Question

The value of gas constant \[R\](in \[cal/mol-K\]) for hydrogen is:
 \[\begin{align}
  & A)\dfrac{1}{2} \\
 & B)2 \\
 & C)4 \\
 & D)1 \\
\end{align}\]

Answer 1

Hint: We know the gas constant \[R\] is a proportionality constant in the ideal gas equation. In S.I. unit, the value of gas constant is given by \[8.314J/mol-K\]. We will convert this value into the \[cal/mol-K\] unit system by changing joules into calorie. The relation between joule and calorie is given by \[1cal=4.184J\].

Complete answer: Firstly, we will try to understand what gas constant is and the value of gas constant in S.I. unit. Gas constant is also known as molar gas constant, universal gas constant or ideal gas constant. The gas constant is defined as the Avogadro constant multiplied by the Boltzmann constant. i.e. \[R={{N}_{A}}k\].

But, we use it as a proportionality constant that relates the energy scale in physics to the temperature scale, when a mole of particles at a particular temperature is considered.
We can calculate the value of \[R\] using the formula, \[R={{N}_{A}}k\].
Where,\[{{N}_{A}}\] is the Avogadro number given as \[6.022\times {{10}^{23}}\] and\[k\] the Boltzmann constant which is equal to \[1.38064J/K\].
Now, \[R={{N}_{A}}k=6.022\times {{10}^{23}}\times 1.38064J/K=8.314J/mol-K\]

Here, the unit \[J/mol-K\] is S.I. unit and it is represented in per mole because Avogadro number is number of particles per mole.

Now, we know the relation between the energy units joule and calorie is given as, \[1cal=4.184J\].
If we convert the value of \[R\] using this relation,
 \[\Rightarrow R=\dfrac{8.314J}{4.184J}\times cal/mol-K=1.967cal/mol-K\]
This is the exact value. But for getting an answer from the given options, we will approximate the value as \[R\approx 2cal/mol-K\]. So, the correct answer is “Option B”.

Note:We can convert the S.I unit of universal gas constant into the desired unit by simply converting the units to their respective required units by just replacing the appropriate values for each unit. Gas constant \[R\] is also expressed in various units as \[0.008206L-atm/mol-K,8.314{{m}^{3}}-Pa/mol-K\] and \[62.36L-torr/mol-K\]. Gas constant is also the proportionality constant in the ideal gas equation which is given as \[PV\propto nT\].