The higher the energy level occupied by an electron in the hydrogen atom, the larger the atom. The diameter of the atom is proportional to n^2, where n=1 labels the lowest or "ground" state, n=3 is the thrid state, and so on. If the atom's diameter is 1 x 10^(-10)m in its lowest energy state, what is its diameter in state number 50? How many unexcited atoms could be fit within this one giant atom?

50^2= 2500, so diameter increases by...

Pichku

To find the diameter of the atom in state number 50, we can use the proportionality relation given: the diameter of the atom is proportional to n^2, where n is the energy level.

Given that the diameter of the atom in its lowest energy state (n=1) is 1 x 10^(-10) m, we can write the proportionality relation as:

Diameter_1 / Diameter_50 = (n_1)^2 / (n_50)^2

Plugging in the values, we have:

1 x 10^(-10) m / Diameter_50 = (1)^2 / (50)^2

Simplifying the equation, we have:

Diameter_50 = (50)^2 * (1 x 10^(-10) m)

Diameter_50 = 2500 x 10^(-10) m

Diameter_50 = 2.5 x 10^(-8) m

Therefore, the diameter of the atom in state number 50 is 2.5 x 10^(-8) m.

To find out how many unexcited atoms could be fit within this one giant atom, we need to calculate the volume of the giant atom and divide it by the volume of an unexcited atom.

Since the diameter of the giant atom is 2.5 x 10^(-8) m, its radius would be half of that:

Radius_giant_atom = 2.5 x 10^(-8) m / 2 = 1.25 x 10^(-8) m

The volume of a sphere is given by the formula:

Volume = (4/3) * π * (radius)^3

Volume_giant_atom = (4/3) * π * (1.25 x 10^(-8) m)^3

Now, we need to calculate the volume of the unexcited atom. Since the diameter of the unexcited atom is 1 x 10^(-10) m, its radius would be half of that:

Radius_unexcited_atom = 1 x 10^(-10) m / 2 = 5 x 10^(-11) m

Volume_unexcited_atom = (4/3) * π * (5 x 10^(-11) m)^3

Finally, to calculate the number of unexcited atoms that could fit within the giant atom, we divide the volume of the giant atom by the volume of an unexcited atom:

Number_of_unexcited_atoms = Volume_giant_atom / Volume_unexcited_atom

Number_of_unexcited_atoms = [(4/3) * π * (1.25 x 10^(-8) m)^3] / [(4/3) * π * (5 x 10^(-11) m)^3]

Simplifying the equation, we have:

Number_of_unexcited_atoms = (1.25 x 10^(-8) m / 5 x 10^(-11) m)^3

Number_of_unexcited_atoms = (250)^3

Number_of_unexcited_atoms = 15,625,000

Therefore, approximately 15,625,000 unexcited atoms could fit within this one giant atom.