Question

At what temperature $(in°C)$do hydrogen molecules have the same $rms$ speed as nitrogen molecules at $100°C$ ?

Short answer

The temperature of hydrogen molecules is$-246°C$

Step-by-step solution

Introduction

Equating the expressions for the $rms$speeds, we can find

$\sqrt{\frac{3k_B{T}_n}{m_n}}=\sqrt{\frac{3k_B{T}_h}{m_h}}$

We can simplify the constants and raise the expression to the second power, thus obtaining

$\frac{T_n}{m_n}=\frac{T_h}{m_h}$

It is therefore clear that the temperature of the hydrogen molecules will be

$T_h=T_n\frac{m_h}{m_n}$

Substitution 

Let us remember that the mass $m$of a molecule of a substance with molar mass $M$will be

$m=\frac{M}{N_A}$

Therefore it is clear that the ratio of molecular masses will be that of the molar masses,

$\frac{m_h}{m_n}=\frac{M_h}{M_n}$

Our final expression therefore becomes

$T_h=T_n\frac{M_h}{M_n}$

Numerically we will have

$T_h=(100+273)\frac{2}{28}\approx \underset{¯}{27\mathrm{K}}$

In degrees Celsius, this temperature belongs to$-246°C$

Step 3:The Value of temperature

The temperature belongs to$-246°C$