First Law for a Closed System Undergoing a Quasi-static process
First Law for a Closed System Undergoing a Quasi-static process
In many occasions it is necessary to consider a system undergoing a process rather than a cycle. The equation, ∮ dQ − ∮ dW = 0 is applicable during the system undergoing a cycle, and algebraic sum of all energies transfer across the system boundary is zero. i.e., ∑ dQ – ∑ dW = 0.
However, if a system undergoes a change of state during which heat and work transfer are involved the net energy transfer, (Q – W) represents the Stored or internal energy of the system. For quasi-static process i.e., for a system undergoing a change of state, the first law can be expressed by the equation,
Q₁₂ − W₁₂ = E₂ − E₁ ……….(i)
where E₂ − E₁ = stored energy which is equal to the sum of internal energy (U), KE (Kinetic Energy) and PE (Potential Energy).
In the absence of motion and gravity
KE = 0 and PE = 0
∴ E₂ − E₁ = U₂ − U₁
Thus the equation (i) can be modified as
Q₁₂ − W₁₂ = U₂ − U₁
or Q₁₂ = U₂ − U₁ + W₁₂ ……..(ii)
The above equation is called Non Flow Energy Equation (NFEE)
Equations (i) and (ii) are referred to as generalised form of the first law of thermodynamics.
Energy of an Isolated System :
An isolated system is one in which there is no interaction of the system with the surroundings. If a system is isolated i.e., enclosed in a rigid adiabatic boundary, the heat (Q) and work (W) transfer are zero. From first law,
dE = 0 ………..(iii)
or E = constant
where E = total energy of system
Equation (iii) is the generalized form of first law for isolated system, and may be stated as the total energy of an isolated system is constant.
Application of NFEE for Quasi-static processes :
The NFEE can be represented in the following form
Q = ∆U + ∫ p dV
where ∆U = change in internal energy (U₂ − U₁)
The evaluation of ∫ p dV for various processes is consider below. It may be noted that ∫ p dV represents work for a closed system undergoing quasi-static process, and boundary should move in order that work may be transferred. While evaluating ∫ p dV , it is considered that there are no viscous effects within the system, and effects due to gravity are negligible.
(i) Constant volume (isochoric) process :
In a constant volume process the working substance is contained in a rigid vessel, hence the boundaries of the system are immovable and no work can be done on or by the system, other than paddle-wheel work input. It will be assumed that constant volume implies zero work unless stated otherwise
Fig. 1 shows the system and states before and after the heat addition at constant volume
W₁₂ = ∫p.dV = 0 ∴ dV = 0
Q₁₂ = U₂ − U₁ = mCᵥ (T₂ − T₁)
where Cᵥ = specific heat at constant volume
m = mass of working substance
p = pressure
T = temperature
U = internal energy
V = volume
(ii) Constant pressure (isobaric) process :
It can be seen from Fig.2(b) that when the boundary of the system is inflexible as in a constant volume process, then the pressure rises when heat is supplied. Hence for a constant pressure process, the boundary must move against an external resistance as heat is supplied; for instance a gas [Fig. 2(a)] in a cylinder behind a piston can be made to undergo a constant pressure process. Since the piston is pushed through a certain distance by the force exerted by the gas, then the work is done by the gas on its surroundings.
Fig. 2 shows the system and States before and after the heat addition at constant pressure
W₁₂ = ᵥ₁ʃⱽ² p.dV = p (V₂ − V₁)
Q₁₂ = (U₂ − U₁) + p (V₂ − V₁)
where Cᵥ = specific heat at constant volume
m = mass of working substance
p = pressure
T = temperature
U = internal energy
V = volume
(iii) Isothermal (constant temperature) process :
A process at a constant temperature is called an isothermal process. When a working substance in a cylinder behind a piston expands from a high pressure to low pressure there is a tendency for the temperature to fall. In an isothermal expansion heat must be added continuously in order to keep the temperature at the initial value. Similarly in an isothermal compression heat must be removed from the working substance continuously during the process.
Fig. 3 shows the system and States before and after the heat addition at constant temperature
This process is represented by the equation, pV = constant, and also referred as hyperbolic process
W₁₂ = ᵥ₁ʃⱽ² p.dV , but p = (p₁ V₁)/V
∴ W₁₂ = p₁ V₁ ᵥ₁ʃⱽ² dV/V = p₁ V₁ logₑ(V₂/V₁)
= p₁ V₁ logₑ(p₁/p₂)
Q = p₁ V₁ logₑ(V₂/V₁) (∴ ∆U = 0)
(iv) Polytropic process; pVⁿ = constant
