State Newton's law of cooling verify with an experiment.

Explanation

(i) It states that the rate of cooling of a body is directly proportional to the temperature differ between the body and the surroundings.
(ii) Consider a spherical calorimeter of mass m whose outer surface is blackened. It is filled with hot water of mass m, the calorimeter with thermometer is suspended from a stand.
(iii) The calorimeter & the hot water radiate heat energy to the surrounding. Using a stop clock, the temperature is noted for every 30 second interval of time see the temperature falls by about `20^(@)C`. The readings are tabulated.
If the temperature falls from `T_(1)` to `T_(2)` in t sec, the quantity of heat energy lost by radiation `Q=(ms+m_(1)s_(1))(T_(1)-T_(2))`, where 'S' is the specific heat capacity of the material of the calorimeter & `S_(1)` - specific heat capacity of water.
Rate of cooling = `("Heat energy lost")/("time "t_(n))`
`therefore (Q)/(E)=((ms+m_(1)s_(1))(T_(1)-T_(2)))/(t)`
Room temperature - `T_(0)`
(v) Average excess temperature of the colorimeter over that of the surroundings = `(T_(1)+T_(2))/(2)-T_(0)`
(vi) Acceleration to Newton's law of cooling `(Q)/(t)prop((T_(1)+T_(2))/(2)-T_(0))`
(vii) Assume the pressure of the gas remains constant during an infinitesimally small outward displacement dy then work done `dW-F.dx=P.A.dx`
`dW=P.dV`
(viii) Total work done by the gas from volume `V_(1)` to `V_(2)` is
`W=underset(V_(1))overset(V_(2))intP.dv`
(ix) But `PV^(gamma)` = constant (k)
`gamma=(C_(P))/(C_(V))`
`therefore W=underset(V_(1))overset(V_(2))intkV^(gamma)dV=k[(V^(1-gamma))/(1-gamma)]_(v_(1))^(v_(2))" "[becauseP=(k)/(V^(gamma))]`
`thereforeW=(k)/(1-gamma)[V_(2)^(1-gamma)-V_(1)^(1-gamma)]`
`W=(1)/(1-gamma)[kV_(2)^(1-gamma)-kV_(1)^(1-gamma)]`
`P_(2)V_(2)^(gamma)=P_(1)V_(2)^(gamma)=k`
(x) Subtract the value of k
`therefore W=(1)/(1-gamma)[P_(2)V_(2)^(gamma),V_(2)^(1-gamma)-P_(1)V_(1)^(gamma),V_(1)^(1-gamma)]`
`W=(1)/(1-gamma)[P_(2)V_(2)-P_(1)V_(1)]`
It `T_(2)` is the final temperature of the gas in adiable expansion, then
`P_(1)V_(1)=RT_(1),P_(2)V_(2)=RT_(2)`
`therefore W=(1)/(1-gamma)[RT_(2)-RT_(1)]`
This is the equation for the work done during adiabating process.