Solution

I got $$R= 10967978.99 m^{-1}$$

then to calculate the electron mass i use the formula
$$R=R_{\infty}(1-\frac{m_{e}}{M})$$

where R - the theoretical value of the Rydberg constant
$$R_{\infty}$$ is the calculated one
$$m_{e}$$ is the electron mass
M- is a unit mass $$1.66\cdot10^{-27}kg$$


I solve the equation to obtain $$m_{e}$$ and I get:

$$m_{e}= M-\frac{RM}{R_{\infty}}$$

but after plugging in the corresponding values I get
$$m_{e} = 1.6598\cdot10^{-27}kg$$ which is not correct...If i compare the calculated value with the theoretical...which should be $$9.11\cdot10^{-31}$$

Can someone tell me where do I make mistake? How to solve it?

 

[Redbelly98]

I'm not sure where you are getting R=10967978.99 m^-1 from, it is not one of the values in the table. Also, what did you use for R_∞? And, did you use the same units m^-1 for both R and R_∞? (The table was showing cm^-1 instead.)

 

[rayman123]

well I am not sure myself if I understood the problem correctly...
What I thought was to start with calculate the Rydberg constat from the wave number which is in the last column of the table. Then using the calculated value for the Rydberg constant compute the electron mass. Do it for every atom which is in this table.
How would you do this? Maybe you have some better ideas?

$$R_{\infty}=1.097 373 1569\cdot10^7 m^{-1}$$

I found on wikipedia one formula for calculating the Rydberg constant

$$R_{m}= \frac{R_{\infty}}{(1+\frac{m_{e}}{M})}}$$

I have solved it to obtain $$m_{e} = (\frac{R_{\infty}}{R_{m}}-1)M$$

Yes I used the same units for both constats. I just multiplied the one from the table by 0.01 to obtain m instead.

then I plug in
$$R_{\infty}$$ as above and for $$R_{m}$$ corresponding value from the table. $$M= 1.6605387\cdot10^{-27}kg$$ but the result is always way much greater that the electron mass...I am stuck here...

 

[Redbelly98]

Okay, it is looking like we are not to use R_∞, i.e. pretend that we don't know it's value.

Let's look at the equation you had earlier,

then to calculate the electron mass i use the formula
$$R=R_{\infty}(1-\frac{m_{e}}{M})$$

If you write that equation out for two different elements in the table*, then you'll have two equations in two unknowns, R_∞ and m_e. And just use the R's from the table and M_nuc for whatever two elements you choose.

*I recommend using the heaviest and the lightest elements in the table.

 

[rayman123]

I am not sure if I follow you...

did you mean to solve it like this?

$$109677.6\cdot10^{-2}=R_{\infty}(1-\frac{m_{e}}{1.66\cdot10^{-27}})$$

$$109732.2\cdot10^{-2}=R_{\infty}(1-\frac{m_{e}}{12\cdot1.66\cdot10^{-27}})$$


and solve it to obtain $$R_{\infty}$$ and $$m_{e}$$?

 

[Redbelly98]

Yes, though we really just need m_e.

Another hint: based on the two equations you just wrote, what is

$$\frac{109677.6\cdot10^{-2}}{109732.2\cdot10^{-2}} \ = \ ?$$​

equivalent to?

 

[rayman123]

those are values from the table 8.6 (converted to meters)
It is equivalent to 0.999502425 I guess we could say 1.
But I still do not get your suggestion of how the problem can be solved...

 

[Redbelly98]

Okay, a couple more comments are in order:

1. We are not converting cm to m, instead we are converting cm^-1 to m^-1. That means the numbers would get multiplied by 10⁺², not 10^-2. This could have been your error before.

2. In addition to calculating the ratio as 0.999502425, you can replace each number in

$$\frac{109677.6\cdot10^{+2}}{109732.2\cdot10^{+2}} \ = \ ?$$​

with the equivalent expressions you had in Post #5. I.e., replace 109677.6·10² with R_∞(1+m_e/1.66·10^-27)