Calculate the bonding energy of two ions

In summary, the conversation discusses differentiating and setting equations equal to zero in order to solve for a variable. The conversation also addresses an error in taking the derivative and provides the correct solution. The final result is that ##r = \sqrt[n-3]{\frac{Bn}{A}}## is the correct solution for the equation being discussed.
  • #1
Jaccobtw
163
32
Homework Statement
$$E_N = \frac{-A}{r} + \frac{B}{r^{n}}$$

Calculate the bonding energy ##E_0## in terms of the parameters A, B, and n using
the following procedure:
1. Differentiate ##E_N## with respect to r, and then set the resulting expression equal
to zero, because the curve of ##E_N## versus r is a minimum at ##E_0##.
2. Solve for r in terms of A, B, and n, which yields ##r_0##, the equilibrium interionic
spacing.
3. Determine the expression for ##E_0## by substituting ##r_0## for r
Relevant Equations
$$E_N = \frac{-A}{r} + \frac{B}{r^{n}}$$
1.) So first I differentiate and set it equal to 0 and get:
$$\frac{A}{r^2} -\frac{Bn}{r^{n-1}} = 0$$

2.) When solving for r, I'm not quite sure how to take away the exponent so I get up to the second to last step:

$$r^{n-3} = \frac{Bn}{A}$$

Would it be:

$$r = \sqrt[n-3]{\frac{Bn}{A}}$$

?

Am I doing this problem correctly?

Thank you
 
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  • #2
Correct. You can note the result as ##r_0##, i.e.,
$$
r_0 = \left( \frac{Bn}{A} \right)^{1/(n-3)}
$$
 
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  • #3
Jaccobtw said:
$$E_N = \frac{-A}{r} + \frac{B}{r^{n}}$$
1.) So first I differentiate and set it equal to 0 and get:
$$\frac{A}{r^2} -\frac{Bn}{r^{n-1}} = 0$$
2.) When solving for r, I'm not quite sure how to take away the exponent so I get up to the second to last step:
$$r^{n-3} = \frac{Bn}{A}$$
Would it be:
$$r = \sqrt[n-3]{\frac{Bn}{A}}\ \ \ ?$$
Am I doing this problem correctly?

Thank you
You made an error in taking the derivative.

Writing ##E_N## as ##\displaystyle \quad \quad E_N= -A\,r^{-1} + B\,r^{-n} \quad## may help.

Then ##\displaystyle \quad \quad \dfrac{E_N}{dr}= A\,r^{-2} - B\,r^{-n-1} = A\,r^{-2} - B\,r^{-(n+1)}##

##\displaystyle \quad \quad \quad \quad \quad \quad = \dfrac{A}{r^2} - \dfrac{nB}{r^{(n+1)}} \quad##
 
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1. What is the definition of bonding energy?

Bonding energy is the amount of energy required to break the bond between two ions in a compound.

2. How is bonding energy calculated?

Bonding energy can be calculated using the equation E = (k * Q1 * Q2)/d, where E is the bonding energy, k is the Coulomb constant, Q1 and Q2 are the charges of the two ions, and d is the distance between them.

3. What factors affect the bonding energy between two ions?

The bonding energy between two ions is affected by the magnitude of the charges of the ions, the distance between them, and the type of bond (ionic, covalent, etc.).

4. Can bonding energy be negative?

Yes, bonding energy can be negative if the two ions have opposite charges and are attracted to each other. This is typically seen in ionic bonds.

5. How does bonding energy impact the stability of a compound?

The higher the bonding energy between two ions, the more stable the compound is. This is because a higher bonding energy means it takes more energy to break the bond, making it less likely for the compound to decompose or react with other substances.

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