What electron transition in a hydrogen atom, ending in the orbit 4, will produce light of wavelength 2170 nm ?

1/wavelength = R(1/16 - 1/x^2)

R=1.09737E7
Solve for x.

To determine the electron transition in a hydrogen atom that produces light of a specific wavelength, we can use the Rydberg formula:

1/λ = R * (1/n₁² - 1/n₂²)

where:
- λ is the wavelength of light emitted or absorbed (in nm)
- R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹)
- n₁ is the initial energy level
- n₂ is the final energy level

In this case, the wavelength of light is given as 2170 nm.

Let's solve the equation for n₁ and n₂:

1/2170 = 1.097 × 10^7 * (1/n₁² - 1/4²)

Simplifying:

1/2170 = 1.097 × 10^7 * (1/n₁² - 1/16)

Now, let's solve for n₁:

1/n₁² - 1/16 = 1/2170 * (1.097 × 10^7)

1/n₁² - 1/16 = 5050

1/n₁² = 1/16 + 5050

1/n₁² = 5050/16

n₁² = 16/5050

n₁ ≈ √(16/5050)

n₁ ≈ 0.08

Since n₁ represents the initial energy level, it cannot be a decimal value or less than 1. Therefore, there is no electron transition in a hydrogen atom with a wavelength of 2170 nm that ends in the 4th orbit.

To determine the electron transition in a hydrogen atom that produces light of a particular wavelength, you can use the Rydberg formula. The Rydberg formula is given as:

1/λ = R * (1/n₁^2 - 1/n₂^2)

Where:
- λ is the wavelength of the emitted light in meters,
- R is the Rydberg constant (approximately 1.097 x 10^7 1/m),
- n₁ is the initial orbit of the electron, and
- n₂ is the final orbit of the electron.

In this case, we are given the wavelength λ as 2170 nm. However, we need to convert it to meters by dividing it by 1,000,000 (since 1 nm = 1/1,000,000 meters). Hence, λ = 2170 nm / 1,000,000 = 2.170 x 10^(-6) meters.

Plugging the values into the Rydberg formula, we can rearrange it to solve for the final orbit n₂:

1/λ = R * (1/n₁^2 - 1/n₂^2)

Rearranging the formula, we get:

1/n₂^2 = (1/λ) / R + 1/n₁^2

Assuming the electron starts in the ground state (n₁ = 1), we can substitute the values and solve for n₂.

1/n₂^2 = (1/2.170 x 10^(-6)) / (1.097 x 10^7) + 1/1^2

Simplifying further:

1/n₂^2 = 0.458 x 10^(7) + 1

1/n₂^2 = 0.458 x 10^(7) + 1

1/n₂^2 = 1.458 x 10^(7)

Taking the square root of both sides:

1/n₂ = √(1.458 x 10^(7))

n₂ = 1 / √(1.458 x 10^(7))

n₂ ≈ 4.43

Since the orbits in hydrogen are discrete, the value of n₂ must be rounded to the nearest whole number, which gives n₂ = 4.

Therefore, the electron transition that produces light of wavelength 2170 nm in a hydrogen atom is from the initial orbit 1 to the final orbit 4.