"will .250 M CuSO4 power a battery the same as .100 M CuSO4? or will it be weaker?"
"power" is a rather loose term here. Power depends on current flow, and the stronger electrolye will allow for better current flow, but this is a very general statement.
I don't know what you mean by weaker or stronger. However, for the Cu half cell, the Nernst equation is
E = Eo - (0.0592/2)*log(Cu/Cu^+2)
Eo for Cu = 0.337 written as a reduction.
As the solution becomes more dilute the denominator is smaller and the fraction is larger. The log of a larger number is greater and that times a negative sign gives that part of the equation a more negative value. That added to a positive number (0.337 for Cu) makes the number smaller. So the voltage of the cell will be smaller (everything else being equal) and assuming the Cu/Cu^+2 is the negative electrode of the cell. The cell will last longer, however, for the higher concn.
To determine if a .250 M CuSO4 solution will power a battery the same as a .100 M CuSO4 solution or if it will be weaker, we need to understand the concept of molarity and its effect on battery performance.
Molarity (M) is a measure of the concentration of a substance in a solution, expressed as the number of moles of a solute dissolved per liter of solution. In this case, CuSO4 is the solute and the solution is water.
In a battery, two reactions occur at the electrodes; one is an oxidation reaction at the anode, and the other is a reduction reaction at the cathode. These reactions involve the transfer of electrons and are driven by the difference in concentration between the two electrode compartments.
Generally, higher concentrations of the solute provide more ions and therefore more available reactants for the electrode reactions, resulting in a stronger battery. Therefore, the .250 M CuSO4 solution is expected to provide more ions for the electrode reactions compared to the .100 M CuSO4 solution, making it potentially stronger.
However, the effect of concentration on battery performance also depends on other factors such as the specific design of the battery, the electrode materials, and the overall electrochemical reactions involved. Therefore, it is not possible to conclusively determine if the .250 M CuSO4 solution will power the battery the same as or weaker than the .100 M CuSO4 solution without considering these additional factors.
To measure the actual performance of the battery, it is recommended to conduct experiments or refer to existing literature or data on similar battery systems to understand the relationship between concentration and battery performance.