Newton's Law of Cooling

Key Questions

  • How do I find #k# in Newton's Law of Cooling?

    Answer:

    Set up an equation with all the knowns and solve for the unknown! Make sure to know your law of cooling too, shown in blue in the Explanation section.

    Explanation:

    Newton's Law of Cooling is given by the formula

    #color(blue)(T(t) = T_s + (T_0 - T_s)e^(-kt)#

    Where

    •#T(t)# is the temperature of an object at a given time #t#
    •#T_s# is the surrounding temperature
    •#T_0# is the initial temperature of the object
    •#k# is the constant

    The constant will be the variable that changes depending on the other conditions. Let's take an example of a question where you would need to find #k#.

    The average coffee temperature at a particular coffee shop is #75˚#C. Marie purchases a coffee from the local coffee shop. After 10 minutes, the drink has cooled to #67˚# C. The temperature outside the coffee shop is steady at #16˚C#. Assuming the coffee follows Newton's Law of Cooling, determine the value of the constant #k#

    Let's identify our variables.

    •#T_0 = 75˚C#
    •#T_s = 16˚C#
    •#t = 10#
    •#T(t) = 67˚C#
    •#k = ?#

    We have

    #67 = 16 + (75 - 16)e^(-10k)#

    #51 = 59e^(-10k)#

    #51/59 = e^(-10k)#

    #ln(51/59) = ln(e^(-10k))#

    #ln(51/59) = -10k#

    #k = -1/10ln(51/59)#

    Use a calculator to get

    #k~~ 0.01457#

    Hopefully this helps!

    Noah G · 15 · May 2 2017
  • What is Newton's Law of Cooling?

    Answer:

    Newton's Law of Cooling :
    #-(dT//dt) prop DeltaT#
    #=>-(d theta //dt) prop Deltatheta#

    Explanation:

    Newton's Law of Cooling states that, if the temperature 'T' of the body is not very different from that of the surroundings '#T_0#', then rate of Cooling '(-dT/dt)'or' #-(d theta//dt)#' is proportional to the temperature difference between them.

    SaiGanesh Gotetti · 1 · Nov 4 2016

Questions

Solving Exponential and Logarithmic Equations