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Chapter 5: Problem 14

(a) What is the electrostatic potential energy (in joules) between two electrons that are separated by \(460 \mathrm{pm} ?\) (b) What is the change in potential energy if the distance separating the two electrons is increased to \(1.0 \mathrm{nm}\) ? (c) Does the potential energy of the two particles increase or decrease when the distance is increased to \(1.0 \mathrm{nm}\) ?

Short Answer

Expert verified
The electrostatic potential energy between two electrons separated by \(460 \, \mathrm{pm}\) is approximately \(-2.50 \times 10^{-18} \, \mathrm{J}\). When the distance between the electrons is increased to \(1.0 \, \mathrm{nm}\), the change in potential energy is approximately \(2.27 \times 10^{-18} \, \mathrm{J}\), indicating that the potential energy increases when the distance is increased to \(1.0 \, \mathrm{nm}\).

Step by step solution

01

Part (a): Electrostatic potential energy between two electrons separated by 460 pm

First, we need to convert the given distance from picometers (pm) to meters (m): \(460 \, \mathrm{pm} = 460 \times 10^{-12} \, \mathrm{m}\) Now, we can use the formula for electrostatic potential energy: \[U = \frac{kq_1q_2}{r}\] Since both charges are electrons, we have: \[U = \frac{k(-e)(-e)}{r}\] Substitute the given values and constants: \[U = \frac{(8.99 \times 10^{9} Nm^2/C^2)((-1.60 \times 10^{-19} C)^2)}{460 \times 10^{-12} m}\] Calculate the electrostatic potential energy: \[U \approx -2.50 \times 10^{-18} \, \mathrm{J}\]
02

Part (b): Change in potential energy

First, convert the new distance from nanometers (nm) to meters (m): \(1.0 \, \mathrm{nm} = 1.0 \times 10^{-9} \, \mathrm{m}\) Calculate the new electrostatic potential energy with the increased distance: \[U' = \frac{k(-e)(-e)}{r'}\] \[U' = \frac{(8.99 \times 10^{9} Nm^2/C^2)((-1.60 \times 10^{-19} C)^2)}{1.0 \times 10^{-9} m}\] Calculate the new electrostatic potential energy: \[U' \approx -2.30 \times 10^{-19} \, \mathrm{J}\] Now, find the change in potential energy: \[\Delta U = U' - U \] \[\Delta U \approx -2.30 \times 10^{-19} \, \mathrm{J} - (-2.50 \times 10^{-18} \, \mathrm{J})\] Calculate the change in potential energy: \[\Delta U \approx 2.27 \times 10^{-18} \, \mathrm{J}\]
03

Part (c): Increase or decrease in potential energy

Compare the initial potential energy \(U\) to the new potential energy \(U'\). Since the initial potential energy was about \(-2.50 \times 10^{-18} \, \mathrm{J}\) and the new potential energy is about \(-2.30 \times 10^{-19} \, \mathrm{J}\), the potential energy has increased. This is because when the distance between two like charges (such as the electrons in this case) increases, the electrostatic force between them decreases, making their potential energy closer to zero.

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Most popular questions from this chapter

When solutions containing silver ions and chloride ions are mixed, silver chloride precipitates $$ \mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \longrightarrow \operatorname{AgCl}(s) \quad \Delta H=-65.5 \mathrm{~kJ} $$ (a) Calculate \(\Delta H\) for the production of \(0.450 \mathrm{~mol}\) of \(\mathrm{AgCl}\) by this reaction. (b) Calculate \(\Delta H\) for the production of \(9.00 \mathrm{~g}\) of $\mathrm{AgCl} . (\mathbf{c})\( Calculate \)\Delta H\( when \)9.25 \times 10^{-4} \mathrm{~mol}$ of \(\mathrm{AgCl}\) dissolves in water.

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In a thermodynamic study, a scientist focuses on the properties of a solution in an apparatus as illustrated. A solution is continuously flowing into the apparatus at the top and out at the bottom, such that the amount of solution in the apparatus is constant with time. (a) Is the solution in the apparatus a closed system, open system, or isolated system? (b) If the inlet and outlet were closed, what type of system would it be?

The standard enthalpies of formation of gaseous propyne $\left(\mathrm{C}_{3} \mathrm{H}_{4}\right),\( propylene \)\left(\mathrm{C}_{3} \mathrm{H}_{6}\right),\( and propane \)\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\( are \)+185.4,+20.4,\( and \)-103.8 \mathrm{~kJ} / \mathrm{mol}$, respectively. (a) Calculate the heat evolved per mole on combustion of each substance to yield \(\mathrm{CO}_{2}(g)\) and $\mathrm{H}_{2} \mathrm{O}(g) .\( (b) Calculate the heat evolved on combustion of \)1 \mathrm{~kg}\( of each substance. \)(\mathbf{c})$ Which is the most efficient fuel in terms of heat evolved per unit mass?
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