Question

At what temperature is the rms velocity of a hydrogen molecule equal to that of an oxygen molecule equal to that of an oxygen molecule at ${47^o}C$ ?
(A) 8OK
(B) -73K
(C) 20K
(D) 3K

Answer 1

Hint : The square root of the mean of squares of each gas molecules' velocity is known as root mean square velocity. The square root of the mean square in mathematics and its applications is known as the root mean square. The RMS (or quadratic mean) is a special instance of the generalised mean with exponent 2. In terms of an integral of the squares of the instantaneous values throughout a cycle, RMS may be defined for a continuously changing function.

Complete Step By Step Answer:
The square root of the mean of squares of the velocity of individual gas molecules is the root mean square velocity (RMS value).
$({v_{rms}} = \sqrt {\frac{{3RT}}{M}} )$
${v_{rms}}$ is the root-mean-square velocity.
M (Kg/mole) is the molar mass of the gas.
R is the constant of a molar gas.
T stands for Kelvin temperature.
As a result, if the r.m.s. velocity of both gases is the same,
$\sqrt {\frac{{{{\mathbf{T}}_{\text{H}}}}}{{{{\text{M}}_{\text{H}}}}}} = \sqrt {\frac{{{{\text{T}}_{\text{O}}}}}{{{{\text{M}}_{\text{O}}}}}}$
$\Rightarrow {{\text{T}}_{\text{H}}} = 320\;{\text{K}} \times \frac{2}{{{\mathbf{32}}}}$
$\Rightarrow T = {\mathbf{20K}}$
Hence option C is correct.

Note :
The square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform, is the RMS value of a collection of values (or a continuous-time waveform). The RMS current is also known as the "value of the direct current that dissipates the same power in a resistor" in physics. A periodic function's RMS across all time is equal to the RMS of one period of the function. The RMS of a sample of evenly spaced observations can be used to estimate the RMS of a continuous function or signal. Cartwright also shows how to calculate the RMS value of various waveforms without using calculus.