Question

Question:

(a)Compute the root-mean-square speed of H₂ molecules in hydrogen at a temperature of 300 K.

(b)Repeat the calculation for SF₆ molecules in gaseous sulfur hexafluoride at 300 K.

Short answer

a) The Root mean square speed of H₂ molecule in hydrogen is$2.72\times {10}^3{\mathrm{ms}}^{-1}$.

b) The Root mean square speed of SF₆ molecules in gaseous sulfur hexafluorideis $226{\mathrm{ms}}^{-1}$.

Step-by-step solution

a) Root mean square speed of H2 molecule in hydrogen

Molar mass of hydrogen is 1.0079 g/mol =$\frac{1\mathrm{kg}}{1000\mathrm{g}}\times 1.0079\mathrm{g}/\mathrm{mol}=1.0079\times {10}^{-3}\mathrm{kg}/\mathrm{mol}$

Temperature (T) = 300 K

The root mean square speed of H₂ molecule in hydrogen is,

$$\begin{gathered} \mathrm{\mu }=\sqrt{\frac{3\mathrm{RT}}{{\mathrm{molarmassofH}}_2}} \\ =\sqrt{\frac{3\times 8.314\frac{\mathrm{J}}{\mathrm{Kmol}}\times 300\mathrm{k}}{1.0079\times {10}^{-3}\mathrm{kg}/\mathrm{mol}}} \\ =\sqrt{\frac{3\times 8.314\frac{{\mathrm{Kgm}}^2{\mathrm{s}}^{-2}}{\mathrm{kmol}}\times 300\mathrm{k}}{1.0079\times {10}^{-3}\mathrm{kg}/\mathrm{mol}}} \\ =\sqrt{7423.95\times {10}^3{\mathrm{m}}^2{\mathrm{s}}^{-2}} \\ =\sqrt{742.39\times {10}^4{\mathrm{m}}^2{\mathrm{s}}^{-2}} \\ =27.24\times {10}^2{\mathrm{ms}}^{-1} \\ =2.72\times {10}^3{\mathrm{ms}}^{-1} \end{gathered}$$

Hence, the Root mean square speed of H₂ molecule in hydrogen is$2.72\times {10}^3{\mathrm{ms}}^{-1}$.

b) Root mean square speed of SF6 molecules in gaseous sulfur hexafluoride

Molar mass of is SF₆is146.06g/mol=$\frac{1\mathrm{kg}}{1000\mathrm{g}}\times 146.06\mathrm{g}/\mathrm{mol}=146.06\times {10}^{-3}\mathrm{kg}/\mathrm{mol}$

Temperature (T) = 300 K

The root mean square speed of SF₆ molecules in gaseous sulfur hexafluorideis,

$$\begin{gathered} \mathrm{\mu }=\sqrt{\frac{3\mathrm{RT}}{\mathrm{molarmassof}SF2}} \\ =\sqrt{\frac{3\times 8.314\frac{\mathrm{J}}{\mathrm{Kmol}}\times 300\mathrm{k}}{146.06\times {10}^{-3}\mathrm{kg}/\mathrm{mol}}} \\ =\sqrt{\frac{3\times 8.314\frac{{\mathrm{Kgm}}^2{\mathrm{s}}^{-2}}{\mathrm{kmol}}\times 300\mathrm{k}}{146.06\times {10}^{-3}\mathrm{kg}/\mathrm{mol}}} \\ =\sqrt{51.229\times {10}^3{\mathrm{m}}^2{\mathrm{s}}^{-2}} \\ =\sqrt{5.122\times {10}^4{\mathrm{m}}^2{\mathrm{s}}^{-2}} \\ =2.26\times {10}^2{\mathrm{ms}}^{-1} \\ =226{\mathrm{ms}}^{-1} \end{gathered}$$

Hence, the Root mean square speed of SF₆ molecules in gaseous sulfur hexafluorideis $226{\mathrm{ms}}^{-1}$.