Question

To what temperature the hydrogen at $327^\circ \,C$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value
A. $ - 123^\circ \,C$
B. $23^\circ \,C$
C. $ - 100^\circ \,C$
D. $0^\circ \,C$

Answer 1

Hint: Here, we will use the concept of the kinetic theory of gases which tells us that the root means square velocity is directly proportional to the square root of the temperature. Now, to calculate the temperature we will divide ${V_{rms1}}$ by ${V_{rms2}}$. Here, the root mean square velocity of the molecules of hydrogen will become half of its previous value.

Complete step by step answer:
Now, it is given in the question that the hydrogen is at $327^\circ \,C = 327+273 = 600\,K$. Here, we will change the temperature from Celsius into Kelvin. Also, the root mean square value of the molecules of the hydrogen becomes half of its previous value. Therefore, if we consider ${V_{rms1}} = V$
Than ${V_{rms2}} = \dfrac{V}{2}$
Now, according to the kinetic theory of gases, the root mean square velocity of voltage is related to the temperature as
${V_{rms}} \propto \sqrt T $
Now, taking ratios of both the root mean square velocities ${V_{rms1}}$ by ${V_{rms2}}$ as is given below
$\dfrac{{{V_{rms1}}}}{{{V_{rms2}}}} = \dfrac{{\sqrt {{T_1}} }}{{\sqrt {{T_2}} }}$
Now, putting the values of ${V_{rms1}}$ , ${V_{rms2}}$ , ${T_1}$ and ${T_2}$ in the above equation, we get
$\dfrac{V}{{\dfrac{V}{2}}} = \dfrac{{\sqrt {600} }}{{\sqrt {{T_2}} }}$
$ \Rightarrow \,\sqrt {{T_2}} = \dfrac{{\sqrt {600} }}{2}$
$ \Rightarrow \,\sqrt {{T_2}} = \dfrac{{24.49}}{2}$
$ \Rightarrow \,\sqrt {{T_2}} = 12.24$
$ \Rightarrow \,{T_2} = 149.81K$
$ \Rightarrow \,{T_2} \simeq 150K$
Now, the above temperature is in kelvin, therefore, we will change the above temperature into Celsius as
${T_2} = \left( {150 - 273} \right)^\circ \,C$
$ \therefore\,{T_2} = - 123^\circ \,C$
Therefore, the hydrogen will be cooled at a temperature $ - 123^\circ \,C$ such that the root mean square value of its molecules becomes half of its previous value.

Hence, option A is the correct option.

Note:Remember here to change the temperature into Kelvin. We are changing the temperature of hydrogen because the temperature in the root mean square velocity will be in kelvin. Also, after calculating the temperature we will change it to Celsius because the answer required is in Celsius.