What electron transition in a hydrogen atom, starting from the orbit 10, will produce infrared light of wavelength 3040 ?
What are the wavelength units? micrometers? nanometers? inches? Just giving a number isnt enough.
Once you have figured that out, use the appropriate Rydberg formula with an upper level quantum number of n = 10 and solve for the lower level.
3040 what? nanometers ?
use Rydberg equation
3040 *10^-9 m * 10^2 cm/m = 3040*10^-7 cm
1/wavelength = 3.29*10^3
1/wavelength in cm =109,678cm^-1 = (1/n1^2 - 1/n2^2)
3.29 *10^3 = 1.1*10^5 (1/n1^2 -1/n2^2)
here n2 = 10
2.99 *10^-2 = (1/n1^2 -.01)
.0299 +.01 = 1/n1^2
.04 = 1/n1^2
n1^2 = 100/4
n1 = 10/2 = 5
so level 10 down to level 5
To determine the electron transition that will produce infrared light of a specific wavelength in a hydrogen atom, we can use the Rydberg formula:
1/λ = R(1/n₁² - 1/n₂²)
Where λ is the wavelength of the emitted light, R is the Rydberg constant (approximately 1.097x10^7 m⁻¹), and n₁ and n₂ are the principal quantum numbers of the initial and final energy levels, respectively.
In this case, we know that the wavelength of the infrared light is 3040 nm (or 3040x10^-9 m), and we need to find the appropriate electron transition starting from orbit 10.
First, we convert the wavelength from nm to meters:
λ = 3040x10^-9 m
Next, we substitute the known values into the Rydberg formula and solve for n₂:
1/λ = R(1/n₁² - 1/n₂²)
1/(3040x10^-9) = (1.097x10^7)(1/10² - 1/n₂²)
Now, let's solve for n₂:
1/(3040x10^-9) = (1.097x10^7)(1/100 - 1/n₂²)
1/(3040x10^-9) = (1.097x10^7)(1/100 - 1/n₂²)
Now, we simplify the equation:
1/(3040x10^-9) = (1.097x10^7)(0.01 - 1/n₂²)
We multiply both sides by n₂² to eliminate the denominator:
n₂²/(3040x10^-9) = (1.097x10^7)(0.01 - 1/n₂²) * n₂²
n₂²/(3040x10^-9) = (1.097x10^7)(0.01n₂² - 1)
n₂²/(3040x10^-9) = 1.097x10^7n₂² - 1.097x10^7
We rearrange the equation:
1.097x10^7n₂² - n₂²/(3040x10^-9) = 1.097x10^7
Now, we can solve this quadratic equation for n₂² using the quadratic formula:
n₂² = [-(-1.097x10^7) ± √((-1.097x10^7)² - 4(1.097x10^7)(-1/(3040x10^-9)))] / (2(1.097x10^7))
n₂² = [1.097x10^7 ± √(1.2049x10^14 + 4.3688 )] / 2.194x10^7
n₂² = [1.097x10^7 ± √(1.2049x10^14 + 4.3688 )] / 2.194x10^7
n₂² = [1.097x10^7 ± √(1.2049x10^14 + 4.3688 )] / 2.194x10^7
n₂² = [1.097x10^7 ± √(1.2049x10^14 + 4.3688 )] / 2.194x10^7
Now, we can solve for n₂ by taking the positive root:
n₂ = √(1.097x10^14 + 4.3688 ) / 2.194x10^7
Calculating this expression will give us the value of n₂.