1.5: Constant volume calorimetry

Constant volume conditions mean that, regardless of the details of the process, the system’s initial volume and final volume are equal, V_i = V_f, and so ΔV = V_f – V_i = 0. If the volume does not change, then there can be no work done (doing work requires the volume to change), so w = 0.

We can now apply these restrictions to the first law:

(the first law; always true)
 

(w = 0 for a constant volume process)

The last equation above is often written as ΔU = q_V, where the subscript V is to remind us that this equation only applies if the volume does not change.

If energy can only move between the system and surroundings as heat (not work), then we have an easy way to calculate changes in internal energy, ΔU.

To measure the heat flow under constant volume conditions, we use a bomb calorimeter.

In a bomb calorimeter (shown above), the reaction is carried out in a sealed vessel (labelled as 'bomb' above) that does not change in volume. Any heat released during the reaction is absorbed by the water surrounding the bomb chamber. Through an experimentally measured value called the heat capacity, we already know exactly how much heat is required to raise the temperature of water by 1 ^oC. Thus, by measuring how much the temperature of the surrounding water changes, we can calculate how much heat was released during the reaction in the bomb.

The total amount of heat released by a reaction in a bomb calorimeter, q_rxn, is related to the heat absorbed by the water, q_water, by
 

The heat for the reaction, q_rxn, is negative compared to the heat for the water, q_water because any heat released by the reaction is absorbed by the water. The heat absorbed by the water is given by the change in temperature, ΔT, multiplied by the heat capacity, C, where the heat capacity is an experimentally determined constant that describes how much heat must be added to increase the temperature of the water by 1 ^oC (or 1 K). Each different substance has a different heat capacity, which also depends on the amount of the substance. Thus, to use this formula we would need to know the heat capacity, C, for our exact substance (in this case water) in its exact amount. Such heat capacities are extensive state variables since they scale with the system size.

Tabulating heat capacities for each substance in each possible quantity would be incredibly tedious. Thus, heat capacities are often converted to intensive state variables by dividing by the number of moles (n) to give the molar heat capacity, (i.e. the heat capacity per mole of substance), or by dividing by the mass (m) to give the specific heat capacity, (i.e. the heat capacity per mass of substance). Since molar and specific heat capacities are intensive state variables, these values are the same regardless of the amount of substance present.
For example, the specific heat capacity of any volume of water is . When using a molar or specific heat capacity, C in the equation above is replaced with C=nC_n or C=mC_sp, where n is the number of moles and m is the mass of substance such that

 

 
Analyze the units of the three given equations to convince yourself that all give heat with units of energy.

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