Question

At what temperature do atoms of helium gas have the same rms speed as molecules of hydrogen gas at$\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$ ?

Short answer

The temperature at which the atoms of helium gas have the same rms speed as molecules of hydrogen gas is $\text{T}=307°\text{C}$.

Step-by-step solution

Concept

The rms speed is the root of the mean of the square of speed of molecules of the gas. Rms value does not represent the averageor the most possible value of speeds.Rms speed of a gas is given as,

${\mathbf{V}}_{\mathrm{rms}}\mathbf{=}\sqrt{\frac{3\mathrm{RT}}{m}}\mathbf{}$

Here,T is the temperature, m is the mass of the molecule of gas and R is the gas constant.

Given Data

1. Temperature of hydrogen gas, $\text{T}=20°=293\text{\hspace{0.17em}K}$
2. The gases are hydrogen and helium
3. As per table 19.1, molar masses of hydrogen and helium are$2.02\text{\hspace{0.17em}kg}/\text{mol}\&4.0\text{kg}/\text{mol}$ respectively.

Calculations

As we know that the rms speed of both molecules is equal

We can write,

$\sqrt{\frac{3RT}{M_{helium}}=}\sqrt{\frac{3RT}{M_{hydrogen}}}$

Squaring on both sides

$\begin{array}{rcl}\frac{3\text{\hspace{0.17em}RT}}{{\text{M}}_{helium}}& =& \frac{3\text{R}\left(293\text{\hspace{0.17em}K}\right)}{{\text{M}}_{hydrogen}}\\ \text{T}& =& \frac{\left(293\text{\hspace{0.17em}K}\right)\left({\text{M}}_{helium}\right)}{{\text{M}}_{hydrogen}}\\ \text{T}& =& \frac{\left(293K\right)(4.0\text{\hspace{0.17em}kg}/\text{mol})}{2.02\text{\hspace{0.17em}kg}/\text{mol}}\\ & & \begin{array}{c}\text{T}=580\text{K}\\ =307°\text{C}\end{array}\\ & & \end{array}$

Conclusion

The temperature of helium, at which the hydrogen and helium molecules have the same speed, is $307°\text{C}$.