Why are you confused? I do not see the confusion...I can be flexible....NO SUBSCRIPT DESIGNATIONS....I can remove Cv and Cp...and set it up as follows:
OK so do it this way:
Heat Capacity of the Calorimeter:
In calorimetry it is often desirable to know the heat capacity of the calorimeter itself rather than the heat capacity of the entire calorimeter system (calorimeter and water). The heat (q) released by a reaction or process is absorbed by the calorimeter and any substances in the calorimeter. If the only other substance in the calorimeter is water, the following energy balance exists:
q = qcal + qw
where qcal is the heat flow for the calorimeter and qw is the heat flow for the water.
Both of these individual heat flows can be related to the heat capacity and temperature change for the substance.
qcal = Ccal ΔT
qw = Cw ΔT
where Ccal is the heat capacity of the calorimeter and Cw is the heat capacity of the water. Because the water and calorimeter are in thermal equilibrium, they both have the same temperature and thus ΔT is the same for both. The consequence is that the heat capacity of the entire system (C) is the sum of the heat capacities for the individual components.
C = Ccal + Cw
The heat capacity is an extensive property; that is, the heat capacity depends upon the amount of substance present. The calorimeter exists as a fixed unit, thus its heat capacity is a fixed value. The amount of water in the calorimeter, however, can vary, and thus the heat capacity of the water can vary. When dealing with variable amounts of material, one often prefers to use an intensive measure of the heat capacity. One common intensive version of the heat capacity is the specific heat capacity (s), which is the heat capacity of one gram of a substance.
s = Cw
Because the mass of water (mw) and the specific heat capacity of water are both known, you can readily calculate the heat capacity of the water. The specific heat capacity of water (sw) is
sw = 4.184 J oC-1 g-1
Overall one can write
C = Ccal + sw mw