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Chapter 6: Problem 73

Calculate \(u_{\mathrm{rms}},\) in meters per second, for \(\mathrm{Cl}_{2}(\mathrm{g})\) molecules at \(30^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The root mean square speed of Cl2 molecules at 30°C is approximately 482 m/s.

Step by step solution

01

Convert temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin by using the formula: \(T(K) = t(°C) + 273.15\). So, we get \(T = 30 + 273.15 = 303.15 K\).
02

Calculating mass of a single molecule

In this step, calculate the mass of a single Cl2 molecule. The molar mass of Cl2 is approximately 70.90 g/mol. Convert it to kg (since the root mean square speed is typically expressed in meters per second), it's equal to \(70.90 \times 10^{-3} kg/mol\). Now, to get the mass of a single molecule, divide the molar mass by Avogadro's number: \(m = (70.90 \times 10^{-3})/(6.022 \times 10^{23}) = 1.178 \times 10^{-25} kg\).
03

Calculate the root mean square speed

Now, calculate the root mean square speed using the formula: \(u_{\mathrm{rms}} = \sqrt{(3kT)/m}\), where \(k\) is the Boltzmann constant equal to \(1.381 \times 10^{-23} m^{2} kg/s^{2} K\). So, we get: \(u_{\mathrm{rms}} = \sqrt{(3 \times 1.381 \times 10^{-23} \times 303.15)/( 1.178 \times 10^{-25})}\). This simplifies to: \(u_{\mathrm{rms}} = 482 m/s\).

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Most popular questions from this chapter

A particular coal sample contains \(3.28 \%\) S by mass. When the coal is burned, the sulfur is converted to \(\mathrm{SO}_{2}(\mathrm{g}) .\) What volume of \(\mathrm{SO}_{2}(\mathrm{g}),\) measured at \(23^{\circ} \mathrm{C}\) and \(738 \mathrm{mm} \mathrm{Hg},\) is produced by burning \(1.2 \times 10^{6} \mathrm{kg}\) of this coal?

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