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Chapter 13: Problem 80

About how long will it take a perfume molecule to diffuse a distance of $5.00 \mathrm{m}\( in one direction in a room if the diffusion constant is \)1.00 \times 10^{-5} \mathrm{m}^{2} / \mathrm{s} ?$ Assume that the air is perfectly still-there are no air currents.

Short Answer

Expert verified
Answer: The time it takes for a perfume molecule to diffuse a distance of \(5.00 \,\mathrm{m}\) in one direction in a room, under the given conditions, is approximately \(1.25\times10^{6} \,\mathrm{s}\).

Step by step solution

01

Understand the problem and the given information

We are given the following information: - Diffusion distance: \(5.00 \,\mathrm{m}\) - Diffusion constant: \(1.00 \times 10^{-5} \,\mathrm{m}^{2}/\mathrm{s}\) Our goal is to determine the time it will take for a perfume molecule to diffuse a given distance in a room, assuming that the air is still.
02

Apply the formula for diffusion

We can use the formula for diffusive spread of particles, which involves the diffusion constant, distance, and time: $$x^{2} = 2Dt$$ Where: - \(x\) is the diffusion distance - \(D\) is the diffusion constant - \(t\) is the time it takes for the particle to diffuse In our case, we are looking for \(t\) and we know \(x = 5.00 \,\mathrm{m}\) and \(D = 1.00 \times 10^{-5} \,\mathrm{m}^{2}/\mathrm{s}\).
03

Solve the equation for time \(t\)

Let's isolate the variable \(t\) by dividing both sides by \(2D\): $$t = \frac{x^{2}}{2D}$$ Now, we plug in the given values for \(x\) and \(D\): $$t = \frac{(5.00 \,\mathrm{m})^{2}}{2(1.00\times10^{-5}\,\mathrm{m}^{2}/\mathrm{s})}$$
04

Calculate the time \(t\)

By performing the calculations, we get: $$t = \frac{25 \,\mathrm{m}^{2}}{2(1.00\times10^{-5}\,\mathrm{m}^{2}/\mathrm{s})}$$ $$t = \frac{25 \,\mathrm{m}^{2}}{2\times10^{-5}\,\mathrm{m}^{2}/\mathrm{s}}$$ $$t = \frac{25 \,\mathrm{m}^{2}}{2\times10^{-5}\,\mathrm{m}^{2}/\mathrm{s}} \times \frac{1\,\mathrm{s}}{2\times10^{-5}\,\mathrm{m}^{2}}$$ $$t = \frac{25}{2\times10^{-5}}$$ $$t = 1.25\times10^{6}\,\mathrm{s}$$ Therefore, it will take approximately \(1.25\times10^{6} \,\mathrm{s}\) for a perfume molecule to diffuse a distance of \(5.00 \,\mathrm{m}\) in one direction in a room, if the diffusion constant is \(1.00 \times 10^{-5} \,\mathrm{m}^{2}/\mathrm{s}\), and there are no air currents.

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