NEET-XII-Physics
24: Kinetic Theory of Gases
- #18At what temperature the mean speed of the molecules of hydrogen gas equals the escape speed from the earth?Ans : Mean speed of the molecule is given by
`` \sqrt{\frac{8\,\mathrm{\,RT\,}}{\,\mathrm{\,\pi M\,}}}``
`` \,\mathrm{\,For\,}\,\mathrm{\,H\,}\,\mathrm{\,molecule\,},M=2\times {10}^{-3}kg``
`` =\sqrt{\frac{4\,\mathrm{\,RT\,}\times {10}^{3}}{\pi }}``
For escape velocity of Earth:
`` \,\mathrm{\,Suppose\,}r\,\mathrm{\,be\,}\,\mathrm{\,the\,}\,\mathrm{\,radius\,}\,\mathrm{\,of\,}\,\mathrm{\,Earth\,}.``
`` v=\sqrt{\frac{2GM}{r}}``
`` \,\mathrm{\,Multiplying\,}\,\mathrm{\,neumerator\,}\,\mathrm{\,and\,}\,\mathrm{\,denominator\,}\,\mathrm{\,by\,}\,\mathrm{\,R\,},\,\mathrm{\,we\,}\,\mathrm{\,get\,}``
`` {v}_{c}=\sqrt{\frac{GM}{{r}^{2}}2r}``
`` g=\frac{GM}{{r}^{2}}``
`` {v}_{c}=\sqrt{2gr}``
`` \sqrt{\frac{4\,\mathrm{\,RT\,}\times {10}^{3}}{\pi }}=\sqrt{2gr}``
`` \Rightarrow \frac{2\times 8.314\times T\times {10}^{3}}{3.142}=9.8\times 6.37\times {10}^{6}``
`` \Rightarrow T\approx 11800K``
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