The Heat Capacity

Have you ever wondered what really happens when you heat an object? The heat capacity is the key to understanding this fundamental phenomenon that connects energy, temperature, and the physical states of matter. This fascinating concept not only explains why water takes longer to heat up than metal but is also crucial in areas such as thermodynamics, engineering, and material sciences.

Learning Objectives

1. Understand the concept of heat capacity as a measure of the amount of energy required to change the temperature of a system.
2. Apply the definitions of specific and molar heat capacity in practical contexts.

TABLE OF CONTENTS:
The Problem of Heat Capacity
Types of Heat Capacities
Exercises

The Problem of Heat Capacity

Discussing heat capacity places us in a peculiar situation. It is not possible to “store heat” in a body as if we were storing water in a bucket with a defined capacity, for example, 5 liters. This is not the meaning behind the words heat or heat capacity. In fact, this misunderstanding comes from a historical inheritance in physics, as in ancient times, heat was thought to be a kind of substance, an idea that we now know is incorrect. Today, we understand that heat is the energy in transit that causes temperature variations, analogous to how work is the energy used to change the state of a system.

Since its creation, the term has had enough time to settle in the common usage of physics. Therefore, even though terms like “capacitance” might seem more appropriate, replacing it now would only mean renaming millions of books without providing any significant benefit.

Nevertheless, and despite everything, the concept of heat and capacity is not particularly complicated. To understand the idea it conveys, it is enough to ask the following question:

What relationship exists between heat and the temperature change \Delta T of an object?

The answer to this question comes in the form of the differential relationship dQ = CdT, where C is the heat capacity.

C:= \dfrac{dQ}{dT}.

Here, as long as we remember that it tells us how much heat is needed to raise the temperature of a given object (it has nothing to do with an object’s capacity to heat other things), we will not risk making a mistake. As can be inferred from the definition of heat capacity, it has units of [J/K].

Types of Heat Capacities

Specific Heat Capacities

When we talk about heat capacity,

there are two different presentations: the usual one, which we have already introduced, and the specific one. The difference between them is that specific heat capacity refers to heat capacity per unit mass, defined as:

c := \dfrac{C}{m}

Where m represents the mass of the body.

If we multiply the specific heat capacity by the molar mass, we obtain the specific molar heat capacity, defined as:

c_{mol} := c \cdot m_{mol}

It is important to note that lowercase letters are always used for specific magnitudes.

Example

The specific heat capacity

of water at room temperature (26^\circ C) is:

c = 4.181 \cdot 10^3 \left[\dfrac{J}{kg \cdot K}\right]

Calculate:

Heat Capacity at Constant Pressure and Volume

When we think of gases, we encounter an additional complication. In this case, we will be trying to find out how much heat we need to apply to a system to raise its temperature by 1[K]. But this can be done in two different ways:

In both cases, we are applying constraints to the system. Under these circumstances, we need to modify our definition of heat capacity according to each case. Thus, we have heat capacities at constant volume and pressure: C_V and C_P, respectively. These magnitudes are expressed in terms of partial derivatives as follows:

C_V = \left(\dfrac{\partial Q}{\partial T}\right)_V

C_P = \left(\dfrac{\partial Q}{\partial T}\right)_P

Example

The heat capacity of helium measured at constant volume is 3.12\left[\dfrac{kJ}{kg \cdot K}\right], and when measured at constant pressure, it is 5.19\left[\dfrac{kJ}{kg \cdot K}\right]. Calculate the molar heat capacities at constant pressure and volume.

Exercises:

1. The world’s oceans contain approximately 10^{21}[kg] of water. Calculate the heat capacity of the world’s oceans. [SOLUTION]
2. The world’s energy consumption is around 13[TW] (and rising) (1TW=10^{12}[W]). Burning a ton of crude oil (approximately 7 barrels) produces around 42[GJ] (1[GJ]=10^9[J]). If the world’s energy consumption relies solely on oil, how many barrels must be burned per second? [SOLUTION]
3. The molar heat capacity of gold is 25.4\left[\dfrac{J}{mol \cdot K}\right]. Its density is 19.3\cdot 10^3 \left[\frac{kg}{m^3}\right]. Calculate the specific heat capacity of gold and its corresponding value per unit volume. [SOLUTION]
4. Two bodies with heat capacities C_1 and C_2 (assumed temperature-independent) and initial temperatures T_1 and T_2, respectively, are put in contact. Show that the final temperature of the body T_f is given by: T_f = \dfrac{C_1 T_1 + C_2 T_2}{C_1 + C_2}

   And if C_1 is much greater than C_2, then we have

   T_f \approx T_1 + \dfrac{C_2}{C_1}(T_2 - T_1)

   [SOLUTION]