daveedka, this is a fun problem!
In reference to what happychem was saying:
Let's say you change your water by 30%, leaving 70% behind with the 'stuff' you added, whether salt or what have you.
Let k be the amount (fractional) of water you leave behind.
Let Ci be the concentration after ith addition of salt or stuff.
Let x be the amount of stuff you add, arbitrary units (mg, mol, drops, whatever).
Let V be your tank volume.
If C0=0 (nothing in there to start with), after 1 addition you get C1 = x/V.
Now you remove (1-k) fraction of water, leaving k fraction behind. The total amount of your stuff left in the tank is k*C1*V. Also, we're adding x more when we do the water change. Therefore C2 = (k*C1*V + x)/ V or
C2 = (k+1) x/V.
Similarly, C3 = (k*C2*V +x)/V and if you use the recursion relation you get
C3 = x*(k^2 + k +1)/V, C4 = x*(k^3 +k^2 + k + 1)/V ...etc.
You can go ad infinitum. The ith addtion/water change gives you
Ci = x * (1 + k + k^2 + .... k^(i-1)) / V
This is a geometric series and can be written in closed form:
Ci = x*(1-k^i)/(1-k)/V
The steady state is given by summing to infinity. You get
Cinfinity = x/(1-k)/V
So now let's put in numbers. If you're doing a 30% water change every day and adding 0.5 g of baking soda to a 55 gallon tank (210L), after many many changes you get
Cinfinity = 0.5 g / (1-0.7)/ 210 L =
0.5 g / 0.3 / 210 L = 0.0079 g/L or ~ 8 ppm of your baking soda.
In reference to what happychem was saying:
Let's say you change your water by 30%, leaving 70% behind with the 'stuff' you added, whether salt or what have you.
Let k be the amount (fractional) of water you leave behind.
Let Ci be the concentration after ith addition of salt or stuff.
Let x be the amount of stuff you add, arbitrary units (mg, mol, drops, whatever).
Let V be your tank volume.
If C0=0 (nothing in there to start with), after 1 addition you get C1 = x/V.
Now you remove (1-k) fraction of water, leaving k fraction behind. The total amount of your stuff left in the tank is k*C1*V. Also, we're adding x more when we do the water change. Therefore C2 = (k*C1*V + x)/ V or
C2 = (k+1) x/V.
Similarly, C3 = (k*C2*V +x)/V and if you use the recursion relation you get
C3 = x*(k^2 + k +1)/V, C4 = x*(k^3 +k^2 + k + 1)/V ...etc.
You can go ad infinitum. The ith addtion/water change gives you
Ci = x * (1 + k + k^2 + .... k^(i-1)) / V
This is a geometric series and can be written in closed form:
Ci = x*(1-k^i)/(1-k)/V
The steady state is given by summing to infinity. You get
Cinfinity = x/(1-k)/V
So now let's put in numbers. If you're doing a 30% water change every day and adding 0.5 g of baking soda to a 55 gallon tank (210L), after many many changes you get
Cinfinity = 0.5 g / (1-0.7)/ 210 L =
0.5 g / 0.3 / 210 L = 0.0079 g/L or ~ 8 ppm of your baking soda.
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