Wavelength, temperature in transition

[akance]

Hi! I really need some help
I'm really really not good at physics...so
Here it is

1) The electron energy of a hydrogen atom is given by -C/n^2 relative to zero energy at infinite separation between the electron and the proton (n is the principle quantum number, and C is a constant). For detection of the n=2 -> n=3 transition(656.3 nm in the Balmer Series), the electron is the ground state of the hydrogen atom needs to be excited first to the n=2 state. Calculate the wavelength (in nm) of the absorption line in the starlight corresponding to the n=1 -> n=2 transition.

2) According to Wien's law, the wavelength corresponding to the maximum light intensity emited from a blackbody at temperature T is given by (wavelength)(Temperature) = 2.9 mmK
Calculate the surface temperature of a star whose blackbody radiation has a peak intensity corresponding to the n=1 -> n=2 excitation of hydrogen

3) The ground state of hydrogen is split into two hyperfine levels due to the interaction between the magnetic moment of the proton and that of the electron. In 1951, Purcell discovered a spectral line at 1420 MHz due to the hyperfine transition of hydrogen in interstellar space. Hydrogen in interstellar space cannot be excited electronically by starlight. However, the cosmic background radiation, equivalent to 2.7K, can cause the hyperfine transition. Calculate the temperature of a blackbody whose peak intensity corresponds to the 1420 MHz transition

Can you please! help me with these 3 problems...! I am so confused!

 

[JoAuSc]

Hi! I really need some help
I'm really really not good at physics...so
Here it is

1) The electron energy of a hydrogen atom is given by -C/n^2 relative to zero energy at infinite separation between the electron and the proton (n is the principle quantum number, and C is a constant). For detection of the n=2 -> n=3 transition(656.3 nm in the Balmer Series), the electron is the ground state of the hydrogen atom needs to be excited first to the n=2 state. Calculate the wavelength (in nm) of the absorption line in the starlight corresponding to the n=1 -> n=2 transition.

2) According to Wien's law, the wavelength corresponding to the maximum light intensity emited from a blackbody at temperature T is given by (wavelength)(Temperature) = 2.9 mmK
Calculate the surface temperature of a star whose blackbody radiation has a peak intensity corresponding to the n=1 -> n=2 excitation of hydrogen

3) The ground state of hydrogen is split into two hyperfine levels due to the interaction between the magnetic moment of the proton and that of the electron. In 1951, Purcell discovered a spectral line at 1420 MHz due to the hyperfine transition of hydrogen in interstellar space. Hydrogen in interstellar space cannot be excited electronically by starlight. However, the cosmic background radiation, equivalent to 2.7K, can cause the hyperfine transition. Calculate the temperature of a blackbody whose peak intensity corresponds to the 1420 MHz transition

Can you please! help me with these 3 problems...! I am so confused!

How much of these problems can you do?

 

[thomate1]

1) The wavelength of the absorption line corresponding to the n=1 -> n=2 transition can be calculated using the Rydberg formula: 1/λ = R(1/n1^2 - 1/n2^2), where R is the Rydberg constant (1.0974x10^7 m^-1), n1=1 and n2=2. Plugging in these values, we get: 1/λ = 1.0974x10^7 (1 - 1/4) = 0.8228x10^7 m^-1. Converting this to nm, we get a wavelength of 121.6 nm.

2) Using the given equation, we can rearrange to solve for the temperature: T = (wavelength)/(2.9 mmK). Plugging in the wavelength from part 1 (121.6 nm), we get a temperature of approximately 4181 K.

3) To calculate the temperature of the blackbody corresponding to the 1420 MHz transition, we can use the formula: T = (wavelength)/(2.9 mmK). The wavelength in this case is given in MHz, so we need to convert it to meters first. 1420 MHz is equivalent to 2.11x10^-4 m. Plugging this into the formula, we get a temperature of approximately 7.28 K. This is very close to the temperature of the cosmic background radiation, which is expected since this is the temperature that can cause the hyperfine transition.

 

[baba89]

1) The energy difference between the n=1 and n=2 states can be calculated using the formula -C(1/nf^2 - 1/ni^2), where nf=3 and ni=2. This gives us an energy difference of -C(1/9 - 1/4) = -3C/36 = -C/12. To excite the electron from the n=1 state to the n=2 state, this energy difference must be supplied by the incoming starlight. We can use the formula E=hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. Setting E equal to -C/12 and solving for λ, we get a wavelength of 121.6 nm.

2) Using Wien's law, we can set the wavelength equal to 656.3 nm (corresponding to the n=1 -> n=2 transition) and solve for the temperature T. This gives us T= 2.9 mmK/656.3 nm = 4422 K.

3) The energy difference between the two hyperfine levels is given by E=hf, where h is Planck's constant and f is the frequency of the spectral line. We can rearrange this equation to solve for f, which gives us f=E/h. Plugging in the energy difference between the two hyperfine levels (1420 MHz) and Planck's constant, we get f= -C/12h. To find the temperature corresponding to this frequency, we can use Wien's law again, setting the wavelength equal to the transition frequency (1420 MHz) and solving for T. This gives us T= 2.9 mmK / 1420 MHz = 2.04 K.