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** Introduction**:

Have you ever thought what happens to wood as it burns? It appears as if the wood might disappear into thin air. As burning wood appears to make energy and destroy the wood, neither is created or destroyed. Instead, energy and matter are changing from one form to the other. Wood includes what we termed as chemical potential energy, which is energy stored in the bonds which hold the chemicals altogether. This stored energy is discharged in the form of light and heat and whenever the wood is burned.

Wood as well includes matter, which is anything that consists of mass and takes up space (that is, volume). The matter in the wood is transformed to different matter, comprising ash and soot, as it burns. The net amount of energy and matter in the wood before burning is equivalent to the energy and matter of the ash, heat, soot and light after burning. In another words, energy and matter are conserved both throughout and after the wood is burned.

This phenomenon of conservation is illustrated by what we state as the first law of thermodynamics, at times termed to as the law of energy conservation.

The empirical conclusion that the heat and internal energy belongs to the general class of energies, assist in extending the law of conservation of mechanical energy that defines that potential and kinetic energies are completely inter-convertible. A thermodynamic system might possess any other forms of energy like electrical energy, surface energy and magnetic energy and so on. Therefore one might arrive at an extended hypothesize that all the forms are energies are inter-convertible. This comprises the basis of the First Law of Thermodynamics that might be represented as follows:

Define: The Energy can neither be created nor destroyed, if it disappears in one form it should re-appear at the similar time in other forms.

** Mathematical representation of First law of Thermodynamics**:

The First Law of Thermodynamics defines that energy can be transformed from one form to the other by the interaction of heat, work and internal energy; however it can't be created nor destroyed, under any situations. Mathematically, this is symbolized as:

ΔU = q + w

Here,

ΔU is the net change in the internal energy of a system

q is the heat exchanged between a system and its surroundings

w is the work done by or on the system

** Internal energy**:

The first law of thermodynamics is the conservation-of-energy principle defined for a system where heat and work are the processes of transferring energy for a system in the thermal equilibrium. 'Q' symbolizes the total heat transfer - it is the sum of the entire heat transfers into and out of the system. 'Q' is positive for total heat transfer into the system. 'W' is the net work done on and by the system. 'W' is positive whenever more work is done via the system than on it. The change in the internal energy of the system, ΔU, is associated to heat and work via the first law of thermodynamics,

ΔU = Q - W

Here, ΔU is the change in internal energy 'U' of the system, 'Q' is the total heat transferred to the system and 'W' is the total work done by the system. We utilize the given sign conventions: if 'Q' is positive, then there is a total heat transfer to the system; if 'W' is positive, then there is total work done via the system. Therefore, positive 'Q' adds energy to the system and positive 'W' takes energy from the system. Therefore ΔU = Q - W. It will be as well noted that if more heat transfer to the system takes place than work done, the difference is stored as the internal energy. The heat engines are a good illustration of this - heat transfer into them occurs in such a way that they can do work.

** Mathematical forms of First law of Thermodynamics**:

The first law of thermodynamics is just a statement of the principle of conservation of energy. This law was first introduced by Julius Robert Mayer in the year 1842 and this great concept was first described by Helmholtz in the year 1847. This law defines that:

The energy of an isolated system remains constant; however it might be changed from one form to the other. Or Energy can neither be created nor destroyed however can be transformed from one form to the other form.

Therefore, the heat supplied to a system is never lost however is partly transformed into internal energy and partially in doing work via the system.

In other words,

Heat supplied = Work done by the system + Increase in the internal energy

Increase in internal energy = Heat supplied - Work done by the system

This statement can be mathematically symbolized as:

dE = q - w

Here, dE is the increase in internal energy in the system, 'q' is the heat supplied and 'w' is the work done by system.

Description: Consider a system symbolized by a state 'A' in the figure. Assume that the conditions are now modified so that the system moves to 'B' by the path-I and then brought back to the state 'A by a different path II. As an effect of the first law of thermodynamics the total or net energy change at 'A' is nil.

