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Question:

The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, is closest to : [Boltzmann Constant kB=1.38×10⁻²³J/K, Avogadro Number NA=6.02×10²⁶/kg, Radius of Earth =6.4×10⁶m, Gravitational acceleration on Earth =10ms⁻²]

650K

104K

800K

3×10⁵K

Solution:

vrms=√(2RT/m)
vescape=√(2gRe)
vrms=vescape
3RT/m=2gRe
3×1.38×10⁻²³×6.02×10²⁶×T/2=2×10×6.4×10⁶
2×T=2×10×6.4×10⁶
3×1.38×6.02×10³
T=4×10⁷×6.4×10⁶
3×1.38×6.02×10³
T=10⁴K
Note : Question gives avogadro Number NA=6.02×10²⁶/ kg but we take NA=6.02×10²⁶/ kmol.