Calculating Kinetic Energy of Recoiling Hydrogen Atom

In summary, the problem involves a hydrogen atom emitting Lyman radiation while at rest in the laboratory. The task is to calculate the recoil kinetic energy of the atom and determine what fraction of the excitation energy of the n = 2 state is carried by the recoiling atom. The equation E=-13.6/n^{2}ev may be relevant and the conservation of momentum must be used to solve the problem. The departing photon carries momentum and the atom must have an equal and opposite momentum, resulting in a nonzero kinetic energy. The sum of the energies of the atom and the photon is equal to the Lyman transition energy.
  • #1
patapat
20
0

Homework Statement


A hydrogen atom at rest in the laboratory emits the Lyman radiation.
(a) Compute the recoil kinetic energy of the atom.
(b) What fraction of the excitation energy of the n = 2 state is carried by the recoiling atom? (Hint: Use conservation of momentum.)

Homework Equations


E=-13.6/n[tex]^{2}[/tex]ev


The Attempt at a Solution


I'm not sure if this equation is relevant, but I'm not sure what they are referring to when they say recoil kinetic energy of the atom.
 
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  • #2
(Hint: Use conservation of momentum.) The departing photon carries momentum. The atom must have an equal and opposite momentum (hence a nonzero kinetic energy). The total energy of the two is the Lyman transition energy.
 
  • #3


I would first clarify any confusion about the terminology used in the problem. The term "recoil kinetic energy" in this context likely refers to the kinetic energy of the atom after it emits the Lyman radiation, due to the conservation of momentum.

To calculate the recoil kinetic energy, we can use the formula:
KE = (1/2)mv^2
where m is the mass of the hydrogen atom and v is its velocity after emitting the radiation.

To find the velocity, we can use the conservation of momentum equation:
m1v1 = m2v2
where m1 is the mass of the hydrogen atom before emission and m2 is the mass of the hydrogen atom after emission (which is essentially the same as m1).

Since the atom is initially at rest, we can set v1 = 0. Therefore, we can rearrange the equation to solve for v2:
v2 = (m1/m2) * v1
Since m1 = m2, we can simplify this to:
v2 = v1

This means that the velocity of the atom after emission is equal to the velocity of the emitted radiation.

To find the velocity of the emitted radiation, we can use the formula for wavelength of the Lyman series:
λ = R*(1/n1^2 - 1/n2^2)
where R is the Rydberg constant and n1 and n2 are the initial and final energy levels, respectively.

In this case, n1 = 2 and n2 = ∞ (since the radiation is being emitted from the excited state to the ground state). So, we can rewrite the formula as:
λ = R*(1/2^2 - 0)
λ = R/4

Since we know the wavelength of the Lyman radiation, we can use the formula for the speed of light to find the velocity:
c = fλ
where c is the speed of light, f is the frequency (which we can assume to be the same as the Lyman series frequency) and λ is the wavelength we just calculated.

Rearranging the formula, we get:
v = fλ
Substituting in the values, we get:
v = (3.29*10^15 Hz) * (R/4)
v = (3.29*10^15
 

1. How is kinetic energy of a recoiling hydrogen atom calculated?

The kinetic energy of a recoiling hydrogen atom can be calculated using the formula KE = 1/2 * m * v^2, where KE is the kinetic energy, m is the mass of the hydrogen atom, and v is the velocity at which the atom is recoiling.

2. What is the mass of a hydrogen atom?

The mass of a hydrogen atom is approximately 1 atomic mass unit (u) or 1.67 x 10^-27 kilograms.

3. How is the velocity of a recoiling hydrogen atom determined?

The velocity of a recoiling hydrogen atom can be determined by measuring the change in momentum of the atom. This can be done using techniques such as time-of-flight spectroscopy or by measuring the Doppler shift of the emitted light from the atom.

4. What is the unit of kinetic energy for a recoiling hydrogen atom?

The unit of kinetic energy for a recoiling hydrogen atom is joules (J), which is a unit of energy in the International System of Units (SI).

5. Can the kinetic energy of a recoiling hydrogen atom be negative?

No, the kinetic energy of a recoiling hydrogen atom cannot be negative. Kinetic energy is a scalar quantity and is always positive. If the velocity of the recoiling atom is negative, it simply means the atom is moving in the opposite direction of its initial motion, but the kinetic energy value will still be positive.

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