A pollution-control device installed in the smokestack of equipment used to run an industrial process measures the root-mean-square speed of pollutant particles given off by the process. If \(v_{\mathrm{rms}}\) exceeds a given limit, the process is shut down to prevent the release of pollutants into the air. (Consider the particles to be ideal and monatomic.) The safety operator knows that the average mass density of the particles involved in this process is \(5600 \mathrm{~kg} / \mathrm{m}^{3}\).

(a) At what threshold should he set the device if the process is running at \(936^{\circ} \mathrm{C}\) and the particle diameter is \(10 \mathrm{~nm}\) ?

(b) As the particles rise and exit the smokestack, the \(10-\mathrm{nm}\) particles coalesce to an average diameter of \(1.0 \mu \mathrm{m}\) and cool to \(190^{\circ} \mathrm{C}\). What is the root-mean-square speed of the particles as they exit the smokestack? (Ignore viscosity and drag.)