Thermo Example 2                                                                                     © 2000 D.P. Maloney

 

Heat is added at a rate of 80 J/sec to a material whose initial temperature is 32^o C.  At this temperature, the material is a solid with a specific heat of 2.4 J/g C^o.  The melting point of this material is 36^oC, its latent heat of fusion is 340 J/g, and the specific heat of its liquid phase is 4.1 J/g C^o.  If there are 18 g of the material present, how long will it take to reach a final temperature of 39^o C?

 

Given:             Rate at which heat is being added: 

                        Initial temperature: 

                        Specific heat of solid phase: 

                        Melting point: 

                        Latent heat of fusion: 

                        Specific heat of liquid phase: 

                        Mass: 

                        Final temperature: 

 

Unknown:       Time to reach final temperature:  t = ? sec.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Physical Principles and/or Ideas:    Heating of a material in a single phase, change of phase, definition of specific heat, and definition of latent heat.

 

Solution:         In this problem, heat is being added to a material at a steady rate.  The material is initially in the solid state so the heat added raises the temperature of the material.  This continues until the temperature reaches 36^o C, the melting point of the material.  Once the material reaches its melting point, the heat added goes into changing the state of the material, i.e. it makes the material melt.  During a change of state, all of the heat added goes to that process and there is no change of temperature during this process.  This is shown in the physics sketch as the flat section.

                        Once all of the material has melted, the temperature can start to increase again.  Now the heat is being added to a liquid.  The specific heat of the liquid phase differs from that of the solid phase, so the straight line for this part of the process has a different slope compared to the first part.

 

                        Since the heat is being added at a constant rate in this situation, what we need to know is the total amount of heat needed.  We can think of the overall process as being composed of three parts:  heating the solid to its melting point, changing the solid to a liquid, and heating the liquid from the melting point to 39^o C.  So, we want to find the heat needed for each of these.  First, heating the solid requires:

 

                                   

 

Next melting the solid requires:

 

           

 

Finally, heating the liquid requires:

 

           

 

So the total is  which at  would take