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Problem 47 Hard Difficulty

(a) How much energy is required to cause an electron in hydrogen to move from the $n=1$ state to the $n=2$ state? $(\mathrm{b})$ If the electrons gain this energy by collision between hydrogen atoms in a high-temperature gas, find the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by $3 k_{\mathrm{B}} T / 2,$ where $k_{\mathrm{B}}$ is the Boltzmann constant.

Answer

a) 10.2 $\mathrm{eV}$
b) $7.88 \times 10^{4} K$

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Top Physics 103 Educators
Elyse G.

Cornell University

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Meghan M.

McMaster University

Video Transcript

in question. A. We have to find the energy required to raise an electron in a hydrogen atom from the ground state to the second energy level. I'm gonna call this energy required to raise the electron, Delta E and from conservation of energy. We know that this energy is equal to the energy of the final energy level. So that's e to mine is the energy of the ground state you want. Now the energy of the ground state is minus 13.6 electoral votes. In the energy of the second under your level is equal to minus 13.6, divided by two square. That's four converts. Okay, because the energy of the end energy level of the hydrogen atom his minus is the actual. It's the energy of a ground states divided by in squared. So this is minus 3.4 electoral votes. So that day is minus 3.4 plus 13.6, and this is 10.2 electoral votes. This is the energy required in question be We have to assume that an electron gains this amount of energy that we calculated in question a delta ah, due to a collision of collisions of the hydrogen atom. Actually, collisions have different hydrogen atoms in ah, certain gas. Okay. And we have to calculate what is the minimum temperature T of the gas. Ah, in the situation. Okay. And what the exercise asks us to remember is that the thermal energy off a gas so e t is given by three have off k b. See, uh, so the kinetic energy of the guess is three houses K B t. And this is eat greater or equal than the energy that was gained by the elections. Don't tell you. Okay, So have a three halves of K B t is greater or equal than delta. So have a tea is greater or equal than Delta E, actually, 2/3 of Delta u K B. So this is 2/3 of Delta is 10 points to electoral votes and K B boats. One constant is 8.6 17. 10% of the minus five electoral votes Calvin to the minus one. So T is greater or equal, then 7.89 times 10 to the fourth Calvin. So the minimum temperature team in is able to 7.89. Time sent the fourth, Cowlings

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Meghan M.

McMaster University