The gas -phase thermal decomposition of one mole of $(CH_3)_3CO_2C(CH_3)_3$ in a constant volume apparatus yields two moles of acetone and one mole of ethane.If the reaction obeys ${1^{st}}$ order then which of the following is true?

Question

$\begin{array}{1 1}(a)\;k_1=\large\frac{1}{t}\normalsize ln\large\frac{P_i}{P_i-P}\\(a)\;k_1=\large\frac{1}{t}\normalsize ln\large\frac{2P_i}{2P_i-P}\\(c)\;k_1=\large\frac{1}{t}\normalsize ln\large\frac{2P_i}{3P_i-P}\\(d)\;k_1=\large\frac{1}{t}\normalsize ln\large\frac{P_i}{3P_i-P}\end{array}$