What wavelenght of radiation is necessary to ionize hydrogen.
To determine the wavelength of radiation necessary to ionize hydrogen, we need to use the concept of ionization energy and the Balmer series. The ionization energy of hydrogen is the minimum amount of energy required to remove an electron from a hydrogen atom, resulting in the formation of a hydrogen ion.
The formula to calculate the wavelength of radiation is given by the Rydberg formula:
1/λ = R * ((1/n1^2) - (1/n2^2))
Where:
- λ is the wavelength of the radiation
- R is the Rydberg constant (equal to 1.097 × 10^7 m^-1)
- n1 and n2 are positive integers representing the energy levels of hydrogen
Since we want to find the wavelength of radiation that will ionize hydrogen, we need to determine the energy level of the ionized hydrogen atom. When an electron is completely removed from a hydrogen atom, it transitions from level n to infinity. Therefore, we can substitute n2 with infinity in the Rydberg formula, which simplifies the equation to:
1/λ = R * (1/n1^2)
To find the wavelength of radiation needed to ionize hydrogen, we need to find the smallest possible value for n1 that satisfies the equation. The value of n1 will depend on the range of the Balmer series, which represents the visible spectrum.
The Balmer series is given by the formula:
1/λ = R * (1/4 - 1/n^2)
Where n takes on integer values greater than 2. We can see that the Balmer series starts at n=3 since plugging in n=2 would result in 1/λ being equal to infinity, which is not physically possible. Thus, the minimum value for n1 in our ionization equation is 3.
Plugging in n1=3 into the simplified Rydberg formula, we get:
1/λ = R * (1/3^2)
1/λ = R * (1/9)
Now, we can plug in the value of the Rydberg constant:
1/λ = (1.097 × 10^7 m^-1) * (1/9)
Simplifying, we find:
1/λ ≈ 1.22 × 10^6 m^-1
Now we can solve for the wavelength by taking the reciprocal of both sides:
λ ≈ 8.2 × 10^-7 m
Therefore, the approximate wavelength of radiation necessary to ionize hydrogen is 8.2 × 10^-7 meters, which corresponds to the ultraviolet region of the electromagnetic spectrum.