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Ne4ueva [31]
3 years ago
6

Practice entering numbers that include a power of 10 by entering the diameter of a hydrogen atom in its ground state, dH = 1.06

× 10⁻¹⁰m, into the answer box.
Express the diameter of a ground-state hydrogen atom in meters using a power of 10.
Physics
1 answer:
Slav-nsk [51]3 years ago
7 0

Answer:

1.06085\times 10^{-10}\ m

Explanation:

h = Planck's constant = 6.626\times 10^{-34}\ m^2kg/s

m = Mass of electron = 9.11\times 10^{-31}\ kg

k = Coulomb constant = 8.99\times 10^{9}\ Nm^2/C^2

e = Charge of electron = 1.6\times 10^{-19}\ C

n = 1 (ground state)

Angular momentum is given by

L=mvr

From Bohr's atomic model we have

L=\dfrac{nh}{2\pi}

mvr=\dfrac{nh}{2\pi}\\\Rightarrow v=\dfrac{nh}{2\pi mr}

The centripetal force will balance the electrostatic force

\dfrac{ke^2}{r^2}=\dfrac{mv^2}{r}\\\Rightarrow \dfrac{ke^2}{r}=mv^2\\\Rightarrow \dfrac{ke^2}{r}=m(\dfrac{nh}{2\pi mr})^2\\\Rightarrow r=\dfrac{n^2h^2}{4\pi^2mke^2}\\\Rightarrow r=\dfrac{1^2\times (6.626\times 10^{-34})^2}{4\pi^2 \times 9.11\times 10^{-31}\times 8.99\times 10^{9}\times (1.6\times 10^{-19})^2}\\\Rightarrow r=5.30426\times 10^{-11}\ m

The diameter is 2\times 5.30426\times 10^{-11}=1.06085\times 10^{-10}\ m

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