What is the unit m^-1?
The question I'm working on says "an electron in the hydrogen atom makes a transition from an energy state of principal quantum number nt to n=1. If the photon emitted has a wavelength of 95 nm, what is the value of nt?
I was trying to use the Rydenberg equation, which is:
(1/wavelength) = R infinity x ((1/n1^2)-(1/n2^2))
My textbook says that R infinity = 1.097 x 10^7 m^-1.
But, what does the m^-1 mean and how do I get my units to be the same so I can solve?
PLEASE help me out if you can! I've tried this problem so many different ways and I can't get a plausible answer!
m^-1 means per meter and since R is measured in per meter the wavelength comes out in meters. So solve as you have suggested and convert the answer for wavelength in meters to nanometers. By the way, you want to substitute 1 for n1 and solve for n2.
The unit "m^-1" refers to reciprocal meters, which is a unit commonly used in physics to represent wave numbers. In the context of the Rydberg equation, it represents the reciprocal of the wavelength of light.
To solve the problem, we need to ensure that the units are consistent. In this case, we need to convert the wavelength given in nanometers (nm) to meters (m) because the Rydberg constant is in reciprocal meters.
To convert nanometers to meters, we can use the conversion factor: 1 nm = 1 x 10^-9 m.
Given that the wavelength is 95 nm, we can convert it to meters as follows:
95 nm x (1 x 10^-9 m/1 nm) = 95 x 10^-9 m
Now that we have the wavelength in meters, we can substitute it into the Rydberg equation along with the known values:
(1/95 x 10^-9 m) = (1.097 x 10^7 m^-1) x ((1/nt^2) - (1/1^2))
The reciprocal of the wavelength on the left side cancels out with the reciprocal meters unit on the right side, leaving just the wave number in m^-1.
Simplifying the equation further, we have:
1.097 x 10^7 m^-1 = (1/nt^2) - 1/1^2
Since 1/1^2 = 1, we can simplify it again:
1.097 x 10^7 m^-1 = (1/nt^2) - 1
Now, we can rearrange the equation to solve for nt:
(1/nt^2) = 1.097 x 10^7 m^-1 + 1
Combining the terms on the right side:
(1/nt^2) = 1.097 x 10^7 m^-1 + 1
(1/nt^2) = 1.097 x 10^7 m^-1 + (1 m^-1)
Since we have the same unit (m^-1) on both sides, we can add them together:
(1/nt^2) = 1.097 x 10^7 m^-1 + 1 m^-1
(1/nt^2) = 1.097 x 10^7 m^-1 + 1 m^-1 = 1.097 x 10^7 m^-1
Now, we can take the reciprocal of both sides to solve for nt^2:
nt^2 = 1/(1.097 x 10^7 m^-1)
Finally, taking the square root of both sides gives us the value for nt:
nt = sqrt(1/(1.097 x 10^7 m^-1))
Now you can evaluate the expression on a calculator or perform the calculation manually to find the value of nt.
I hope this explanation helps you solve the problem!