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Work Done During a Change of Volume of a Gas
Consider a quantity of gas in a container which has a frictionless piston.  
The volume occupied by the gas is changed by ΔV, as show in the diagram below.  
   
 
   
The pressure, p, acting on the surface of the piston produces a force, F.  
From the definition of pressure  
 
During the change in volume, this force does work, w, given by  
 
but  
 
Therefore, the work done by the force is given by  
 
   
When the volume of a gas increases, work is done by the gas (so, if no energy is supplied, the temperature of the gas will decrease).  
   
When the volume of a gas decreases, work is done on the gas by an external force (so, if the energy is not allowed to escape, the temperature of the gas will increase).  
   
The Principal Specific Heat Capacities of a Gas  
The above result shows that if the temperature of a gas is increased at constant volume, no work is done.  
However, if the temperature is increased and the gas is allowed to expand during the heating, work will be done.  
In this case, extra energy will have to be supplied to do this work.  
   
For this reason, if we want to talk about the specific heat capacity of a gas, we must specify the conditions under which the gas is heated.  
   
We therefore define two principal specific heat capacities of a gas, given the symbols, cv and cp  
   
cv is the quantity of energy needed to change the temperature of unit mass of a gas by 1C, at constant volume  
   
and, if we allow the volume to change, we can maintain a constant (or very nearly constant) pressure, hence  
   
cp is the quantity of energy needed to change the temperature of unit mass of a gas by 1C, at constant pressure.  
   
It should be clear that cp > cv and that the difference between them is equal to the work done during the heating at constant pressure.  
 
   
Quantities of substance are often measured in moles rather than grams or kilograms (especially when considering gases).   
Similar logic is used to define the principal molar heat capacities of a gas.  
   
These molar heat capacities are usually represented by capital letter, Cp and Cv and the same relation exits between them   
   
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