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Internal Energy and the First Law of Thermodynamics
The total energy possessed by all the particles of a gas is called the internal energy of the gas, U.  
 
If work is done, either by the gas (during expansion) or on the gas (during compression by an external force) the internal energy can be changed.  
 
If the gas is in an insulating container, the change in internal energy is equal to the work done  
 
If the gas is in a conducting container then this tells us that energy can enter or leave the gas without work being done.  
Energy entering or leaving the gas by this other means will be represented by Q and the process by which the energy enters (or leaves) is called heating (or cooling) the gas.  
So, if we want to represent a situation in which the internal energy of a gas is changed without doing any work, we have
 
and, in the more general case in which the internal energy is changed by working and heating, we can write  
 
This simple equation is a statement of the first law of thermodynamics and is really just a thermodynamic version of the principle (or law) of conservation of energy.  
When using this equation, Q is positive when the gas is heated (negative for cooling).  
Similarly, work done on the gas is positive and work done by the gas is negative.  
   
A useful form of the equation is  
 
see here for explanation of this substitution.  
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