Special Case $3.$ Batch and Flow Systems for Gases in General $($varying $,T,$

$$ which react according to

$aA+bBrR,a+br$

Pick one reactant as the basis for determining the conversion. We call this the

key reactant. Let A be the key. Then for ideal gas behavior,

$x_A=\frac{1-\frac{C_A}{C_{A0}}(\frac{T{}_0}{T_0})}{1+{}_A\frac{C_A}{C_{A0}}(\frac{T{}_0}{T_0})}or\frac{C_A}{C_{A0}}=\frac{1-x_A}{1+{}_A{x}_A}(\frac{T_0}{T{}_0})$

$x_A=\frac{\frac{C_{B0}}{C_{A0}}-\frac{C_B}{C_{A0}}(\frac{T{}_0}{T_0})}{\frac{b}{a}+{}_A\frac{C_B}{C_{A0}}(\frac{T{}_0}{T_0})}or\frac{C_B}{C_{A0}}=\frac{\frac{C_{B0}}{C_{A0}}-\frac{b}{a}{x}_A}{1+{}_A{x}_A}(\frac{T_0}{T{}_0})$

How can I dervice xa and CA$/$CA$0?$

is it came from ideal gas equation?