Question

At what temperature, will rms velocity of the second member of the homologous series ${C_n}{H_{2n - 2}}$ be the same as that of oxygen at ${527^o}C$?
A. ${1000^o}C$
B. ${727^o}C$
C. ${1727^o}C$
D. ${1044^o}C$

Answer 1

The rms velocity stands for root mean square velocity which is the velocity at which all the molecules have the same total kinetic energy as in case of their actual velocity. Its value is equal to the square root of the summation of squares of velocities divided by the number of values.
Formula used-
Rms velocity $= \sqrt {\dfrac{{3RT}}{M}}$
Where, R is the universal gas constant, T is absolute temperature and M is the molecular mass of the gas considered.

Complete answer: Homologous series: A series of compounds with similar physical and chemical properties. The compounds of homologous series only differ by $- C{H_2}$ group. The formula for the given homologous series is ${C_n}{H_{2n - 2}}$ and it is the general formula of alkynes, where value of $n = 2,\,3,\,4,\,5,\, \cdot \cdot \cdot$
For the second member of the given homologous series, the value of $n = 3$. Therefore, the molecular formula for the second member of given homologous series will be ${C_3}{H_{2 \times 3 - 2}}$ i.e., ${C_3}{H_4}$ and its chemical name is propyne.
Molecular mass of propyne $= 3 \times 12 + 4$
$\Rightarrow M = 40\;gmo{l^{ - 1}}$
Substituting the value in the formula of rms velocity:
${v_{{C_3}{H_4}}} = \sqrt {\dfrac{{3RT}}{M}}$
$\Rightarrow {v_{{C_3}{H_4}}} = \sqrt {\dfrac{{3RT}}{{40}}} \,\,\,\,\, - (i)$
As per question, temperature at which oxygen gas is present $= {527^o}C$
$\Rightarrow {T_{{O_2}}} = 527 + 273$
$\Rightarrow {T_{{O_2}}} = 800\,K$
Molecular mass of oxygen $= 32\,gmo{l^{ - 1}}$
Substituting the value in the formula of rms velocity:
${v_{{O_2}}} = \sqrt {\dfrac{{3RT}}{M}}$
$\Rightarrow {v_{{O_2}}} = \sqrt {\dfrac{{3R \times 800}}{{32}}}$
$\Rightarrow {v_{{O_2}}} = \sqrt {3R \times 25} \,\,\,\,\, - (ii)$
As the rms velocities of oxygen and propyne are considered to be equal, so from equation (i) and (ii) the temperature of propyne can be calculated as follows:
${v_{{C_3}{H_4}}} = {v_{{O_2}}}$
$\Rightarrow \sqrt {\dfrac{{3RT}}{{40}}} \, = \sqrt {3R \times 25}$
Squaring both sides:
$\Rightarrow \dfrac{{3RT}}{{40}}\, = 3R \times 25$
$\Rightarrow T = 1000\,K$
Converting the unit of temperature into degree Celsius:
$\Rightarrow T = 1000 - 273$
$\Rightarrow T = {727^o}C$
Hence, at ${727^o}C$ the rms velocity of the second member of the homologous series ${C_n}{H_{2n - 2}}$ will be the same as that of oxygen at ${527^o}C$.
So, option (B) is the correct answer.

Note:
It is important to note that rms velocity is the maximum velocity which can be attained by the molecules of gases. The order of velocity for a molecule of gas is ${v_{rms}} > {v_{avg}} > {v_{mps}}$ i.e., rms velocity is the maximum whereas most probable speed is the minimum velocity a gaseous molecule can attain.