`E_(n)=-(2,18times10^(-18))(Z^(2))/(n^(2))`
Electrons follow paths, called orbits around the nucleus of an atom. According to the Bohr model, the energy in joules of an atom's nth orbit containing a single electron is given by the formula shown above, where Z is the atomic number of the atom and n is the orbit number. Which of the following equations gives the atomic number of an atom given the energy of its nth orbit (assuming there is a single electron is that orbit)?

Options

A. `Z=10^(9)sqrt((-n^(2)E_(n))/(2.18))`
B. `Z=(1)/(10^(9))sqrt((-n^(2)E_(n))/(2.18))`
C. `Z=10^(9)sqrt((-2.18E_(n))/(n^(2)))`
D. `Z=(1)/(10^(9))sqrt((-2.18E_(n))/(n^(2)))`

Answer

A

Explanation

Difficult: Hard
Category: Passport to Advanced Math/ Exponents
Strategic Advice: A question like this requires patience and concentrations. As with any equation, use inverse operations to solve for (isolate) the desire variable.
Getting to the Answer: To solve the equation for Z, start by multiplying both sides of the equation by the only expression that's in the denominator on the right hand side `(n^(2))`. Then, divide by everything that Z is being multiplied by `(-2.18times10^(-18))`. To remove negative exponent, you and move `10^(-18)` to the numenator (but not 2.18 because it is not being raised to a negative power):
`E_(n)=-(2.18times10^(-18))(Z^(3))/(n^(2))`
`n^(2)(E_(n))=[-(2.18times10^(-18))(Z^(2))/(n^(2))]n^(2)`
`n^(2)E_(n)=-(2.18times10^(-18))Z^(2)`
`(n^(2)E_(n))/(-2,18times10^(-18))=(-(2.18times10^(-18))Z^(2))/(-(2.18times10^(-18)))`
`-(n^(2)E_(n)times10^(18))/(2.18)=Z^(2)`
You're asked for Z (not `Z^(2)`),so take the square root of both sides of the equation to find theat `Z=sqrt(-(n^(2)E_(n)times10^(18))/(2.18))`. This is not one of the answer choices, but you can take the square root of `10^(18)` by writing it as `(10^(9))^(2)`, which results in the equation `Z=10^(9)sqrt(-(n^(2)E_(n))/(2.18))`. This means (A) is correct.
Note: You can recall that you can't take the square root of a negative number, but electron energy is given as a negative number, which means you're multiplying a negative number by another negative number under the radical, resulting in a positive quantity.