Question

The ratio between the root mean square velocity of ${H_2}$ at $50K$ and that of ${O_2}$ at $800K$ is
A. $4$
B. $2$
C. $1$
D. $0.25$

Answer 1

Hint: To evaluate the ratio of the two compounds, we will first determine the ${V_{rms}}$ also known as the root mean square velocity of the two compounds. We have the molecular weight and temperature of the two compounds which is required for further calculation.

Complete step by step answer:
Given data contains,
Temperature of ${H_2}$ is $50K$ and of ${O_2}$ is $800K$ .
We know:
Molecular weight of ${H_2}$ is $2$ and that of ${O_2}$ is $32$.
Since ${V_{rms}}$ (root mean square velocity) is proportional to square root of temperature upon molecular weight. The expression can be written as
${V_{rms}} \propto \sqrt {\dfrac{T}{M}} $
Where,
T-Temperature
M-Molecular weight
Hence, the ratio can be written as,
\[\dfrac{{\left( {{V_{rms}}} \right){H_2}}}{{\left( {{V_{rms}}} \right){O_2}}} = \sqrt {\dfrac{{{T_{{H_2}}}}}{{{M_{{H_2}}}}}} \times \sqrt {\dfrac{{{M_{{O_2}}}}}{{{T_{{O_2}}}}}} \]
Where,
${T_{{H_2}}}$ -Temperature of hydrogen
${T_{{O_2}}}$ -Temperature of oxygen
${M_{{H_2}}}$ -Molecular weight of hydrogen
${M_{{O_2}}}$ -Molecular weight of oxygen
Now we can substitute the given values we get,
\[\dfrac{{\left( {{V_{rms}}} \right){H_2}}}{{\left( {{V_{rms}}} \right){O_2}}} = \sqrt {\dfrac{{50}}{2}} \times \sqrt {\dfrac{{32}}{{800}}} \]
On simplification we get,
\[\dfrac{{\left( {{V_{rms}}} \right){H_2}}}{{\left( {{V_{rms}}} \right){O_2}}} = 1\]

Hence option C is correct.

Note: We must need to remember that to determine the ratio we always use the root mean square velocity formula, since root mean square velocity is directly proportional to square root of temperature and inversely proportional to square root of molecular weight. Generally, if we see this example here, we have ${H_2}$ which has lower molecular weight and lower temperature, whereas ${O_2}$ has higher molecular weight and higher temperature. We must remember that to make the calculation easier we can compare the ratio by just checking the ratio of molecular weight to the temperature of both the compounds. And then when we inverse their arrangement and take a square root we get the exact answer.