Writing equations for adiabatic process - Thermodynamics Video Lecture | Physical Chemistry

so what is adiabatic process it is a
thermodynamic process
in which
there is no heat exchange between system
and surrounding
so let us say we have a system and we
have surrounding and we have a adiabatic
process
so in this process basically there is no
heat exchange mathematically this can be
written as Q is equals to zero where Q
denotes heat exchange between system and
surrounding so heat exchange between
system and surrounding
now we are interested to find equation
for adiabatic process so what is the
equation for adiabatic process and this
can be done by using first law of
thermodynamics so what is fast law of
thermodynamics that is Delta e is equals
to Q plus W where Delta e denotes change
in internal energy
q denotes heat exchange and W denotes
work we know for adiabatic process Q is
equals to 0 so we can write fast law of
thermodynamics for adiabatic process
that is Delta e is equals to Q plus W so
this becomes Delta e is equals to zero
plus W because for adiabatic process Q
is equals to zero so we will have simply
Delta e is equals to W so for adiabatic
process
we can write first law of thermodynamics
that is Delta e is equals to W now let
us write expression for work and
internal energy W is written as minus P
DV where P denotes pressure and V
denotes volume and change in internal
energy Delta e is given by n CV DT so
where we have n is number of moles CV is
molar heat capacity at constant volume
and T is the temperature basically see
we can also be written as R by gamma
minus 1 where R is the gas constant and
gamma is the molar heat capacity ratio
that is CP by CV so gamma is simply CP
by CV so this expression that is CV is
equal to R by gamma minus 1 can easily
be proved that is we know CP minus CV is
equals to R and we also know CP by CV is
equals to gamma that is CP is equals to
gamma C V so from Equation 1 we can
write that is gamma CB minus CV is
equals to R so if I take C be common we
will have gamma minus 1 is equals to R
so from here we can write C V is equals
to R by gamma minus 1 that is the same
thing
so now we have for adiabatic process we
have written that as Delta e is equals
to W and Delta e is given by n CV DT and
work is given by minus PDV and CV we can
write that is R by gamma minus 1 now we
have three variable temperature volume
and pressure so basically we have this
time three variable we can change into
two variable if I apply ideal gas
equation that is PV is equal to NRT so
from here we can write T is equals to or
basically we can write P is equals to
NRT by V so we will eliminate pressure
so we will have pressure can be written
as NRT by V now in our cancels so we
will have DT by gamma minus 1 and this
is DV by V into temperature now let us
separate the variable that I take to
this side so basically we have
separation of variable
so we will have DT by T is equals to
gamma minus one DV by V basically we
have a minus sign also here so this is
the minus sign so we will have DT by T
one minus gamma DV by V now let us
integrate both side from t1 to t2 and
volume is from v1 to v2 so this
integration will be Ln t2 by t1 and this
side we will have 1 minus gamma ln v2 by
v1 if I take 1 minus gamma inside the
log it will become power so Ln t2 by t1
is ln v2 by v1 raised to the power 1
minus gamma R we can write Ln t2 by t1
minus ln v2 by v1 1 minus gamma is
equals to 0 or this can also be written
as Ln t2 by t1 be 2 raised to the power
1 minus gamma V 1 raised to the power 1
minus gamma is equals to Ln 1 because 1
log of 1 is 0 so this we can write Ln t2
v2 gamma minus 1 t 1 v1 gamma minus 1 is
equals to Ln 1 so we will have this is
equals to 1
so we can write t 1 B 1 gamma minus 1 is
equals to T 2 V 2 gamma minus 1 so from
here we can see adiabatic process will
have a quiz on T V gamma minus 1 this is
equals to constant so this is the
equation for adiabatic process
now we can also write same equation in
terms of pressure and volume because we
know PV is equal to NRT so temperature
is goes to PV by NR so if I plug the
value of temperature in this equation we
will have PV by NR V to the power gamma
minus 1 is constant so this becomes P V
gamma is equal to constant multiplied by
n into R so this is another constant so
we can write PV gamma is equals to
constant it is also equation for
adiabatic process so either we can write
T V gamma minus 1 is equals to constant
or we can write PV gamma is equals to
constant