Page 95 - Mathematics Class - XII

PROJECT 1

AIM
Formation of differential equation to explain the process of cooling of boiled water to a given room temperature.

MATERIAL REQUIRED

Y Gas burner Y Water Y Beaker Y Thermometer watch

PRE-REQUISITE KNOWLEDGE

1. Concept of differentiation and integration. 2. Knowledge about measuring time and temperature.

PROCEDURE AND OBSERVATION
1. Take 1 litre of water in a beaker and boil it on burner.
2. Take the temperature of boiling water as 100°C.
3. Note the temperature of room as 25°C.
4. Now, keep the boiled water in breaker for cooling.
5. Now, prepare a chart for temperature of water by taking a definite time intervals as 1 hour.

Time (t) at an Temperature of Room temperature Difference in
interval of 1 hour water (T) in °C (S) in °C temperature (T – S)°C

Here, T denotes the temperature of boiled water at time t, S denotes the room temperature remains constant
through the experiment.
Now, according to Newton’s law of cooling that the temperature of hot body changes at a rate which is proportional
to the difference of the temperature of the hot body and surrounding temperature.

According to law, dT ( TS) or dT KT S ( )
dt dt

Here, K is the constant of proportionality and minus sign shows decrease in temperature.

Now, dT KT S ( ) or dT Kdt

dt TS
Now, taking integration of both sides of above equation.

Now, dT Kdt or log |T – S| = –Kt + C .............. (1)

TS
Equation (1) is the required differential equation.

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