Define heat capacity. What are `C_(p) and C_(v)`? Show that `C_(p)-C_(v)=R.`

Explanation

Heat capacity (C): "Heat capacity (C) of a substance is defined as the amount of heat required to raise its temperature through one degree".
(or)
It is the ratio of heat absorbed (q) to the resulting increase in temperature (dT)
`C=q/(d T)`
Heat capacity (C) is a state function. Hence to evaluate .C., the conditions such as volume or constant pressure have to be specified in order to define the.path. Thus there are two different types of heat capacities. These are :
i) Heat capacity at constant volume `(C_V)`
ii) Heat capacity at constant pressure `(C_p).`
From first law of thermodynamics,
`q=dE +W=dE +P.dV (W=PV)`
`C=q/(d T)=(dE +p.dV)/(dT)`
If the absorption of heat takes place at constant volume.
`C_(v) (i.e,) dV=0`
`C_(V)=(Q_(V))/(d T)=(delta E+O)/(delta T)=((delta E)/(delta T))_(V)`
Definition for `C_(v)` : Heat capacity of a system at constant volume `(C_(V))` may be defined as rate of change of internal enerny with temperature at constant volume.
If heat is absorbed by the system at constant pressure, heat capacity is denoted by `.C_(p).` and is called heat capacity at constant pressure.
`C_(P)=(q_(P))/(d T)=((delta E)/(delta T))_(P) +P ((delta V)/(delta T))_(P) ........(1)`
But H=E+PV
Differentiating this equation w.r.t. T at constant pressure, we get
`((delta H)/(delta T))_(P)=((delta E)/(delta T))_(p)+((delta V)/(delta T))_(P) …..(2)`
From equations (1) and (2), we get
`C_(p)=((delta H)/(delta T))_(P)`
Definition of `C_P` : Heat capacity of a system at constant pressure `(C_P)` may be defined as ra change of enthalpy with temperature at constant pressure.
Derivation of `C_p-C_V=R`
H= E +PV (For an ideal gas)
differentiating w.r.t. T.
`(d H)/(d T)=(d E)/(d T)=(d(PV))/(d T)`
`rArr (dH)/(dT)=(dE)/(dT)+(d(RT))/(dT) [PV=RT]`
`(dH)/(dT)=(dE)/(dT)+R`