Question

Hot water was poured into an empty pop can and allowed to cool in surroundings whose temperature is 20 °C. The initial temperature of the water is 92 °C. After 12½ min the water temperature had dropped by 10 deg C. Assume the rate of change of the water temperature is directly proportional to the difference in the temperature between the water and its surroundings.

a. What is the value of the proportionality constant?

b. What is the temperature of the water after 83 min?

c. When is the rate of change of temperature at its largest?

d. When is the rate of change of temperature at its smallest?

Answer 1

Initial Temperature of hot water = 92 C

Surroundings temperature = 20 C

after 12. min, Temperature = 92 - 10 = 82 C

Rate of change of temperature is proportional to difference between the temperature of water and surroundings,

-dT/dt = K*(T - T_surr),

Here negative sign depicts that the temperature decreases as the system moves forward.

on integrating,

ln( T - T_surr) = -Kt + C,

K = proportionality constant and C = integration constant

at t = 0 ; T = T0 ( initial temperature of water).

ln(T0 - T_surr ) = C

ln[( T - T_surr) /(T0 - T_surr )] = -Kt

At 12.5 min, T = 82 C. Substituting this in the above equation,

ln [ (82 - 20)/(92 - 20)] = -K*12.5

K = 0.0112 min^-1

b) at , t = 83 min

ln[( T - T_surr) /(T0 - T_surr )] = -Kt;

[( T - T_surr) /(T0 - T_surr )] = e^-kt

(T - 20)/(92 - 20) = e^-0.0112*83

T = 48.41 C

c) Rate of change of temperature is largest when the temperature difference between the hot water and surroundings is largest. This is at the initial condition when time , t = 0 min.

d) Rate of change of temperature is smallest when the temperature difference between the hot water and surroundings is smallest. This is at the condition when the temperature of hot water approaches the temperature of surroundings.usually at time, t = infinity.