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Problem 67 Hard Difficulty

The average thermal energy due to the random translational motion of a hydrogen atom at room temperature is $(3 / 2) k T$ Here $k$ is the Boltzmann constant. Would a typical collision between two hydrogen atoms be likely to transfer enough energy to one of the atoms to raise its energy from the $n=1$ to the $n=2$ energy state? Explain your answer. [Note: Earth's free hydrogen is in the molecular form $\mathrm{H}_{2}$ . However, the above reasoning is similar for atomic and molecular hydrogen.]

Answer

10.2 $\mathrm{eV}$

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Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Meghan M.

McMaster University

Video Transcript

for this question. We're asked if a typical typical collision energy between 200 Adams will likely transfer enough energy from one atom to the other to raise its energy from an equals one and equals two energy states. Um and then explain your answer here. So we're going to assume that the energy of the collision all transfer is from, uh, from this three halves. Katie, work a. Here is the bolt Been constant. T is the temperature start. Ruto 30 is about 300 Kelvin so we can call this, uh, value e subsea for collision all and this is equal to three house Katie, we're again K's The Bolton Constant and T is the temperature. I'm also gonna multiply this by the conversion between jewels and electron volts because I want to put my answer an electron volts to easily compare it to the energy that I calculate from going from the in equals one to the n equals two state. So that conversion is one electron volt is equal to 1.6 times 10 to the minus 19 jewels. Okay, so carrying out this operation, we find that this clinic collision all energy is equal to zero 0.39 electron volts. So now the see if that is enough to raise it from the in equals one of the n equals two orbital. We can, uh we can calculate the energy required to raise an electron from the n equals one of the n equals two orbital. So we'll call this Yeah, 12 from to go from the 1 to 2 orbital. So you want to is going to be equal to the equation that governs. This is negative 13.6 electron volts, which is the ground state energy of hydrogen multiplied by one over two squared. Our final state, minus 1/1 squared our initial state. So carrying out this operation, we find that this energy required is equal to 10 0.2 electron volts. So it's pretty clear here that the thermal energy at room temperature is three orders of magnitude too small to excite the hydrogen atom. So we can say from these calculations, it is apparent at the thermal energy is to small to excite the Adam from n equals 12 and equals two. And that is a solution to our question so we can go ahead and box it in

University of Kansas
Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Meghan M.

McMaster University