) From Newton’s law of cooling show that temperature of a hot liquid, when allowed to cool, falls
exponentially.
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Answers
Answer:
Greater the difference in temperature between the system and surrounding, more rapidly the heat is transferred i.e. more rapidly the body temperature of body changes. Newton’s law of cooling formula is expressed by,
T(t) = Ts + (To – Ts) e-kt
Where,
t = time,
T(t) = temperature of the given body at time t,
Ts = surrounding temperature,
To = initial temperature of the body,
k = constant.
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Answer:
In the late of
17
th century British scientist Isaac Newton studied cooling of bodies. Experiments showed that the cooling rate approximately proportional to the difference of temperatures between the heated body and the environment. This fact can be written as the differential relationship:
d
Q
d
t
=
α
A
(
T
S
−
T
)
,
where
Q
is the heat,
A
is the surface area of the body through which the heat is transferred,
T
is the temperature of the body,
T
S
is the temperature of the surrounding environment,
α
is the heat transfer coefficient depending on the geometry of the body, state of the surface, heat transfer mode, and other factors.
As
Q
=
C
T
,
where
C
is the heat capacity of the body, we can write:
d
T
d
t
=
α
A
C
(
T
S
−
T
)
=
k
(
T
S
−
T
)
.
The given differential equation has the solution in the form:
T
(
t
)
=
T
S
+
(
T
0
−
T
S
)
e
−
k
t
,
where
T
0
denotes the initial temperature of the body.
Thus, while cooling, the temperature of any body exponentially approaches the temperature of the surrounding environment. The cooling rate depends on the parameter
k
=
α
A
C
.
With increase of the parameter
k
(for example, due to increasing the surface area), the cooling occurs faster