Use the Balmer's equation to determine:
(a). The frequency in per second of the radiation corresponding to n=5.
(c). The value of "n" corresponding to the Balmer series line at 380nm.
(a) According to Balmer's equation, the frequency (v) of radiation corresponding to a given energy level (n) can be calculated using the formula:
v = R * (1/ n₁² - 1/ n₂²)
where R is the Rydberg constant and n₁ and n₂ are the energy levels.
If we want to determine the frequency for n = 5, we need to compare it with another energy level. Let's set n₁ = 2, as the Balmer series corresponds to transitions to and from the second energy level (n₂ = 2) in the hydrogen atom.
Plugging these values into the equation, we get:
v = R * (1/2² - 1/5²)
To make calculations easier, we'll use the speed of light constant (c) to relate frequency (v) to wavelength (λ) using the equation: c = λ * v.
(b) The value of "n" corresponding to the Balmer series line at 380 nm.
The wavelength (λ) is given as 380 nm. We can use the equation:
λ = c / v
to relate wavelength (λ) with frequency (v). We know the speed of light constant (c) and can solve for frequency.
Let's solve part (a) first to find the frequency corresponding to n = 5, and then we'll use the frequency to solve part (b) for the value of "n" corresponding to 380 nm.
Note: The Rydberg constant (R) has a value of 1.097 × 10⁷ m⁻¹.
Now let's calculate the answers:
(a) Frequency (v) of the radiation corresponding to n = 5:
v = R * (1/2² - 1/5²)
v = 1.097 × 10⁷ m⁻¹ * (1/4 - 1/25)
v = 1.097 × 10⁷ m⁻¹ * (6/25)
v = 2.667 × 10⁶ m⁻¹
(b) Now, let's use the frequency to solve for the value of "n" corresponding to the Balmer series line at 380 nm:
λ = c / v
v = c / λ
v = (3 × 10⁸ m/s) / (380 × 10⁻⁹ m)
v ≈ 7.895 × 10¹⁴ Hz
Using this frequency, let's solve for "n" using Balmer's equation:
7.895 × 10¹⁴ Hz = 1.097 × 10⁷ m⁻¹ * (1/2² - 1/n²)
Rearranging the equation and solving for "n":
(1/2² - 1/n²) = (7.895 × 10¹⁴ Hz) / (1.097 × 10⁷ m⁻¹)
(1/n²) = 1/4 - (7.895 × 10⁶ m⁻¹) / (1.097 × 10⁷ m⁻¹)
(1/n²) = 1/4 - 0.720
(1/n²) = 1/4 - 0.720
(1/n²) = -0.520
n² = -1.923
n ≈ 1.39 (approximately)
Therefore, the value of "n" corresponding to the Balmer series line at 380 nm is approximately 1.39.