Why is the slope steeper in the given diagram?

[NTesla]

Homework Statement

I'm reading the book: Sears and Zemansky. In the topic: Calorimetry and Phase change, I came across a line(in reference to the given diagram, pasted below) stating: If the rate of heat input is constant, the line for the solid phase (ice) has a steeper slope than does the line for the liquid phase (water). I've tried to find the answer on the internet, but couldn't find it. Kindly help.

Relevant Equations

Maybe this question doesn't involve any equations, as far as i understand, or maybe don't understand.

I've tried to figure out the why of the question. But all i can think of is, somehow, the specific heat capacity of ice is more than the specific heat capacity of liquid water, but how and why is it so, that i'm not sure of.

 

[kuruman]

Hint: You can stick your finger in a glass filled with liquid water but not in a glass filled with solid ice. Why not?

 

[nasu]

But all i can think of is, somehow, the specific heat capacity of ice is more than the specific heat capacity of liquid water,

Why would you think this? It is easy to check if it's true or not. Have you?

 

[Lnewqban]

I've tried to figure out the why of the question. But all i can think of is, somehow, the specific heat capacity of ice is more than the specific heat capacity of liquid water, but how and why is it so, that i'm not sure of.

The actual values are opposite to your statement.
The specific heat capacity of ice is about half than the specific heat capacity of liquid water.

Copied from:
https://en.wikipedia.org/wiki/Specific_heat_capacity#Definition

"In thermodynamics, the specific heat capacity (symbol c) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature."

"For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.

Specific heat capacity often varies with temperature and is different for each state of matter. Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C; but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1."

If the rate at which the substance absorbs heat is constant, the phase having a lower value of specific heat capacity (less heat needed "to cause an increase of one unit in temperature") should show a steeper slope in a temperature-versus-time diagram.

In practical terms, it is quicker to provide certain ΔT to a mass of ice than similar ΔT to the same mass of water, having the same source of heat for both.

Therefore, the text in the book is correct: "the line for the solid phase (ice) has a steeper slope than does the line for the liquid phase (water)."

I believe that the line that represents heating of the gaseous phase in the posted diagram should be steeper than the line representing heating water, as the specific heat of steam oscillates between 1.5 and 2.3.

Please, see:
https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html

 

[NTesla]

The actual values are opposite to your statement.
The specific heat capacity of ice is about half than the specific heat capacity of liquid water.

Copied from:
https://en.wikipedia.org/wiki/Specific_heat_capacity#Definition

"In thermodynamics, the specific heat capacity (symbol c) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature."

"For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.

Specific heat capacity often varies with temperature and is different for each state of matter. Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C; but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1."

If the rate at which the substance absorbs heat is constant, the phase having a lower value of specific heat capacity (less heat needed "to cause an increase of one unit in temperature") should show a steeper slope in a temperature-versus-time diagram.

In practical terms, it is quicker to provide certain ΔT to a mass of ice than similar ΔT to the same mass of water, having the same source of heat for both.

Therefore, the text in the book is correct: "the line for the solid phase (ice) has a steeper slope than does the line for the liquid phase (water)."

I believe that the line that represents heating of the gaseous phase in the posted diagram should be steeper than the line representing heating water, as the specific heat of steam oscillates between 1.5 and 2.3.

Please, see:
https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html

Yes, you are right. I did mean to write that I thought that the specific heat capacity of water is more than that of ice. But by mistake, ended up writing the opposite.

But, now I'm wondering why is it so. Why is the specific heat capacity of ice lower than that of water. It could possibly be because of the kind of bonds formed in ice. Can someone elaborate on why is it so, or maybe let me know link to any webpage or any resource for further reading on this topic.

 

[Lnewqban]

But, now I'm wondering why is it so. Why is the specific heat capacity of ice lower than that of water. It could possibly be because of the kind of bonds formed in ice. Can someone elaborate on why is it so, or maybe let me know link to any webpage or any resource for further reading on this topic.

I don't know the reasons behind the different values of specific heat capacity for the different phases of the same substance, but I have found the following in the same linked Wikipedia article:

"The temperature of a sample of a substance reflects the average kinetic energy of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the equipartition theorem."

"An isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy. Thus, heat capacity per mole is the same for all monatomic gases (such as the noble gases)."

"A polyatomic gas molecule (consisting of two or more atoms bound together) can store heat energy in kinetic energy, but also in rotation of the molecule and vibration of the atoms relative to each other (including internal potential energy).

These extra degrees of freedom or "modes" contribute to the specific heat capacity of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat energy is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number degrees of freedom of the molecules."

"For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at standard ambient temperature, quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3R per mole of atoms in the solid, although in molecular solids, heat capacities calculated per mole of molecules in molecular solids may be more than 3R."

"Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example, the heat capacity of liquid water is twice that of ice at near the same temperature."

 

[kuruman]

"Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example, the heat capacity of liquid water is twice that of ice at near the same temperature."

Which brings us back to my hint in post #2. When one adds energy in the form of heat where does it go when you have ice as opposed to water?

 

[Lnewqban]

https://courses.lumenlearning.com/suny-physics/chapter/14-3-phase-change-and-latent-heat/

 

[haruspex]

Which brings us back to my hint in post #2. When one adds energy in the form of heat where does it go when you have ice as opposed to water?

Isn't the question where is the heat not going in the ice?
The answer is outlined in the Wikipedia article @Lnewqban quotes, https://en.wikipedia.org/wiki/Specific_heat_capacity#Derivations_of_heat_capacity, under "Solid phase".