hey all!
i have a question i was hoping some of you could unravel. specifically, in thermodynamics i understand in a quasi-static situation we can right work as:
[tex]W=\int PdV[/tex] where [itex]W[/itex] is work, [itex]P[/itex] is pressure, and [itex]V[/itex] is volume.
my book defines polytropic to be [tex]PV^n = constant[/tex]
it then writes the following polytropic, quasi-static equality when [itex]n=1[/itex] , which is where i am lost:
[tex]W=\int PdV = \int P \frac{{V_1}}{V} dV = P{V_1} \int \frac{{dV}}{V} = P{V_1} ln(\frac{{V_2}}{V_1}) [/tex]
specifically, if [itex]PV_1[/itex] is constant, then if we pull it out of the integral how is it we integrate over [itex]\frac{1}{V}[/itex] ? why doesn't the equality implode here: [tex] \int P \frac{{V_1}}{V} dV = P{V_1} \int \frac{{dV}}{V}[/tex] if [itex]V_1[/itex] is a constant why isn't [itex]V[/itex]? any help is greatly appreciated!
i have a question i was hoping some of you could unravel. specifically, in thermodynamics i understand in a quasi-static situation we can right work as:
[tex]W=\int PdV[/tex] where [itex]W[/itex] is work, [itex]P[/itex] is pressure, and [itex]V[/itex] is volume.
my book defines polytropic to be [tex]PV^n = constant[/tex]
it then writes the following polytropic, quasi-static equality when [itex]n=1[/itex] , which is where i am lost:
[tex]W=\int PdV = \int P \frac{{V_1}}{V} dV = P{V_1} \int \frac{{dV}}{V} = P{V_1} ln(\frac{{V_2}}{V_1}) [/tex]
specifically, if [itex]PV_1[/itex] is constant, then if we pull it out of the integral how is it we integrate over [itex]\frac{1}{V}[/itex] ? why doesn't the equality implode here: [tex] \int P \frac{{V_1}}{V} dV = P{V_1} \int \frac{{dV}}{V}[/tex] if [itex]V_1[/itex] is a constant why isn't [itex]V[/itex]? any help is greatly appreciated!