How can I calculate the work of bringing an ion of negative charge Q = -2 from infinite distance to within 10Å of a protein of negative charge -10, at 300K?

 

This is a integral i think.
\(
W = -q\int_a^b \! E \, ds
\)
where the limit a is -infinity and b 10angstrom

q is charge, E is the electric field. Integral over the distance.
You can calculate the electric field, then you need to integrate it and solve for the limits. I vaguely remember solving a similar problem like this.

This integral should converge, and actually the answer will be the electric field integrated, times the charge evaluated at 10 angstrom since the integral should be zero when evaluated at infinity.

 

Doing work in a conservative force field is calculated using a line integral \(\int \mathbf F \cdot \mathbf {ds}\). Since the force on an electric charge is qE, the formula becomes (I'll let you place the signs appropriately)

\(\int {q\mathbf E \cdot \mathbf ds} \)

Since you'll no doubt model the protein as a point charge (a distributed charge would probably be too hard to solve for a homework problem, in general), you'll use a linear path in from infinity; this will result in the (implied) Riemann integral given by steinar96. A possibly pedantic distinction, perhaps, but you'll be able to handle more complex cases when the integration path isn't so simple.

 

This problem can also be solved from consideration of voltage. A conservative field has the property that the path integral between two points is independent of the path taken, as pointed out by someonesdad. Hence, the work done will be the difference in potential energy of the two points. The voltage potential (potential energy per unit charge) in the field of a point charge is well known, and calculation of the potential energy at the two radii (10 Å and infinity) is easy.