How Long Until Coffee Cools to 70° Using Newton's Law?

In summary, the temperature of a cup of coffee, initially heated to 100°, drops to 92 degrees after one minute and to 95° after another minute when left in a room with a constant temperature of 68°. It is impossible to determine how much time must pass before the temperature of the coffee drops to 70° without knowing the initial temperature of the coffee and the amount of latent heat it possesses.
  • #1
baileyyc
4
0
a cup of coffee is heated to 100° then left in a room with a constant temperature of 68° after a minute the temperature of the coffee has dropped to 95° after another minute to 92 degrees. how much time must pass before the temperature of the coffee has dropped to 70°?
 
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  • #2
Welcome to PF!

Hi baileyyc! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
  • #3
this is how I tried to solve the problem:

y'=k(y-68)
dy/dt=k(y-68)
∫(1/y-68)dy=∫kdt
e^ln|y-68|=e^kt+c1
y-68=e^kt+c1
y=68+ce^kt
100=68+ce^k(0)
100=68+c
c=32

when t=1, y=95 when t=2, y=92
95=68+32e^k(1) 92=68+32e^k(2)
27=32e^1k 24=32e^2k
ln(27/32)=k ln(24/32)=2k
k ~ -.1699 k ~ -.1438

plugging k into the equation plugging k into the equation
y=68+32e^-.1699t y=68+32e^-.1438t

when y=70 when y=70
70=68+32e^-.1699t 70=68+32e^-.1438t
2=32e^-.1699t 2=32e^-.1438t
ln(2/32)=-.1699t ln(2/32)=-.1438t
t~16.32 minutes t~19.28 minutes

I don't know which one is the answer.. can you help me?
 
  • #4
baileyyc,

Are you sure the problem is stated correctly ? The reason I ask is because with those times and temperatures the thermal time constants differ by 18% and that is why you are getting two different answers.

Thanks
Matt
 
  • #5
yes I am typing it correctly
 
  • #6
baileyyc said:
I don't know which one is the answer.. can you help me?

Hi baileyyc! :smile:

I think this must be a trick question :frown:

if the coffee is at 100º, then it must be boiling, so it'll have some latent heat, but you don't know how much.

So I'd ignore the first minute, and calculate on the basis of the second minute only. :wink:
 

Related to How Long Until Coffee Cools to 70° Using Newton's Law?

1. How does Newton's Law of Cooling work?

Newton's Law of Cooling is a mathematical equation that describes the rate at which an object cools down to the temperature of its surrounding environment. It states that the rate of change of an object's temperature is proportional to the difference between its temperature and the temperature of its surroundings.

2. What are the main assumptions of Newton's Law of Cooling?

The main assumptions of Newton's Law of Cooling are that the object is in a closed system with constant environmental temperature, and that the object's temperature is only affected by heat transfer through convection or radiation.

3. Can Newton's Law of Cooling be used for all types of cooling processes?

No, Newton's Law of Cooling is only applicable to cooling processes where the temperature difference between the object and its surroundings is relatively small. It also assumes that the object has a uniform temperature throughout and that the surrounding environment is well-mixed.

4. How can Newton's Law of Cooling be applied in real-life situations?

Newton's Law of Cooling can be used in various practical applications, such as predicting the cooling rate of hot beverages, determining the cooling time of heated food, and monitoring the temperature changes in industrial processes.

5. Is Newton's Law of Cooling accurate in all situations?

While Newton's Law of Cooling provides a good approximation for many cooling processes, it may not be accurate in situations where there are significant temperature differences or when the object's temperature changes rapidly. Other factors, such as humidity and air flow, can also affect the accuracy of this law.

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