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 n2, where n = 1 labels the lowest, or "ground" state, n = 2 is the second state, n = 3 is the third state, and so on. If the atom's diameter is 1 multiplied by 10-10 m in its lowest energy state, what is its diameter in state number 52? 1 m How many unexcited atoms could be fit within this one giant atom?
<<If the atom's diameter is 1 multiplied by 10^-10 m in its lowest energy state, what is its diameter in state number n = 52? >>
52^2 times higher than the n = 1 diameter
<< How many unexcited atoms could be fit within this one giant atom?>>
[52^2]^3 = 52^6 = ____
To determine the diameter of the atom in state number 52, we can use the relationship stated in the question, which is that the diameter of the atom is proportional to n^2, where n represents the state number.
To find the diameter in state number 52, we can use the following formula:
diameter in state 52 / diameter in lowest energy state = (state 52)^2 / (lowest energy state)^2
Let's calculate it step by step:
diameter in state 52 / (1 x 10^(-10) m) = (52^2) / (1^2)
Now, let's solve for the diameter in state 52:
diameter in state 52 = (52^2) x (1 x 10^(-10) m)
diameter in state 52 = 2704 x (1 x 10^(-10) m)
diameter in state 52 = 2704 x 10^(-10) m
diameter in state 52 = 2.704 x 10^(-7) m
Therefore, the diameter of the atom in state number 52 is 2.704 x 10^(-7) meters.
Now, let's move on to the second part of the question: How many unexcited atoms could be fit within this one giant atom?
To calculate the number of unexcited atoms that can fit within the state 52 atom, we can use the formula for the volume of a sphere and divide the volume of the state 52 atom by the volume of an unexcited atom in the lowest energy state (given that the diameter is proportional to n^2).
The volume of a sphere is given by the formula:
Volume = (4/3) x π x (radius)^3
The radius in this case will be half of the diameter.
Let's calculate the number of unexcited atoms that can fit within the state 52 atom:
Number of unexcited atoms = (Volume of state 52 atom) / (Volume of an unexcited atom in lowest energy state)
First, we calculate the volume of the state 52 atom:
Volume of state 52 atom = (4/3) x π x (radius of state 52 atom)^3
Volume of state 52 atom = (4/3) x π x ((1/2) x diameter of state 52 atom)^3
Volume of state 52 atom = (4/3) x π x ((1/2) x 2.704 x 10^(-7) m)^3
Now, let's calculate the volume of an unexcited atom in the lowest energy state:
Volume of unexcited atom in lowest energy state = (4/3) x π x ((1/2) x diameter of lowest energy state atom)^3
Volume of unexcited atom in lowest energy state = (4/3) x π x ((1/2) x 1 x 10^(-10) m)^3
Finally, we can calculate the number of unexcited atoms that can fit within the state 52 atom:
Number of unexcited atoms = (Volume of state 52 atom) / (Volume of unexcited atom in lowest energy state)
Number of unexcited atoms = [(4/3) x π x ((1/2) x 2.704 x 10^(-7) m)^3] / [(4/3) x π x ((1/2) x 1 x 10^(-10) m)^3]
Number of unexcited atoms = [(1/8) x (2.704 x 10^(-7))^3] / [(1/8) x (1 x 10^(-10))^3]
Number of unexcited atoms = (2.704 x 10^(-7))^3 / (1 x 10^(-10))^3
Number of unexcited atoms = (2.704^3) x (10^(-7))^3 / (1^3) x (10^(-10))^3
Number of unexcited atoms = 19.621 x 10^(9)
Therefore, approximately 19.621 billion unexcited atoms can fit within the state 52 atom.