Question

Which of the following is not the unit of the Universal Gas Constant$?$
(i) $erg\,mo{l^{ - 1}}\,{K^{ - 1}}$
(ii) $J\,mo{l^{ - 1}}\,{K^{ - 1}}$
(iii) $Pa\,{m^2}\,mo{l^{ - 1}}\,{K^{ - 1}}$
(iv) $Pa\,{m^3}\,mo{l^{ - 1}}\,{K^{ - 1}}$

Answer 1

The universal gas constant is denoted by the symbol $R$ and is equivalent to the Boltzmann constant $\left( {{k_b}} \right)$, but expressed in units of energy per temperature increment per mole. From here check which of the given options match the criteria as mentioned, the one which does not is the correct option.

Complete step-by-step answer: The Universal gas constant is also known as the molar gas constant or the ideal gas constant and is denoted by the symbol $R$. It is equivalent to the Boltzmann constant $\left( {{k_b}} \right)$, but is expressed in units of energy per temperature increment per mole, i.e. the pressure–volume product, rather than energy per temperature increment per particle.
The universal gas constant $R$is defined as the Avogadro ’s number multiplied by the Boltzmann constant and can be expressed as,
$R = {N_A} \times {k_b}$, where ${N_A}$is the Avogadro’s Number and has a value of $6.022 \times {10^{23}}\,mo{l^{ - 1}}$and ${k_b}$ is the Boltzmann constant and has a value of $1.3806 \times {10^{ - 23}}\,J\,{K^{ - 1}}$.
Hence the value of $R\, = \,8.314\,J\,mo{l^{ - 1}}\,{K^{ - 1}}$.
Now, $Joule$ is the SI unit of energy and $erg$is the CGS unit of energy hence Universal Gas Constant, $R$ can also be expressed in terms of $erg\,mo{l^{ - 1}}\,{K^{ - 1}}$.
Also, Energy $=$ Pressure $\times$ Volume.
Hence $J\, = \,Pa\,{m^3}$, where $Pa$is the unit of pressure and ${m^3}$ is the unit of volume.
Hence Universal Gas Constant, $R$ can also be expressed in terms of $Pa\,{m^3}\,mo{l^{ - 1}}\,{K^{ - 1}}$.
Therefore $R$ cannot be expressed in terms of $Pa\,{m^2}\,mo{l^{ - 1}}\,{K^{ - 1}}$.
Hence the correct answer is (iii) $Pa\,{m^2}\,mo{l^{ - 1}}\,{K^{ - 1}}$.

Note: You must read the question properly as it has mentioned which one is not correct. Many students in a hurry do not read properly and mark the wrong option. Also do take care of the conversions and formulas required for converting from one unit to another in order to avoid mistakes.