At -230 °C temperature will He atoms have the same rms value as [tex]N_{2}[/tex] molecules at 25°C .
Calculation ,
Given temperature = 25°C = 25+273 = 298 K
[tex]V_{rms}[/tex] =√3RT/M
[tex]V_{rms}[/tex] =√3R[tex]T_{1}[/tex]/[tex]M_{He}[/tex] ......( i )
[tex]V_{rms}[/tex] =√3R[tex]T_{2}[/tex]/[tex]M_{N_{2}[/tex] ......( ii )
Equalize equation ( i ) and ( ii )
√3R[tex]T_{1}[/tex]/[tex]M_{He}[/tex] = √3R[tex]T_{2}[/tex]/[tex]M_{N_{2}[/tex]
[tex]T_{1}[/tex]/[tex]M_{He}[/tex] = [tex]T_{2}[/tex]/[tex]M_{N_{2}[/tex]
[tex]T_{1}[/tex] = [tex]M_{He}[/tex] × [tex]T_{2}[/tex]/[tex]M_{N_{2}[/tex] = 4 ×298 K/28 = 42.57 K
Temperature in°C = 42.57 - 273 = -230 °C
Therefore , at -230 °C temperature will He atoms have the same rms value as [tex]N_{2}[/tex] molecules at 25°C .
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