Specific heat: The first alphabet of heat transfer

This is a four-page article on specific heat which by definition ends in one line. There is so much to know inside this one-line definition. The author has made an attempt to explain them in simple words. Everything in this note is fundamental to heat transfer.

The post covers the following: [1] Why specific heat is resistant to heat transfer? [2] Where does the heat go when a substance is heated? Who are the recipients of energy in a molecule? [3] Thermal energy [4] Kinetic, potential and internal energies of the molecule [5] Molecular motion and the modes of motion [6] Equal distribution of energy to molecules [7] Cp and CV and relation between Cp and Cv [Cp> Cv, Cp-Cv = R, Cp/Cv = Y] and their meaning

Specific heat: Resistance to heat transfer A substance cannot increase its temperature until it has met the energy demand of its molecules. In other words, the energy supply to molecules is pre-taxed at the source before the substance has internally reached thermal equilibrium. A substance can increase its temperature only when internally it is in a state of thermal equilibrium. Every substance has different types of molecules, different numbers of molecules, and different arrangements of molecules; therefore, every substance has its own demand for a certain quantity of heat [energy] to reach internal thermal equilibrium, therefore, every substance offers resistance to heat transfer until its energy demand has been met before increasing temperature. This resistance to heat transfer is the specific heat.

In one line, the specific heat is the resistance to heat transfer.

Specific heat: Definition The specific heat capacity (symbol cp) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample. This implies that it is the amount of energy that must be added, in the form of heat, to one unit of mass of the substance in order to cause an increase of one unit in temperature. The specific heat capacity often varies with temperature and is different for each state of matter. The specific heat of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat at constant pressure) than when is heated in a closed vessel that prevents expansion (specific heat at constant volume). These two values are usually denoted by Cp and Cv, their quotient y = Cp/Cv is the heat capacity ratio.

The specific heat capacity of a substance, usually denoted by c, is the heat capacity C of a sample of the substance, divided by the mass m of the sample. Heat capacity is defined as the amount of heat required to raise the temperature of a mass of a substance by 1°C. The heat capacity for one gram of substance is called its specific heat

c = C / m = 1/m x dQ/dT

where dQ} represents the amount of heat needed to uniformly raise the temperature of the sample by a small increment dt.

Like the heat capacity of an object, the specific heat of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure p applied to it. Therefore, it should be considered a function c (p,T ) of those two variables              

However, the dependency of c on starting temperature and pressure can often be ignored in practical contexts, e.g., when working in narrow ranges of those variables. In those contexts, one usually omits the qualifier (p, T)}, and approximates the specific heat by a constant c suitable for those ranges.

Before proceeding further let us see why specific heat is resistance to heat transfer.

This graph is a representation of how the average energy requirement of a molecule increases with temperature increase before it can reach thermal equilibrium.

Three questions

[1] Why specific heat is resistance to heat transfer?

[2] Where does the heat go when a substance is heated? Who are the recipients of energy in a molecule?

[3] What is temperature?

Thermal energy:

 All molecules at temperatures above absolute zero possess thermal energy; this generates the randomized kinetic energy associated with the various motions of the molecules like translational, vibrational, and rotational motions as a whole, and also the atoms within them. Molecules are both vehicles for storing and transporting energy, and the means of converting it from one form to another when the formation, breaking, or rearrangement of the chemical bonds within them is accompanied by the uptake or release of heat. Polyatomic molecules possess potential energy in the form of chemical bonds.

The thermal energy of a substance is its randomized kinetic energy

Molecular motions in a substance

The kinetic energy in a molecule

A matter is made up of particles that are constantly moving. Kinetic energy is a property of a moving object or particle and depends not only on its motion but also on its mass. The translation of motion is motion along a path from one place to another. In addition to translation, molecules composed of two or more atoms can possess other kinds of motion. Because a chemical bond acts as a kind of spring, the two atoms in H2 will have a natural vibrational frequency. In more complicated molecules, many different modes of vibration become possible, and these all contribute a vibrational term KE (vib) to the total kinetic energy. Finally, a molecule can undergo rotational motions which give rise to a third term KE (rot). Thus, the total kinetic energy of a molecule is the sum as follows

KE total = KE translation + KE vibration + KE rotation

Potential energy in a molecule

The total potential energy of the molecule is the sum of the repulsions between like charges and the attractions between electrons and nuclei

PE total = PE electron-electron + PE nucleus-nucleus + PE nucleus-electron

Internal energy of a molecule

The total energy of the molecule is its internal energy U.

U = KE total + PE total.

This concept is very important. In an isothermal system ?U = 0.

Where does the heat go when a substance is heated? Who are the recipients of energy in a molecule?

When a substance is heated the average energy is divided equally among all the independent components of motion in a system until the substance/system has reached thermal equilibrium.

What does it mean?

