Deriving equations

$w=-P\Delta V$
and
$w=-\int_{V_{1}}^{V_{2}}PdV=-nRTln\frac{V_{2}}{V_{1}}$

904638389
Posts: 16
Joined: Fri Sep 25, 2015 3:00 am

Deriving equations

In lecture, Lavelle put a huge emphasis on deriving the different equations in this section dealing with reversible and irreversible expansion. Everytime it seemed like the steps were scattered and hard to follow, so can someone please explain the steps or post a link where it explains the deriving?

Phoebe Hao 1J
Posts: 20
Joined: Fri Sep 25, 2015 3:00 am

Re: Deriving equations

For irreversible:

W=FxD
F=PxA
==>w=PxAxD ; since AxD gives units3 (units cubed), this results in a volume term
==>w=PxdeltaV ; since the energy of the system has decreased (energy used as work of expansion), the term becomes negative
==>w=-PdeltaV

For reversible:
since it is isothermal, T is a constant

w=-PdeltaV
PV=nRT ==> P=nRT/V
w=-(nRT/V)(deltaV)
w= integral of -(nRT/V)dV with lower limit as V1 and upper limit as V2 ; since the volume is changing
since -nRT does not change, it is a constant==> you can pull it out to the front of the integral
==> with the lower and upper limits as stated before: -nRT integral (1/V)dV
the integral of of any argument u: du/u is ln(u)
so==> -nRT integral (1/V)dV is -nRT (ln(V))] (lower and upper limits)
this leads to: -nRT(ln(V2) - ln(V1))
when dealing with logarithms, subtracting can be rewritten as division (i.e. (ln(V2) - ln(V1)) ==> ln(V2/V1)
so, the w term for reversible is:
w=-nRTln(V2/V1)