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The main difference between a true bomb calorimeter and the coffee cup calorimeter is that the bomb is much more sophisticated and thus gives more accurate calculations. Other than that, the function of each is the same. As far as how it works, I think all we should know for this course is that a bomb calorimeter is a type of constant-volume calorimeter used in measuring the heat of combustion of a particular reaction.
A coffee cup calorimeter is good for measuring heat flow in a solution however it can’t be used for gaseous reactions because the gas would escape from the styrofoam cup. Although they both have the same function, the bomb calorimeter is generally used to measure heat flows for higher temperature reactions and gases .
A bomb calorimeter's outstanding feature is its ability to measure heat under constant volume, in contrast to the coffee cup calorimeter which measures heat under constant pressure (enthalpy). The bomb calorimeter accomplishes this distinction, hopefully, by containing the experiment within a box (usually steel) whose walls are rigid enough to contain any changes in pressure the system may induce. A sealed thermometer measures the temperature of the system within the container, and some form of trigger is wired through the container's walls to allow the scientist to initiate the experiment. If the walls are not rigid enough, the built-up pressure within the container (if the reaction is exothermic, because an increase in temperature under constant volume leads to an increase in pressure) will cause the entire system to explode violently like a bomb (in fact, the difference between a bomb calorimeter and say a pipe bomb or a pressure cooker bomb -- like the one used in the Boston marathon bombing -- is whether it is meant to explode or not); however, if the calorimeter works as intended, it will not explode.
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