Question
Correct gas equation is
A.
$$\frac{{{V_1}{T_2}}}{{{p_1}}} = \frac{{{V_2}{T_1}}}{{{p_2}}}$$
B.
$$\frac{{{p_1}{T_1}}}{{{V_1}}} = \frac{{{p_2}{V_2}}}{{{T_2}}}$$
C.
$$\frac{{{p_1}{V_1}}}{{{p_2}{V_2}}} = \frac{{{T_1}}}{{{T_2}}}$$
D.
$$\frac{{{V_1}{V_2}}}{{{T_1}{T_2}}} = {p_1}{p_2}$$
Answer :
$$\frac{{{p_1}{V_1}}}{{{p_2}{V_2}}} = \frac{{{T_1}}}{{{T_2}}}$$
Solution :
If temperature, volume and pressure of fixed amount ( say $$n$$ mole ) of a gas vary from $${T_1},{V_1}$$ and $${p_1}$$ to $${T_2},{V_2}$$ and $${p_2}$$ respectively. Then, ideal gas equation for two states can be written as
$$\eqalign{
& {p_1}{V_1} = nR{T_1} \cr
& {\text{or}}\,\,\frac{{{p_1}{V_1}}}{{{T_1}}} = nR\,\,\,...{\text{(i)}} \cr
& {p_2}{V_2} = nR{T_2} \cr
& {\text{or}}\,\,\frac{{{p_2}{V_2}}}{{{T_2}}} = nR\,\,\,...{\text{(ii)}} \cr
& {\text{On combining Eqs}}{\text{. (i) and (ii), we get}} \cr
& \frac{{{p_1}{V_1}}}{{{T_1}}} = \frac{{{p_2}{V_2}}}{{{T_2}}} \cr
& {\text{or}}\,\,\,\frac{{{p_1}{V_1}}}{{{p_2}{V_2}}} = \frac{{{T_1}}}{{{T_2}}} \cr} $$
So, it called combined gas equation.
Hence, (C) answer is correct.