Internal energy, first law of thermodynamics, heat capacities and enthalpy

Internal Energy of a System

Every substance is associated with a definite amount of energy which depends upon its chemical nature as well as upon its temperature, pressure and volume. This energy includes not only the translational kinetic energy of the substance molecules but also other molecular energies such as rotational and vibrational energies. The total of all possible kinds of energy of a system is called its internal energy. The kinetic and potential energy of nuclei and electrons within the individual molecules also contribute to the total energy. The internal energy of a system is a state function and is independent of the path by which it is obtained.

It is not possible to calculate the absolute value of internal energy of asystem. It is the change in internal energy accompanying chemical or physical process generally concerned. Internal energy of a substance in a system is a definite quantity and it is a function only of the state.

Suppose, a system is subjected to change of state from A to B. Let E_A and E_B be internal energies of the system in state A and B respectively. Then the change in internal energy of the system is given by \(\Delta\) E.

\(\Delta\)E = E_B-E_A

Since, E_A and E_B are definite quantity, the change in internal energy is also a definite quantity, irrespective of the path or the manner in which the change is brought about. If E_B > E_A then, \(\Delta\) E is +ve, and if E_A > E_B, then \(\Delta\) is –ve.

A system may transfer energy to or from the surrounding as work or heat or both.

First Law of Thermodynamics

First law of thermodynamics is nothing but the law of conservation of energy. According to which, energy can neither be created nor be destroyed, although ot can be transformed from one form to another. Thus, whenever energy in one form disappears, an equal amount of energy in some other form must appear. It is impossible to construct a perpetual motion machine, i.e., a ,machine which cannot do work without expenditure of energy.

The equivalence of heat and mechanical work, established by Joule experimentally , follows a consequence of the first law. Suppose there is no such equivalence . Then it may be possible at first to convert certain amount of heat, say ‘x’ calories, into a certain amount of mechanical wprk and then, in the reverse process to transform the same amount of mechanical work into heat, producing ‘y’ cals of heat, where y> x. Then, the original state will have been restrored but heat energy, equivalent to ‘y’ cals, will have been created. By repeating the above cycle, energy can be continuously created and in this way a perpetual motion machine can be constructed. Hence, there must be exact equivalence of heat and mechanical work.

The law of conservation is now modified because energy can be created by destruction of mass, according to Einstein’s equation :

E = mc²

where E is energy produced, ‘m’ is mass and ‘c’ is velocity of light. Thus , the first law may now be stated as : ‘ the total mass and energy of an isolated system remains constant.’

Mathematical Statement of First Law of Thermodynamics

Let E_A be the energy of a system in its state A and E_B be the energy in its state B. Suppose, when the system undergoes a change from state A to state B absorbs heat ‘q’ from the surroundings and also performs some work, mechanical or electrical, equal to ‘W’. The absorption of heat by a system tends to raise the energy of the system. The performance of work by the system, on the other hand, tends to lower the energy of the system because performance of work requires expenditure of energy. Hence, the change of internal energy, \(\Delta\) E, accompanying the above process will be given by

\(\Delta\)E = E_B-E_A = q-W

\(\Delta\)E = q-W …………………………… (1)

The equation is a mathematical statement of the first law of thermodyanamics. In all energy transformations, q is given +ve sign when heat is absorbed and –ve sign when heat is evolved by a system; ‘W’ is given a positive sign when work is done by a system and a negative sign when wprk is done on a system.

For an infinitesimally small change, the equation (1) may be expressed as :

dE = dq-dW

where dE is the small increase in energy and, ‘dq’ and ‘dW’ represent small quantities of heat absorbed and external work done by the system respectively.

Enthalpy or Heat Content

When a process is carried out at constant volume, the heat content of the system is same as internal energy (E), as no PV- work is done. But when a process is carried out at constant pressure, the system also expands energy in doing PV- work. Therefore, the total heat content of system at constant pressure is equivalent to the internal energy (E) plus the PV-energy. This is called enthalpy ( Greek : en = in, thalpos = heat) of the system and is represented by symbol ‘H’. Thus, enthalpy is defined by equation

H = E + PV

where P and V are pressure and volume of the system respectively.

Since both P and V are function of state as E, H must also be a function of state. Thus, if H₁ and H₂ are the enthalpies of system in the initial and final states, then we may write

H₁ = E₁ + P₁V₁ …………………………………………..(2)

and, H₂ = E₂ + P₂V₂ …………………………………………………….. (3)

Hence,

H₂-H₁ = \(\Delta\) H = (E₂-E₁) + (P₂V₂-P₁V₁)

or, \(\Delta\)H = \(\Delta\)E + (P₂V₂-P₁V₁) …………………………………..(4)

Equation (4) is the most general definition of \(\Delta\)H. When the pressure remains constant throughout the process, then

\(\Delta\)H = \(\Delta\)E + P(V₂-V₁)

or, \(\Delta\)H = \(\Delta\)E + P\(\Delta\)V ……………………………… (5)

Thus, the change in enthalpy at constant pressure is equal to the increase in internal energy plus any pressure-volume work done. Hence, at constant pressure \(\Delta\)H represents the heat absorbed by a system in going from an initial state to a final state provided the only work done is PV- work. When, initial pressure and final pressure are not the same, \(\Delta\)H is calculated not by equation (5) but by equation (4).

According to the first law of thermodynamics

\(\Delta\)E = q-W

Since at constant pressure

W = P.\(\Delta\)V = q_p ……………………………. (6)

From equations (6) and (5), we have

\(\Delta\)H = q_p

Hence, increase in enthalpy equals the heat absorbed at constant pressure when no work other than P.\(\Delta\)V work is done.

The change in enthalpy, \(\Delta\)H, is positive if H₂ > H₁ and the process or reaction will be endothermic whereas \(\Delta\)H is negative if H₁ > H₂ and the reaction will be exothermic.

Heat Capacities

The heat capacity, C , may be defined as the amount of heat required to raise the temperature of a system through 1⁰C. Thus, if ‘dq’ is the heat absorbed by the system and the resulting increase of temperature is ‘dT’, then

C = dq/dT ………………………………….. (1)

Heat capacities may be measured either at constant volume or at constant pressure. These are designated as C_V and C_P respectively.

We know that,

dq = dE+ PdV …………………………………..(2)

Therefore

C = dE + P.dV/dT ………………………….. (3)

When the volume is held constant, dV = 0 and, hence

C_v = (dE/dT)_v …………………………………(4)

The subscript ‘v’ refers to constant volume.

At constant pressure, equation (3) can be written as

C_p = (dE/dT)_p + P ( dV/dT)_p ………………………. (5)

We know that

H = E + PV

Differentiating above with respect to ‘T’ at constant ‘P’, we get

(dH/dT)_p = (dE\dT)_p + P(dV/dT)_p .........................(6)

From equations (5) and (6), we have

C_p = (dH/dT)_P

Thus, the heat capacity of the system at constant volume is defined as the increase in internal energy of the system per unit degree rise of temperature and that at constant pressure is defined as the increase in heat content ( enthalpy ) of the system per unit degree rise of temperature.

For one mole of a gas, heat capacities are known as molar heat capacity.

Reference :

Glasstone, Samuel. Textbook of Physical Chemistry. New Delhi: Macmillan India Ltd, 1996.

Kapoor, L.,. Textbook of Physical Chemistry, Vol I and Vol II. Macmillan India Ltd, 1992.

Puri, B.R., Sharma, L.R., and Pathania M.S.,. Principles of Physical Chemistry. Jalandhar: Vishal Publishling Co., , 2008.