Problem 5.

An ice cube having a mass of 50 grams and an initial temperature of -10 degrees Celsius is placed in 400 grams of 40 degrees Celsius water. What is the final temperature of the mixture if the effects of the container can be neglected?

Solution:

In this problem we need to use the energy conservation law. Namely, the energy transferred from the ice cube is equal to the energy transferred to the water.

Initially we have two systems: (1) ice cube and (2) water. The systems have different initial temperatures. In the final state the temperatures of the systems are the same – thermal equilibrium.

To find the final temperature we need to write the energy conservation law: the energy transferred from the system (1) [to the system (2)] is equal to the energy transferred to the system (2) [from the system (1)].

We introduce the final temperature of the systems: . We assume that the final temperature is greater than 0. It means that in the final state the system (1) (ice) becomes water. This is our assumption – if after the calculations we obtain that the temperature is less than 0, then we need to repeat the calculations with an assumption that the temperature is less than 0 or equals to 0.

Now we need to write the energy conservation law.

(I) energy (heat) transferred from system (1) has three contributions:

Process (1): the heat required to change the temperature of 50 g = 0.05 kg ice is determined by the specific heat of the ice:

The specific heat of ice is 2060 J/kg C, the initial temperature is (-10) and the final temperature is 0. Then the change of the temperature is 10:

Process (2): the heat required to melt the ice is determined by the specific latent heat of fusion:

The specific latent heat of fusion is 334000 J/kg. Then

Process (3): The ice now becomes the water and we need to increase the temperature of the water from 0 to the final temperature . The heat required to increase the temperature of the water is determined by the specific heat of the water (the mass of the water is equal to the mass of the ice):

The specific heat of the water is 4186 J/kg C. Then

Then the total heat transferred from the ice is

(II) energy (heat) transferred to the system (2) has only one contribution: We just need to decrease the temperature of the water from the initial temperature 40 degrees to the final temperature . The mass of the water is .

The heat required to decrease the temperature of the water is determined by the specific heat of the water:

The specific heat of the water is 4186 J/kg C, the change of the temperature of the water is . Then

Then the energy conservation takes the form:

From this equation we can find the final temperature of the system: