Question

The correct gas equation is:
A. \[\dfrac{{{P_1}{V_1}}}{{{P_2}{V_2}}} = \dfrac{{{T_1}}}{{{T_2}}}\]
B. \[\dfrac{{{V_2}{T_2}}}{{{P_1}}} = \dfrac{{{V_1}{T_1}}}{{{P_2}}}\]
C. \[\dfrac{{{P_1}{T_1}}}{{{V_1}}} = \dfrac{{{P_2}{T_2}}}{{{V_2}}}\]
D. \[\dfrac{{{V_1}{V_2}}}{{{T_1}{T_2}}} = {P_1}{P_2}\]

Answer 1

Hint: The gas is said to be ideal if it obeys the ideal gas law. An ideal gas is free from the intermolecular force of attraction and it has almost negligible or less volume. The gas that deviates from ideal gas is known as real gas.

Complete step by step answer:
The ideal gas law is derived from the combination of Charles's law and Boyle's law and Avogadro's law.
Charles's law states that "At constant pressure, the volume is directly proportional to temperature". It can be written as,
\[V\alpha T(const.P)\]
Boyle's law states that "PV is constant at constant temperature". It can be written as,
\[PV = constant\]
Avogadro's law states that "An equal volume of all gases under the same conditions of temperature and pressure contains an equal number of molecules". It can be written as,
\[{N_1} = {N_2}\]
By combining the above three laws, the ideal gas equation can be written as,
\[PV = nRT\]
The ideal gas equation helps to conclude the fact that the volume of a gas depends on temperature, pressure, and the number of moles present in it.
Let us have two gases of different volume, pressure, and temperature. The ideal gas equation for two different gas can be written as,
\[{P_1}{V_1} = nR{T_1}.......(1)\]
\[{P_2}{V_2} = nR{T_2}.......(2)\]
Dividing equation (1) and (2), we get,
\[ \Rightarrow \dfrac{{{P_1}{V_1}}}{{{P_2}{V_2}}} = \dfrac{{nR{T_1}}}{{nR{T_2}}}\]
Since R is gas constant and let us assume that the number of moles is the same for both gases.
\[ \Rightarrow \dfrac{{{P_1}{V_1}}}{{{P_2}{V_2}}} = \dfrac{{{T_1}}}{{{T_2}}}\]
Thus, the correct gas equation is \[\dfrac{{{P_1}{V_1}}}{{{P_2}{V_2}}} = \dfrac{{{T_1}}}{{{T_2}}}\]

So, the correct answer is Option A.

Note: In general, the higher the pressure and lower the temperature, the ideal gas tends to deviate i.e. it behaves like a real gas. The deviations of an ideal gas can be measured by using the compressibility factor (z). The compressibility factor is the ratio of the product of pressure and volume to the product of pressure and volume of an ideal gas.
i.e. \[z = \dfrac{{PV}}{{{{(PV)}_{ideal}}}}\]
For an ideal gas, the compressibility factor is unity. i.e. \[z = 1\]