The time derivative of the amount of salt in the basin is equal to the salt outflow rate and the time derivative of the amount of water is the sum of the water inflow and outflow rates,
\dot{N} _s(t) = \Omega ^{out}_{s} (t).
\dot{N} _w(t) = \Omega ^{in}_{w} + \Omega ^{out}_{w} (t).
where Ω^{in}_{w} is the fresh water inflow rate (positive), and Ω^{out}_{s} and Ω^{out}_{w} are the salt and water outflow rates (negative). Since a matter flow is an extensive quantity, the constant salty water outflow rate Ω^{out}_{sw} is the sum of the salt outflow rate Ω^{out}_{s} (t) and the water outflow rate Ω^{out}_{w} (t),
\Omega ^{out}_{s w} = \Omega ^{out}_{s}(t) + \Omega ^{out}_{w}(t).
Water and salt are assumed to be thoroughly mixed in the basin so that the salt concentration can be considered homogeneous. Thus, the salt outflow rate Ω^{out}_{s} (t) is equal to the product of its concentration c_s (t) in the basin and of the salty water outflow rate Ω^{out}_{sw} ,
Ω^{out}_{s} (t) = c_s (t) Ω^{out}_{s w}.
Applying this equation for Ω^{out}_{s} (t) in the balance equation for the salt in the basin, using the definition of the molar concentration,
c_s (t) = \frac{N_s (t)}{N_s (t) + N_w (t)} .
and dividing by N_s (t), we obtain,
\frac{\dot{N}_s(t) }{N_s (t)} =\frac{\Omega ^{out}_{s w} }{N_s (t) + N_w (t)}.
Adding up the first two balance equations, we obtain the balance equation for the salty water in the basin,
\dot{N}_s(t) + \dot{N}_w(t) = Ω^{in}_{w}+Ω^{out}_{s w}.
Since the term on the right hand side is constant, we integrate this equation with respect to time from t = 0 onwards
N_s (t) + N_w (t) = (Ω^{in}_{w} +Ω^{out}_{s w} )t + N_s (0) + N_w (0).
Applying this into the equation for \dot{N}_s(t) /N_s (t), we obtain,
\frac{\dot{N}_s(t) }{N_s (t)} =\frac{\Omega ^{out}_{s w} }{(\Omega ^{in}_{w}+\Omega ^{out}_{sw} )t + N_s(0) +N_w(0)}.
The time integral of this equation is given by,
\ln \biggl(\frac{N_s(t)}{N_s(0)}\biggr) =\frac{\Omega ^{out}_{sw} }{\Omega ^{in}_{w} + \Omega ^{out}_{sw} } \ln \Biggl(\frac{(\Omega ^{in}_{w}+\Omega ^{out}_{sw} )t +N_s(0)+N_w(0)}{N_s(0)+N_w(0)}\Biggr).
The exponential of this integrated expression yields,
N_s(t) = N_s(0) \biggl(1+\frac{(\Omega ^{in}_{w} + \Omega ^{out}_{s w} )t}{N_s(0) + N_w (0)} \biggr)^{\frac{\Omega ^{out}_{s w} }{\Omega ^{in}_{w} + \Omega ^{out}_{s w} } }.
Applying the equations for N_s (t) and N_s (t) + N_w (t) in the expression for the molar salt concentration c_s (t), we obtain,
c_s(t) = \frac{N_s(0)}{(\Omega ^{in}_{w} +\Omega ^{out}_{s w} )t + N_s (0)+ N_w (0)}\biggl(1 + \frac{(\Omega ^{in}_{w} + \Omega ^{out}_{s w} )t}{N_s (0) + N_w (0)} \biggr)^{\frac{\Omega ^{out}_{s w} }{\Omega ^{in}_{w} + \Omega ^{out}_{s w} } }.
This can be recast as,
c_s(t) = \frac{N_s(0)}{ N_s (0)+ N_w (0)}\biggl(1 + \frac{(\Omega ^{in}_{w} + \Omega ^{out}_{s w} )t}{N_s (0) + N_w (0)} \biggr)^{-\frac{1 }{1+\Omega ^{out}_{ sw} / \Omega ^{in}_{w} } }.