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

• Can Newton's Law of Cooling be used to describe heating?
• How do I find `k` in Newton's Law of Cooling?
• What is `k` in Newton's Law of Cooling?
• What is `e` in Newton's Law of Cooling?
• What is `c` in Newton's Law of Cooling?
• Can Newton's Law of Cooling be used to find an initial temperature?
• A pie is removed from a 375°F oven and cools to 215°F after 15 minutes in a room at 72°F. How long (from the time it is removed from the oven) will it take the pie to cool to 72°F?
• You place a cup of 205°F coffee on a table in a room that is 72°F, and 10 minutes later, it is 195°F. Approximately how long will it be before the coffee is 180°F?
• A body was found at 10 a.m. in a warehouse where the temperature was 40°F. The medical examiner found the temperature of the body to be 80°F. What was the approximate time of death?
• Knowing `T-T_s=(T_0 - T_s)e^(kt)`, A pan of warm water (46dgC) was put in a refrigerator. 10 minutes later, the water's temperature was 39dgC; 10 minutes after that, it was 33dgC. Use Newton's law of cooling to estimate how cold the refrigerator was?