Calculate the total entropy change for the following reversible processes:
a. Isothermal b. Adiabatic

Explanation

Entropy change for a process is given by:
`Delta_("sys")S = nC_(V) "In"(T_(2))/(T_(1)) +nR "In"(V_(2))/(V_(1))`,
`Delta_("surr")S =(-q_("system"))/(T_("sorroundings"))`
a. For isothermal process: `T_(2) = T_(1)`
`rArr Delta_(sys)S = nR"In"(V_(2))/(V_(1))` and
`Delta_(surr)S = (-q_(system))/(T_(surr.))=(-nRT"In"(V_(2))/(V_(1)))/(T) =- nR"In"(V_(2))/(V_(1))`
`[ :'q_("system")=nRT "In"(V_(2))/(V_(1))"for isothermla process"]`
`rArr Delta_(total)S = Delta_(sys)S +Delta_(surr)S = 0`
b. For adiabatic process:
`Delta_(sys)S = nC_(V) "In"(T_(2))/(T_(1)) +nR "In"(V_(2))/(V_(1))`
`=(nR)/((gamma-1)) "In"(T_(2))/(T_(1)) +nR "In" (V_(2))/(V_(1))`
Also, `TVgamma^(-1)=` constant
`rArr T_(1)V_(1)^(gamma-1) = T_(2)V_(2)^(gamma-1) rArr (T_(2))/(T_(1)) = ((V_(1))/(V_(2)))^(gamma-1)`
`rArr Delta_(sys)S = (nR)/((gamma-1))In In[((V_(1))/(V_(2)))^(gamma-1)] +nR"In" (V_(2))/(V_(1)) = 0`
and `Delta_(surr)S = (-q_(sys))/(T_(surr)) = 0 [ :' q = 0` in adiabatic process]
`rArr Delta_("total")S = Delta_("sys")S +Delta_("surr")S = 0`