A container of hot water at 80 deg C cools to 79 deg C in 15 seconds when it is placed in a room that is at 20 deg C. Use Newton's law of cooling to estimate the time it will take for the container to cool from 70 deg C to 69 deg C.
To estimate the time it will take for the container to cool from 70 deg C to 69 deg C using Newton's law of cooling, we need to determine the cooling constant in the equation and apply it to the given information.
Newton's law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. The equation can be written as:
dT/dt = -k(T - Ts)
where dT/dt is the rate of change of temperature, k is the cooling constant, T is the temperature of the object, and Ts is the temperature of the surroundings.
First, we need to find the cooling constant, k. The given information states that the container of hot water cools from 80 deg C to 79 deg C in 15 seconds.
dT/dt = -k(T - Ts)
Substituting the known values:
1/15 = -k(80 - 20)
Simplifying the equation:
1/15 = -60k
Solving for k:
k = -1/900
Now that we have determined the cooling constant, we can estimate the time it will take for the container to cool from 70 deg C to 69 deg C.
dT/dt = -k(T - Ts)
Substituting the known values:
dT/dt = -(-1/900)(70 - 20)
Simplifying the equation:
dT/dt = 1/900 * 50
dT/dt = 1/18
To find the time it takes for the temperature to change by 1 degree, we can use the reciprocal of dT/dt.
dt/dT = 18
Therefore, it will take approximately 18 seconds for the container to cool from 70 deg C to 69 deg C.