Solve works with symbols, not with numbers

The reason is the precedence of simplifications undertaken by Solve for symbolic vs. numeric values. It turns out that for the latter case Solve tries another route of solution and gets trapped into a very inefficient procedure and eventually gives up.

Let us simplify your problem. In fact, there is a single relevant parameter mo/RT

xN=79900/(6197399567/2500000)//N
(*   32.2313 *)

The respective equation is

eq=x+2Log[n/(1+n)]-Log[-1+2/(1+n)]

Let us approximate xN in 3 different ways and see what happens.

• Integer (32)

---

val=Rationalize[xN,1]
Solve[eq==0/.x->val,n]
Out[1]= 32
Out[2]= {{n->1/Sqrt[1+E^32]}}

Fine, this work as expected. $$\frac{1}{\sqrt{1+e^{32}}}$$

• Simplest rational number (129/4)

---

val=Rationalize[xN,0.1]
Solve[eq==0/.x->val,n]
Out[3]= 129/4
Out[4]= {{n->Sqrt[(-1+E^(129/4)-E^(129/2)+E^(387/4))/(-1+E^129)]}}

Here, the solution is still possible, but it has a completely different (and equivalent form) $$\sqrt{\frac{-1+e^{129/4}-e^{129/2}+e^{387/4}}{-1+e^{129}}}$$

• Finally (290/9)

---

val=Rationalize[xN,0.01]
Solve[eq==0/.x->val,n]
Out[5]= 290/9
Out[6]= {{n->1/Sqrt[(1+E^290)/(1-E^(290/9)+E^(580/9)-E^(290/3)+E^(1160/9)-E^(1450/9)+E^(580/3)-E^(2030/9)+E^(2320/9))]}}

or in human-readable form $$\frac{1}{\sqrt{\frac{1+e^{290}}{1-e^{290/9}+e^{580/9}-e^{290/3}+e^{1160/9}-e^{1450/9}+e^{580/3}-e^{2030/9}+e^{2320/9}}}}$$

If we continue with more and more accurate val the expression get more and more complicated until it reaches the point that MA cannot handle.