Fig: Internal energy as a function of state

** Internal Energy as a function of State**:

If it is supposed that the energy comprised in path I is more than in the returning path II, then some amount of energy would have increased in the system on its own accord. This is against the first law of thermodynamics and therefore it should be concluded that the total energy change of a system will based on the initial and final states however not on the path followed. Let 'E_{A}' symbolizes the energy in the state 'A' and 'EB' in the state 'B', the increase in energy on passing from 'A' to 'B' might be represented by:

dE = ΔE = E_{B} - E_{A}

which is independent of the path taken. The quantity is termed as the internal or intrinsic energy. Whenever a system changes from one state to the other it might gain or lose energy as heat and work. Assume that, the heat absorbed via a system is 'Q'. If in a change from 'A' to 'B' the energy constant of the system is raised by 'AE', the work done being 'W', then according to first law:

ΔE = Q - W

The equation above is a form of first law of thermodynamics. Therefore the difference between the heat absorbed and total work done by the system is equivalent to the increase in the energy content.

For infinitesimal change, the equation above might be put as:

dE = dQ - dW

Or dE = q - w

Here, 'dE' is the small increase in energy and 'q' and 'w' symbolizes small quantities of heat absorbed and external work done via the system, correspondingly.

If as an outcome of a series of processes the system returns to its original state then its energy content remains unchanged so that ΔE should be zero. In such a situation, it is obvious that work done is equivalent to heat absorbed in the process.

That is,

q = w

This equation is an expression of the impossibility of perpetual motion of the first type that is, creation of energy out of nothing. This law consists of the given limitations:

a) Why heat can't be completely transformed into work.

b) It doesn't describe the spontaneity of the process.

c) It puts no limitation regarding direction flow of heat.

** Isothermal expansion**:

We can compute the change in thermodynamic properties such as q, w, ΔU, ΔH with the assistance of first law of thermodynamics. The expansion can be isothermal that can be reversible or irreversible or it can be adiabatic which can as well be reversible or irreversible. This is illustrated below:

Calculation of ΔU:

Throughout the expansion of isothermal process, temperature remains constant all through the process. As the internal energy 'U' for an ideal gas based only on the temperature. As process is isothermal, therefore the temperature is constant. This signifies internal energy 'U' remains constant of the gas.

That is ΔU = 0

Calculation of ΔH:

As we are familiar that H = U + PV

Therefore ΔH = Δ (U + PV) = ΔU + ΔPV

= ΔU + ΔnRT

As, for an isothermal process, temperature is constant and also internal energy U remains constant of the gas. That is ΔT as well as ΔU are equal to zero,

Hence, ΔH = 0

Calculation of q and w:

From the first law of thermodynamics, ΔU = q + w

As for isothermal process, ΔU = 0

Hence, W= - q

This implies in an isothermal expansion, the work is done at the expense of the heat absorbed.

The magnitude of 'w' and 'q' based in which the expansion process is taken out, that is, whether it is taken away reversibly or irreversibly.

*Isothermal reversible expansion:*

The reversibly expansion of the gas occurs in a finite number of infinitesimally small intermediate steps. To begin by the external pressure, P_{gas} is arranged equivalent to the internal pressure of the gas P_{ext} and the piston remains stationary.

If P_{ext} is reduced via an infinitesimal amount DP and the gas expands reversibly and the piston moves via a distance dl.

As dP is so small, for all practical purposes therefore, P_{ext }= P_{gas} = P

The work done by gas in one infinitesimal step (dw) can be represented as:

dw = P x A x dl (here, A = cross-section area of piston)

P x dV

Here, dV is the increase in volume. The net amount of work done via the isothermal reversible expansion of the ideal gas from V_{1 }to V_{2} is, thus,

w = V_{1}∫V_{2} P. dV

By the ideal gas equation,

P = nRT/V

w = _{V1}∫^{V2} {nRT/V} dV

nRT _{V1}∫^{V2} (dV/V)

On integration, we obtain:

As P_{1}V_{1} = P_{2}V_{2}

V_{2}/V_{1} = P_{1}/P_{2}

w = nRt ln (P_{1}/P_{2})

Isothermal irreversible expansion:

As the process is Isothermal,

Thus, ΔU=0 and ΔH = 0 (as for an isothermal expansion of an ideal gas ΔU= 0 and ΔH = 0)

As work done in the reversible isothermal expansion is represented by:

w= - nRT ln (V_{2}/ V_{1})

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