Heat capacity: practical or logical meaning (question for student of an high school without calculus)

The thermal or heat capacity $$C$$ of a body (or more generally of any system) is defined as the ratio between the heat exchanged between the body and the environment and the resulting temperature variation:

$$C=\frac{Q_{\text{heat}}}{\Delta T}$$

Thus, the heat capacity depends neither on the substance nor on the amount of substance we are heating.

the heat capacity, structurally, is an amount of heat related to the temperature variation (the average velocity is the ratio between a displacement and a time interval)? We could also define it as the ability of a body subjected to a certain heat to increase more easily its temperature, considering that $$C$$ represents the slope of a line in a temperature-heat graph or vice versa?

When we say that the heat capacity of the water is of $$4186\, J/K$$, can we say that water holds the heat more than other liquids or solids or that it releases it more easily?

Possible connection to the question topic: Why and who has established that $1\, cal \equiv4.186\, J$?

• $C$ in your equation is the heat capacity not the specific heat capacity. It is the heat capacity of an object having a mass m and a specific heat of $c$. Jan 19, 2021 at 22:16
• @BobD Yes $C$ it is the heat capacity not the specific heat capacity :-). Generally for the specific heat capacity I use $c_s$. Jan 19, 2021 at 22:20
• But your post says "the heat capacity depends neither on the substance nor on the amount of substance we are heating". In fact it does. It's the product of the specific heat for the material and its mass. Jan 19, 2021 at 22:22
• @BobD In my textbook it says: We can therefore define a new quantity, the specific heat $c_s$, which depends only on the type of substance and not on the quantity. So what are the correct definitions you need to give to heat capacity and specific heat? Could you please add an answer with the reasons for it? Thank you, very much much. Jan 19, 2021 at 22:29

In thermodynamics, heat capacity is not defined in terms of heat Q at all. The specific heat capacities at constant volume and at constant pressure are defined precisely in terms of the partial derivatives of specific internal energy and specific enthalpy with respect to temperature. These are certainly physical properties of the substance involved:

$$C_V=\left(\frac{\partial U}{\partial T}\right)_V$$ $$C_P=\left(\frac{\partial H}{\partial T}\right)_P$$

• Unfortunately, I cannot use derivatives as my 17 year old students do not even know them. I already have a huge struggle to make them understand very low level exercises. Thank you veryyyyyyyyy much. Jan 20, 2021 at 23:14
• @Sebastiano I believe what you should take from this answer is that heat flow is not the only way to change temperature. Doing work on an object (for example, hammering a nail into a board) will change the temperature, and the specific heat will affect the work/temperature change ratio. Jan 20, 2021 at 23:42

When we say that the heat capacity of the water is of $$4186\text{ }\mathrm{J/K}$$, [...]

But we don't say that. The value of $$4186$$ is the specific heat capacity $$c_p$$: $$c_p=4186\text{ }\mathrm{Jkg^{-1}K^{-1}}$$ $$c_p$$ in $$\text{Joule}$$ per $$\text{kg}$$ and per $$\text{K}$$.

[...] can we say that water holds the heat more than other liquids

Compared to other substances (or mixtures of substances) with a lower specific heat capacity, yes.

On request of the OP:

Specific heat capacity is the amount of heat energy needed to heat $$1\text{ }\mathrm{kg}$$ of the material by one $$1\text{ }\mathrm{K}$$:

$$c_p(T)=\frac{1}{m}\Big(\frac{\text{d}Q}{\text{d}T}\Big)_T$$

If $$c_p$$ is constant over an interval $$\Delta T$$, then we can write:

$$c_p=\frac{\Delta Q}{m\Delta T}$$

For a uniform object made of a material of specific heat capacity $$c_p$$ and of mass $$m$$ its heat capacity is:

$$C=mc_p$$

• Thank you very much for your fast answer: generally for the specific heat capacity I use $c_s$. So how should the two definitions of heat capacity and specific heat be given appropriately? Obviously if I eat a piece of cake that is still moist I will burn faster than eating a piece of cake that contains less water. Jan 19, 2021 at 22:24
• Obviously if I eat a piece of cake that is still moist I will burn faster than eating a piece of cake that contains less water. Here you are referring to the calorific value of a substance: the amount of heat freed when a substance is digested or burned. It is independent of the $c_p$. The calorific value of pure water is zero.
– Gert
Jan 19, 2021 at 22:28
• So I am asking you please if you could edit your answer which I appreciated and clarify this concept for me of the the calorific value $Q$? As I wrote to @BobD, so what are the correct definitions you need to give to heat capacity and specific heat? Could you please add an answer with the reasons for it? Jan 19, 2021 at 22:31
– Gert
Jan 19, 2021 at 22:41
– Gert
Jan 19, 2021 at 22:57

Thus, the heat capacity depends neither on the substance nor on the amount of substance we are heating.

That's not true. The heat capacity depends on both the substance and the mass of the substance. The specific heat capacity is independent of the mass, but not independent of the substance. It can also depend on the phase of the substance (ice is different from liquid water and steam). And it can depend on temperature.

As far as the relationship between heat capacity and specific heat capacity at a certain temperature (in your notation): $$C = mc_s.$$

Finally, objects do not contain or hold heat. They have internal energy. That internal energy is related to the temperature and the phase. We change the internal energy by means of work and/or heat. Heat is energy transferred due to differences in temperature. It is possible for the temperature of an object to change without any heat flow.

Substances with a large specific heat capacity will require more energy input to warm, and will lose more energy in cooling than something with a small specific heat. That's why we will use water to cool a hot piece of metal.

• Thank you to you and all user for to have had an answer. Thank you very much...again. Jan 21, 2021 at 12:42