Make p-V equation for an adiabatic process.

Answer

A, C

Explanation

In adiabatic process, `dQ=0`
and `dW=-dU`
`:.` `pdV=-C_VdT` (for `n=1`)
`:.` `dT=-(pdV)/(C_V)` …(i)
Also, for 1 mole of an ideal gas,
`d(pV)=d(RT)`
or `pdV+Vdp=RdT`
or `dT=(pdV+Vdp)/(R)`…(ii)
From Eqs. (i) and (ii), we get
`C_VVdp+(C_V+R)pdV=0`
or `C_VVdp+C_ppdV=0`
Dividing this equaiton by PV, we are left with
`C_V(dp)/(p)+C_p(dV)/(V)=0`
or `(dp)/(p)+gamma(dV)/(V)=0`
or `int(dp)/(p)+gamma int(dV)/(V)=0`
or `In(p)+gamma In (V)=` constant
We can write this in the form
`pV^gamma=`constant