Coffee Cooling
An example of first order ordinary differential equations
Does white coffee cool faster if one adds the milk immediately or if one delays adding the milk
until later? Let
Tc  the initial temperature of the coffee (no milk)
Tm  the temperature of the milk when added
TR  the ambient (room) temperature
T  t   the temperature of the coffee (with milk after it has been added) at time t
h = the time to halve the temperature difference between coffee and air (the half life)
k = the cooling coefficient   ln 2   h
t1  the time at which the milk is added [assuming instant complete mixing]
Vc  the volume of the coffee
Vm  the volume of the milk
Newton’s law of cooling is
dT
 k T  TR 
dt
the solution of which, (before the milk is added), is
T  t   TR  Tc  TR  ekt  TR  Tc  TR  2t / h
If the milk is added immediately (at t = 0), then
 V T  VmTm

T  t   TR   c c
 TR  ekt
 Vc  Vm

If the milk is added at some subsequent time t  t1 , then
T  t1   TR  TC  TR  ekt1
T t

1


VcT  t1   VmTm
Vc  Vm
After the milk is added

 t  t1  ,

T  t   TR  T  t1   TR e

k t t1 

 V T  T  T  ekt1  V T

c
R
c
R
m m
 k t t1 

 T  t   TR 
 TR  e


Vc  Vm


 TR 
Vc Tc  TR  ekt
 V T  VmTm
 k t t1 
  c R
 TR  e
Vc  Vm
 Vc  Vm

Cooling Coffee
Page 2 of 2
Define the difference between these two temperatures at time t  t1 to be
T  (temp. if milk added later) – (temp. if milk added immediately), that is
 VcTR  VmTm  ekt
 VcTR  VmTm  k t t1 
 k t t1  
T 
 TR  ekt  e
  
e
Vc  Vm


 Vc  Vm 

 VcTR  Vm TR  Tm  TR   ekt
Vc  Vm
 VcTR  Vm TR  Tm  TR   k t t1 
 k t t1  
 TR  ekt  e
  
e
Vc  Vm




  Vc  Vm  TR  Vm Tm  TR   ekt
 Vc  Vm  TR  Vm Tm  TR   k t t1 
 k t t1  
 TR  ekt  e
  
e
Vc  Vm
Vc  Vm




V T  T 
V T  T  k t t1 
 k t t1 
k t t1 
  TR ekt  m m R ekt  TR ekt  TR e
 TR e
 m m R e
Vc  Vm
Vc  Vm


T  
Vm TR  Tm   k t t1 
 ekt 
e
Vc  Vm 

If the milk is kept at room temperature, Tm  TR  , then this simplifies to T  0 .
Therefore the time t1 at which one adds milk at room temperature to cooling coffee has no effect
at all on the subsequent temperature of the milk/coffee mixture! The temperature is

TR  Tc  TR  e kt

T t   
 VcTc  VmTR

 TR  e kt
 TR  
 Vc  Vm


 t  t1 
 t  t1 
This independence disappears if the milk is kept at a temperature other than room temperature.
If the milk is colder than room temperature, then delaying its addition cools the coffee down
faster. If the milk is warmer than room temperature, then immediate addition maximizes the
cooling rate.
An Excel file illustrates this situation.
This work is inspired by the article “Keeping the Coffee Warm, The Mathematics of Cooling”,
by Stella Dudzic, Mathematics in Schools, vol. 42(3), pp. 9-10, 2013 May.
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