Unformatted text preview: erved, and as its temperature nears room temperature, the coﬀee cools ever
more slowly. Of course, if we take an item from the refrigerator and let it sit out in the kitchen,
the object’s temperature will rise to room temperature, and since the physics behind warming and
cooling is the same, we combine both cases in the equation below.
Equation 6.6. Newton’s Law of Cooling (Warming): The temperature T of an object at
time t is given by the formula
T (t) = Ta + (T0 − Ta ) e−kt ,
where T (0) = T0 is the initial temperature of the object, Ta is the ambient temperaturea and
k > 0 is the constant of proportionality which satisﬁes the equation
(instantaneous rate of change of T (t) at time t) = k (T (t) − Ta )
a That is, the temperature of the surroundings. If we re-examine the situation in Example 6.1.2 with T0 = 160, Ta = 70, and k = 0.1, we get,
according to Equation 6.6, T (t) = 70 + (160 − 70)e−0.1t which reduces to the original formula given.
The rate constant k = 0.1 indicates the coﬀee is cooling at a rate equal to 10% of the diﬀere...
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