When a substance is heated while its potential energy is internal to its atoms -atom chemical bonds do not change [ assuming no bonds break] its kinetic energy increases. Along with the translational motion, rotational and vibrational motion set up as temperature increases with their own demand of share of energy. Every type of motion of molecule gets an equal share of it until there is thermal equilibrium in the molecule. The temperature of a substance can only increase when the substance has reached a state of internal thermal equilibrium. Without getting into complex theories [ equipartition of energy/degrees of freedom etc], the more complex the molecule in terms of its structure [number of atoms] the more is its internal energy requirement and therefore the more is its resistance to heat transfer, and to sum all, the more is its specific heat. In short, for a monoatomic molecule, the kinetic energy consists only of the translational energy of the individual atoms. Monoatomic particles do not vibrate, and their rotational energy can be neglected because the atomic moment of inertia is so small. Also, they are not electronically excited to higher energies except at very high temperatures. Therefore, practical internal energy changes in a monoatomic ideal gas may be described solely by changes in its translational kinetic energy.

Two questions

[1] Why there are two specific heats, Cp and Cv

[2] What is the relationship between Cp and Cv

Specific heat constant pressure: Cp

Cp is the amount of heat energy released or absorbed by the unit mass of the substance with the change in temperature at constant pressure. In another word, it is the heat energy transfer at a constant pressure between a system and surrounding.

Specific heat constant volume: Cv

During the small change in the temperature of a substance, Cv is the amount of heat energy released or absorbed by the unit mass of the substance with the change in temperature at constant volume. In another word, it is the heat energy transfer between a system and surrounding when there is no change in volume. 

What do they mean?

Cp: When heat is transferred at a constant temperature with volume increasing, in a typical case with gas, the gas expands. In order to create space for expanding the gas has to do mechanical work to push the surrounding. This is additional energy required by the gas compared to a situation if there is no expansion of gas and the energy gets spent only to increase the kinetic energy of gas molecules.

Cv: At constant volume, since there is no expansion, there is no work, heat just goes to increase the kinetic energy of molecules.  For gases, Cp> Cv - this is the first relationship between Cp and Cv

A typical example

It typically explains that Cp>Cv. This is a graph of ethanol Cp and Cv at gas-liquid equilibrium pressure. The green line representing Cp shows a steady increase in specific heat with an increase in temperature while the red line is almost flat. For an ideal gas Cp-Cv = R [ Universal gas constant] This is the second relationship between Cp and Cv.

What does Cp = Cv+R mean?

The specific heat constants for constant pressure and constant volume processes are related to the gas constant for a given gas.

Without getting into complexities, it simply means that the specific heat capacity of a gas at constant pressure is greater than its specific heat capacity at constant volume by R or 8.314 J/kgK.

Relationship of specific heat ratio Cp/Cv = Y [ Gamma] This is the third relationship between Cp and Cv

Explanation:

The heat capacity ratio, also known as the adiabatic index, is the ratio of the heat capacity at constant pressure (CP) to heat capacity at constant volume (CV). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas

As explained above, Cp is always greater than Cv because the amount of heat supplied at constant pressure is utilized in two ways [1] increasing the internal energy (and hence temperature since internal energy is a function of temperature), [2] for doing work. On the other hand, the amount of heat supplied at constant volume is used only for increasing the internal energy. Therefore, for increasing the temperature by unity you need more heat at constant pressure.

 The internal energy of any gas is directly proportional to the temperature of the gas. It is the number of independent ways in which a molecule of gas can move in space in three axes, x,y, and z, this is technically the degrees of freedom [DOF] of a gas. As the number of atoms in a molecule increases their freedom to move in other modes like vibrational and rotational mode other than just a linear translational motion also increases. Without making it complex, monoatomic gases have DOF=3, diatomic gases have DOF=5 and triatomic gases have DOF =6

Each degree of freedom contributes 1/2kT per atom to the internal energy.

For monatomic ideal gases with N atoms, its total internal energy U is given as U=3/2NkT. For diatomic gases, U=5/2NkT, k is Boltzmann constant

Y [Gamma] = 1 + 2/[DOF]

Higher the DOF the smaller the Cp/Cv = Gamma ratio. As Cp approaches Cv , the capacity for doing work by a gas reduces.

Cp/Cv ratio for monoatomic, Diatomic, triatomic is 1.67,1.4,1.33 respectively. A polytropic process is one where the pressure and volume of a system obey the equation, PV^n= Constant. Where P represents the pressure, V represents the volume, n represents the polytropic index. For isentropic processes, n = γ = Cp/Cv, 

What does Cp/Cv signify?

Cp/Cv is a measure of the capacity of gas to do work.

The purpose of supplying heat to a system is to get the maximum return in terms of mechanical work. If the system consists of gases with high DOF, poly-atomic molecules with higher specific heat in a comparative situation, more heat gets locked in the internal energy, [H = U + W] than what is available to do work W

Credit: Google