Greetings dear chemists!
I got a nice exam problem on wich i am stuck for hours, well maybe the problem is with me. I know the problem can be solved. So here we go.
We got a weak monoprotic acid, we don't know its concentration, dissociation degree nor the dissociation constant. If we dilute it 3 times its volume, then the dissociated ions grow with an unknown constant, lets call it $y$. If we dilute the diluted solution again by 5 times its volume, then the dissociated ions grow 4 times. Whats the dissociation degree of the undiluted solution?
Yeah woah that looks tough. So lets give it my try.
Lets call the concentrations in order of dilution by $c_0, c_1, c_2$ and the dissociation degree by $\alpha_0,\alpha_1,\alpha_2$, and the dissociation constant of course $K_s$.
I managed to get three equations for this problem, but i feel i missed something or messed up my algebra someway i dont understand.
$$c_0=?,\alpha_0=?,K_s=?$$ $$\alpha_0=\frac{[A^-]}{c_0}$$ $$HA=H^++A^-$$ $$c_1=\frac{c_0}{3},\qquad c_2=\frac{c_1}{5} \to c_2=\frac{c_0}{15}$$ $$\alpha_1=y\alpha_0,\qquad \alpha_2=4\alpha_1 \to \alpha_2=4y\alpha_0$$ We call forth Ostwald; $$K_s=\frac{c_0\alpha_0^2}{1-\alpha_0}\qquad \qquad \qquad \qquad (1)$$ $$K_s=\frac{c_1\alpha_1^2}{1-\alpha_1}=\frac{c_0y^2\alpha_0^2}{3(1-y\alpha_0)}\qquad \qquad(2)$$ $$K_s=\frac{c_2\alpha_2^2}{1-\alpha_2}=\frac{c_016y^2\alpha_0^2}{15(1-4y\alpha_0)}\qquad \qquad(3)$$ Equating the first and the second equation, i could get an expression for $y$. By setting $\beta=\frac{3}{1-\alpha_0}$ and ignoring the negative root $$y^2+y\beta\alpha_0-\beta=0$$ $$y=\frac{-\beta\alpha_0+\sqrt{\beta^2\alpha^2+4\beta}}{2}$$ My problem starts when i equate the second and the third equation; $$\frac{c_0y^2\alpha_0^2}{3(1-y\alpha_0)}=\frac{c_016y^2\alpha_0^2}{15(1-4y\alpha_0)}$$ $$5-20y\alpha_0=16-16y\alpha_0$$ $$11+4y\alpha_0=0$$ From wich i clearly see that im on the wrong track, since $\alpha_0$ and $y$ cant be negative, please help me on solving this nightmare. Maybe i missed something obvious, i dont really know that im on the right method to solve this either.