Part E The ionic molecule potassium bromide (KBr), made up of a positive potassium ion (K+) of charge +e= 1.60 x 10-19 C and a negative bromine ion (Br-) of charge -e = -1.60 x 10-19 C, has an electric dipole moment of 3.50 x 10-29 Cm. Calculate the distance between the two ions. Express your answer with the appropriate units. d = 2.19×10-10 m Submit Previous Answers Part F Correct At a certain point along the axis that connects the ions, the electric field due to the KBr molecule has magnitude 8.00 x 104 N/C. How far from the center of the molecule is this point? Express your answer with the appropriate units. x = 1.99×10-8 m please show steps for how to find f and provide all the work The ionic molecule potassium bromide (KBr), made up of a positive potassium ion (K+) of charge +e=1.60×10-19C and a negative bromine ion (Br-) of charge -e=-1.60×10-19C, has an electric dipole moment of 3.50×10-29C·m. Calculate the distance between the two ions. Express your answer with the appropriate units. Part F At a certain point along the axis that connects the ions, the electric field due to the KBr molecule has magnitude 8.00×104 N/C. How far from the center of the molecule is this point? Express your answer with the appropriate units.
The correct answer and explanation is:
Solution and Explanation
Part E: Finding the Distance Between the Ions
The electric dipole moment (pp) is given by: p=q⋅dp = q \cdot d
where:
- p=3.50×10−29p = 3.50 \times 10^{-29} C·m (given),
- q=1.60×10−19q = 1.60 \times 10^{-19} C (charge of each ion),
- dd is the distance between the two ions.
Rearrange the formula to solve for dd: d=pqd = \frac{p}{q}
Substituting the values: d=3.50×10−29 C\cdotpm1.60×10−19 Cd = \frac{3.50 \times 10^{-29} \text{ C·m}}{1.60 \times 10^{-19} \text{ C}} d=2.19×10−10 md = 2.19 \times 10^{-10} \text{ m}
Thus, the distance between the potassium and bromine ions is 2.19×10−102.19 \times 10^{-10} m.
Part F: Finding the Distance Where the Electric Field is Given
For a dipole, the electric field along the axial line at a distance xx from the center of the dipole is given by: E=14πϵ0⋅2px3E = \frac{1}{4\pi\epsilon_0} \cdot \frac{2p}{x^3}
Rearrange to solve for xx: x3=2p4πϵ0Ex^3 = \frac{2p}{4\pi\epsilon_0 E} x=(2p4πϵ0E)13x = \left( \frac{2p}{4\pi\epsilon_0 E} \right)^{\frac{1}{3}}
where:
- p=3.50×10−29p = 3.50 \times 10^{-29} C·m,
- E=8.00×104E = 8.00 \times 10^4 N/C (given),
- ϵ0=8.85×10−12\epsilon_0 = 8.85 \times 10^{-12} C²/N·m².
Substituting the values: x3=2×(3.50×10−29)4π(8.85×10−12)(8.00×104)x^3 = \frac{2 \times (3.50 \times 10^{-29})}{4\pi (8.85 \times 10^{-12}) (8.00 \times 10^4)} x3=7.00×10−291.112×10−8x^3 = \frac{7.00 \times 10^{-29}}{1.112 \times 10^{-8}} x3=6.29×10−21x^3 = 6.29 \times 10^{-21}
Taking the cube root: x=(6.29×10−21)13x = (6.29 \times 10^{-21})^{\frac{1}{3}} x≈1.99×10−8 mx \approx 1.99 \times 10^{-8} \text{ m}
Thus, the distance from the center of the dipole where the electric field is 8.00×1048.00 \times 10^4 N/C is 1.99×10−81.99 \times 10^{-8} m.
Explanation
A dipole consists of two equal and opposite charges separated by a distance. The electric dipole moment is a measure of this charge separation. The electric field due to a dipole is strongest along its axis and weakens with distance.
In Part E, we determined the distance between the potassium and bromine ions by using the definition of dipole moment. Since dipole moment is simply charge multiplied by separation distance, we solved for dd by dividing the given dipole moment by the charge.
In Part F, we used the equation for the electric field along the axis of a dipole to find the distance from the center of the dipole where the given electric field strength occurs. This equation involves cube dependence, so solving for xx required taking the cube root.
This result shows that the electric field from a dipole drops off rapidly with distance (∝1/x3\propto 1/x^3), meaning the effect of the dipole weakens significantly as we move away.
Now, I will generate an image to visualize the dipole and electric field.
Here is the generated image illustrating the potassium bromide (KBr) dipole, showing the separation distance, dipole moment, and electric field at the specified location. Let me know if you need any modifications or further explanations!
