Question

The temperature of an ideal gas is increased from \[{\text{120 K}}\] to \[{\text{480 K}}\]. If at\[{\text{120 K}}\], the root mean square speed of gas molecules is ${\text{v}}$ then at \[{\text{480 K}}\], it will be:
A.${\text{4v}}$
B.${\text{2v}}$
C.$\dfrac{{\text{v}}}{2}$
D.$\dfrac{{\text{v}}}{4}$

Answer 1

Hint:The root mean square speed of the gas molecules is defined at the square root of the average of the squares of the velocities of the molecules divided by the number of molecules in the sample of the gas. We shall substitute the values in the equation and compare them.
Formula Used: ${{\text{V}}_{{\text{rms}}}}{\text{ = }}\sqrt {\dfrac{{{\text{3kT}}}}{{\text{m}}}} $, where T is the temperature of the gas molecules in absolute scale and k is called the Boltzmann’s constant.

Complete step by step answer:
According to the question, the root mean square speed of the molecules at\[{\text{120 K}}\] is ${\text{v}}$, that can be expressed as:
\[{\text{v = }}\sqrt {\dfrac{{{{3 \times k \times 120}}}}{{\text{m}}}} \],
Therefore, when the temperature is increased to\[{\text{480 K}}\], the root mean square speed becomes,
${\text{Vrms = }}\sqrt {\dfrac{{{{3 \times k \times 480}}}}{{\text{m}}}} $
Now, \[{\text{v = }}\sqrt {\dfrac{{{{3 \times k \times 120}}}}{{\text{m}}}} \] ,
Therefore the root mean square speed of the gas at\[{\text{480 K}}\]= ${\text{Vrms = v}}\sqrt {\text{4}} {\text{ = 2v}}$

Hence, the correct answer is option B.

Note:
The average speed of the molecules is equal to the sum of the speeds of all the molecules divided by the number of the molecules. While the average square speed of the molecules is defined as the sum of the squares of speeds of the molecules divided by the total number of the molecules.
There is an alternative definition of the root mean square speeds on the molecules which is mathematically expressed as ${{\text{V}}_{{\text{rms}}}}{\text{ = }}\sqrt {\dfrac{{{\text{3RT}}}}{{\text{M}}}} $hence it can be said that the Boltzmann constant is equal to ${\text{k = }}\dfrac{{\text{R}}}{{{{\text{N}}_{\text{A}}}}}$where ${{\text{N}}_{\text{A}}}$is the Avogadro number of particles that is equal to $6.023 \times {10^{23}}$particles and R is the universal gas constant.
The root mean square directly takes into account both the molecular weight and the temperature of the molecules both of which affect the kinetic energy of the molecules.