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Chapter 6: Problem 138

If the Kelvin temperature of a sample of ideal gas doubles (e.g., from 200 K to 400 K), what happens to the root-mean-square speed, \(u_{\mathrm{rms}}\) ? (a) \(u_{\mathrm{rms}}\) increases by a factor of \(\sqrt{2} ;\) (b) \(u_{\mathrm{rms}}\) increases by a factor of \(2 ;(\mathrm{c}) u_{\mathrm{rms}}\) decreases by a factor of 2 (d) \(u_{\mathrm{rms}}\) increases by a factor of \(4 ;\) (e) \(u_{\mathrm{rms}}\) decreases by a factor of 4.

Short Answer

Expert verified
The root-mean-square speed, \(u_{\mathrm{rms}}\) increases by a factor of \( \sqrt{2} \).

Step by step solution

01

Identify the relationship between temperature and rms speed

We know that \(u_{rms} = \sqrt{ \frac {3kT} {m} }\). This equation shows us that the rms speed is proportional to the square root of the temperature.
02

Calculate the change in rms speed due to temperature doubling

If we double the temperature i.e., \( T \rightarrow 2T \), the new rms speed \( u_{new} \) will be \( u_{new} = \sqrt{ \frac {3k(2T)} {m} } \). This can also be written as \(u_{new} = \sqrt{2} \cdot \sqrt{ \frac {3kT} {m} } \) = \( \sqrt{2} \cdot u_{rms} \). This shows us that the rms speed increases by a factor of \( \sqrt{2} \).
03

Choose the correct option

Based on our calculations, the root mean square speed increases by a factor of \( \sqrt{2} \) which corresponds to option (a).

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Most popular questions from this chapter

Hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2},\) is used to disinfect contact lenses. How many milliliters of \(\mathrm{O}_{2}(\mathrm{g})\) at \(22^{\circ} \mathrm{C}\) and \(752 \mathrm{mmHg}\) can be liberated from \(10.0 \mathrm{mL}\) of an aqueous solution containing \(3.00 \% \mathrm{H}_{2} \mathrm{O}_{2}\) by mass? The density of the aqueous solution of \(\mathrm{H}_{2} \mathrm{O}_{2}\) is \(1.01 \mathrm{g} / \mathrm{mL}\) $$2 \mathrm{H}_{2} \mathrm{O}_{2}(\mathrm{aq}) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(1)+\mathrm{O}_{2}(\mathrm{g})$$

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