Question

A glass full of hot milk is poured on the table. It begins to cool gradually. Which of the following is incorrect then?
A. The rate of cooling is constant till the milk attains the temperature of the surroundings.
B. The temperature of milk falls off exponentially with time.
C. While cooling, there is a flow of heat from milk to the surroundings as well as from surroundings to the milk but the net flow of heat is from milk to the surroundings and that is why it cools.
D. All three phenomena, conduction, convection and radiation are responsible for the loss of heat from milk to the surroundings.

Answer 1

Hint: This question can be solved by directly applying the principle of Newton's law of cooling. This can be applied to check the validity of each of the four statements.
Newton’s law of cooling states that –
The rate of heat loss of a body is directly proportional to the difference among the temperatures of the body and the surroundings.

Complete step by step answer:
Newton's law of cooling states that the rate of heat loss of a body is directly proportional to the difference among the temperatures of the body and the surroundings.
The mathematical form of the Newton’s law of cooling is –
$\dfrac{{dT}}{{dt}} = k\left( {T - {T_0}} \right)$
where $\dfrac{{dT}}{{dt}}$ is the rate of change of temperature in differential form, $T$ is the temperature of the body, ${T_0}$ is the temperature of the surroundings and k is a constant which depends on the area and nature of the body.
By applying Newton's law of cooling to the statements, we will be able to verify the correctness of the statements.
Let us consider the statements one-by-one:

Statement – A: The rate of cooling is constant till the milk attains the temperature of the surroundings.
The mathematical form of Newton’s law of cooling is,
$\dfrac{{dT}}{{dt}} = k\left( {T - {T_0}} \right)$
When the milk is split on the floor initially, the temperature difference between the milk and the surroundings is very high. Hence, the rate of heat transfer will be very high. But, as the temperature of the hot milk falls, the difference of temperature between milk and surroundings decreases, and hence, the rate of heat transfer further decreases.
Thus, we can see that the rate of heat transfer will not be a constant.
Hence, this statement is incorrect.

Statement – B: The temperature of milk falls off exponentially with time.
The mathematical form of Newton’s law of cooling is,
$\dfrac{{dT}}{{dt}} = k\left( {T - {T_0}} \right)$
By solving the above differential equation, we get –
$T = {T_i} + \left( {{T_0} - {T_i}} \right){e^{ - kt}}$
where ${T_i}$ = initial temperature.
Here, we can see the exponentially decreasing function of time. Hence, we can say that the temperature of milk decreases exponentially with time.
Hence, this statement is correct.

Statement – C: While cooling, there is a flow of heat from milk to the surroundings as well as from surroundings to the milk but the net flow of heat is from milk to the surroundings and that is why it cools.
The heat transfer is dependent on temperature.
Since, both the milk and the surroundings have a finite temperature, when they are in contact, there is some heat energy exchange among them. The only difference is the difference in the rate of heat transfer. Since the milk is at a higher temperature than the surroundings, the flow of heat is higher than that of the reverse case. Hence, the net flow is from the milk to the surroundings, resulting in cooling of the milk.
Hence, this statement is correct.

Statement – D: All three phenomena, conduction, convection and radiation are responsible for the loss of heat from milk to the surroundings.
When milk is spilt on the table, there are three modes in which heat is transferred to the surroundings as described:
i) The heat is carried away by the table through the mode of conduction.
ii) The heat is carried away by the air through convection.
iii) There is heat radiated from the hot milk, since the temperature of the milk is higher.
Hence, one can observe all the three modes of heat transfer, irrespective of their speeds.
Hence, this statement is correct.

Therefore, Information in Statement-A is incorrect, so option A is the required answer.

Note:
Newton's law of cooling implies that the heat transfer is higher if the temperature difference of the body and surroundings is higher. Hence, you can see that the rate of cooling of a hot object becomes very slow during the winter season where the temperature of the air around, is significantly lower than in summer.