# Boundary Work for a Polytropic Process

A

#### polytropic process

is a process in which:
C & δ are constants
Where :

for a

:
For an

(PV = nRT), the

#### boundary work

equation can be further simplified:
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### Ch 4, Lesson A, Page 19 - Boundary Work for a Polytropic Process

• A polytropic process is a process for which P times V raised to some constant power, delta, is equal to a constant, C.
• This is a process path because it provides a relationship between P and V that applies throughout the process.
• We can rearrange our definition of a polytropic process to get P = C times V raised to the minus delta pawer.
• Then all we need to do is plug this back into our equation for boundary work: Wb = the integral of P dV from V1 to V2.
• This integral isn’t too bad because C is a constant.
• The integral of V to the minus delta dV is just V to the minus delta plus 1, divided by the quantity minus delta plus 1.
• When you evaluate this from V1 to V2 things get kind of messy.
• If you group terms in a creative way, this equation can be simplified.
• Notice that C times V2 raised to the minus delta power is just P2.
• Similarly, C times V1 raised to the minus delta power is just P1.
• This result is clean and easy to use.
• But it can be simplified even further when the gas in the system can be considered to be an ideal gas.
• For an ideal gas, P1 V1 is just nRT1 and P2 V2 is nRT2.
• The result is that the boundary work is n times R times the change in temperature divided by the quantity 1 minus δ.
• That pretty much concludes our introduction to boundary work.
• This is probably the most important form of work in this course, but it is not the only one.
• Flip the page to see some other forms of work that will play important roles in your introduction to thermodynamics.

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