How Newton's Law of Cooling cools your Champagne

Here's a picture of a champagne fridge, and a perfect place for our next application of mathematical modelling with differential equations:

Image Source

Newton's Law of Cooling states that the rate of change of the temperature of an object over time, is proportional to the difference between its temperature and the surroundings, provided that this difference is not too large.

So by this definition, we have the differential equation:

where:

• T(t) is the temperature of the object in question
• T_s is the temperature of the surroundings
• k a constant of cooling

We can pretty much solve this problem by separating the variables straight away and going through the motions. However, it would be more advantageous if we first made a change of variable, since T_s is a constant.

So, if we let...

Then...

...as T_s disappears in the differentiation of y with respect to t. Thus our differential equation becomes...

This is the same form of equation as our exponential growth and decay models, and thus the solution is:

Now, let's have a look at our champagne example...

A bottle of champagne at room temperature of 25℃ is placed in a refrigerator a 3℃. After half an hour, the champagne has cooled to 15℃.

• What is the temperature of the champagne after an hour?
• How long does it take to cool it down to 4℃?

Ok, so once inside the fridge, the starting temperature of the champagne is T(0) = 25℃. Thus...

We are given that after half an hour, the temperature is 15℃. Thus...

Rearranging and using our logarithm laws to find the cooling constantk...

Thus...

So, after 1 hour, the temperature will be...

Great. Now how long till the temperature reaches 4℃?

A plot of the temperature versus time is shown below

Created with: www.desmos.com/calculator

As we can see, the champagne temperature at the time we put it in the fridge is 25℃. As time goes on, its temperature approaches a maximum of 3℃.

That will complete this tutorial. Below is a list of tutorials I've created so far on Differential Equations:

1. Introduction to Differential Equations - Part 1

2. Differential Equations: Order and Linearity

3. First-Order Differential Equations with Separable Variables - Example 1

4. Separable Differential Equations - Example 2

5. Modelling Exponential Growth of Bacteria with dy/dx = ky

6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky

7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky

8. Mixing Salt & Water with Separable Differential Equations

9. How Newton's Law of Cooling cools your Champagne

Please give me an Upvote and Resteem if you have found this tutorial helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

I would really appreciate any small donation which will help me to help more math students of the world.

Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3
Tip me at PayPal: https://paypal.me/MasterWu