Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 2 to the level n = 1.

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

Look up R. That's the Rydberg constant. It's about 1.09 E7

Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 5 to the level n = 1.

To calculate the wavelength, we can use the Rydberg formula:

1/λ = R * (1/n₁² - 1/n₂²)

Where:
λ is the wavelength of the spectral line
R is the Rydberg constant (approximated to 1.097 x 10^7 m⁻¹)
n₁ is the initial energy level (n = 2 in this case)
n₂ is the final energy level (n = 1 in this case)

Let's substitute the values into the formula:

1/λ = (1.097 x 10^7 m⁻¹) * (1/2² - 1/1²)

Simplifying:

1/λ = (1.097 x 10^7 m⁻¹) * (1/4 - 1)

1/λ = (1.097 x 10^7 m⁻¹) * (-3/4)

1/λ = -3.27 x 10^6 m⁻¹

Now, let's solve for λ by taking the reciprocal of both sides:

λ = -1 / (-3.27 x 10^6 m⁻¹)

λ = 3.06 x 10⁻⁷ m

To convert this to nanometers, we multiply by 10⁹:

λ = 3.06 x 10⁻⁷ m * 10⁹ nm/m

λ ≈ 306 nm

Therefore, the wavelength of the spectral line produced is approximately 306 nanometers.

To calculate the wavelength of the spectral line produced when an electron undergoes a transition in a hydrogen atom, we can use the Rydberg formula:

1/λ = R * (1/n₁² - 1/n₂²)

Where:
λ is the wavelength of the spectral line.
R is the Rydberg constant, approximately 1.097 × 10⁷ m⁻¹.
n₁ is the initial energy level of the electron.
n₂ is the final energy level of the electron.

In this case, the electron transitions from n = 2 to n = 1. We can substitute these values into the formula and calculate the wavelength:

1/λ = 1.097 × 10⁷ m⁻¹ * (1/1² - 1/2²)
= 1.097 × 10⁷ m⁻¹ * (1 - 1/4)
= 1.097 × 10⁷ m⁻¹ * (3/4)
= 8.2275 × 10⁶ m⁻¹

Now that we have the reciprocal of the wavelength, we can calculate the wavelength by taking the reciprocal of this value:

λ = 1 / (8.2275 × 10⁶ m⁻¹)
= 1.214 × 10⁻⁷ m

To convert this wavelength to nanometers, we multiply by a conversion factor of 10⁹:

λ = 1.214 × 10⁻⁷ m * 10⁹ nm/m
= 121.4 nm

Therefore, the wavelength of the spectral line produced when an electron in a hydrogen atom undergoes the transition from n = 2 to n = 1 is approximately 121.4 nm.