In the red blood cell, glucose is transported into the cell against its concentration gradient.
The energy for this transport is supplied by the hydrolysis of ATP:

ATP + H2O → ADP + Pi at 298K

Assume the overall transport reaction is:

ATP + H2O + 2glucose(out) → ADP + Pi + 2glucose(in).

(a) At 37 °C, under conditions where [ATP], [ADP], and [Pi] are each held constant at 100 mM by cell metabolism, what is the maximum value of [glucose(in)]/[glucose(out)]? You can assume the activity coefficients of all species are 1.
(b) In an actual cell, the glucose inside the cell may exhibit non-ideal behavior. How would this affect the activity coefficient? Would this increase or decrease the maximum concentration gradient obtainable?

a)   the maximum value of   = 4,200

b) Due to non ideal behavior; In an actual cell, the glucose inside the cell may have an activity coefficient much less than 1 .

Since,   , we conclude that the   ratio will be bigger which implies that the maximum concentration gradient obtainable will also increase.

Explanation:

The hydrolysis of ATP can be written as follows:

Free energy available from ATP hydrolysis can be expressed as:

Given that:

[ATP], [ADP], and [Pi] are each held constant at 100 mM =

R(rate constant) = 8.314 J/mol

Temperature (T) = 37 °C = (37+273)K = 310 K

replacing our value into the above equation; we have:

We therefore conclude that; is the energy available for the transport of glucose.

However, if the glucose transport energy is equal to free energy available out of ATP hydrolysis;

Then:    

As Such ;  +42.9 kJ/mole = -

From the question; we Assume  that  the activity coefficients of all species = 1

Thus K= 1

    (since no energy input at equilibrium)

Making the subject of the formula and taking the square of the exponential; we have:

    4125.74

 = 4,200        (to the nearest thousand )

Thus, the maximum value of   = 4,200

b) Due to non ideal behavior; In an actual cell, the glucose inside the cell may have an activity coefficient much less than 1 .

The expression for glucose transport reaction can be written as:

=

=

= 4200

 =

From the foregoing; we can conclude that , and as such; the   ratio will be bigger which implies that the maximum concentration gradient obtainable will also increase.