Question

The correct value of the gas constant R is close to:
A \[{\text{0}}{\text{.082 litre}} - {\text{atm K}}\]
B \[{\text{0}}{\text{.082 litre}} - {\text{atm }}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}\]
C \[{\text{0}}{\text{.082 litre}} - {\text{at}}{{\text{m}}^{ - 1}}{\text{ Kmo}}{{\text{l}}^{ - 1}}\]
D \[{\text{0}}{\text{.082 litr}}{{\text{e}}^{ - 1}} - {\text{at}}{{\text{m}}^{ - 1}}{\text{ Kmo}}{{\text{l}}^{ - 1}}\]

Answer 1

Hint:Write the expression for the ideal gas equation. Substitute units for all quantities (except the ideal gas constant R) in the ideal gas equation and calculate the unit of the ideal gas constant R.

Complete answer:
Write the ideal gas equation as shown below:
\[PV = nRT\]
Here, P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is absolute temperature.
Rearrange the ideal gas equation to obtain an expression in terms of ideal gas constant.
\[R = \dfrac{{P \times V}}{{n \times T}}\]
To obtain the unit of the ideal gas constant, substitute the unit atm for P, L for V, mol for n and K for T in the above expression.
\[
R = \dfrac{{P \times V}}{{n \times T}} \\
\Rightarrow R = \dfrac{{{\text{atm}} \times {\text{L}}}}{{{\text{mol}} \times {\text{K}}}} \\
\Rightarrow R = {\text{ litre}} - {\text{atm }}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}
 \]
As you know, the volume of one mole of an ideal gas at STP is \[22.4{\text{ L}}\]
Substitute one atmosphere for pressure, \[22.4{\text{ L}}\] for volume, one mole for number of moles and \[273.15{\text{ K}}\] for temperature in the rearranged ideal gas equation and calculate the value of R.
\[
  R = \dfrac{{P \times V}}{{n \times T}} \\
\Rightarrow R = \dfrac{{{\text{1 atm}} \times 22.4{\text{ L}}}}{{{\text{1 mol}} \times 273.15{\text{ K}}}} \\
\Rightarrow R = 0.082{\text{ litre}} - {\text{atm }}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}
 \]
Thus, the correct value of the gas constant R is close to \[{\text{0}}{\text{.082 litre}} - {\text{atm }}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}\]

Hence, the correct option is the option (B)

Note:
In the unit for ideal gas constant R, the term litre atmosphere is in the numerator. This is because, in the expression for the ideal gas constant, the \[R = \dfrac{{P \times V}}{{n \times T}}\]
pressure and volume are in the numerator. Similarly, the term moles kelvin is in the denominator. This is because, in the expression for the ideal gas constant \[R = \dfrac{{P \times V}}{{n \times T}}\], the number of moles and the absolute temperature are in the denominator.