Fundamentals of Thermodynamics Assignment: System Analysis and Laws

This assignment solution provides a comprehensive overview of thermodynamics, addressing the first law, system properties, and various thermodynamic processes. It begins by defining thermodynamic systems, including closed, open, adiabatic, and isolated systems, and explains their characteristics. The solution then explores different thermodynamic processes such as isobaric, isochoric, isothermal, and adiabatic processes, detailing how they relate to the first law of thermodynamics. The document further explains how equations of state are derived from first principles, particularly focusing on the ideal gas law, and how these equations are applied to different systems. The solution also covers the derivation of the equations for work from first principles and demonstrates how to calculate the index of expansion in polytropic processes, providing detailed calculations and assumptions for specific scenarios. References to key literature are included to support the analysis.

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(a) Thermodynamic systems and their Properties
Thermodynamics refers to a field of science which handles energies that are contained and
exchanged by vapors and gases system, their transformations with regard to the heat as well as
work alongside their relationship with the features of the system. Hence thermodynamics deals
with the changing of energy from a form to another mostly heat into work and work into heat
(Gaskell and Laughlin, 2017).
Thermodynamic systems define the space or other forms of matter contained within a defined
boundary which could be virtual or real on which the attention is directed for investigation for
example a gas contained within the inner surface of a vessel may be deemed as a thermodynamic
system. This system is responsible for conversion of energy.

A thermodynamic system could be of different types including:
 Closed system: A system with the boundaries closed otherwise continuous in such a way
that there no mass entering or leaving the system. The energy transfer is however going
on across the system boundary between the system
 Open system is a system with no closed boundaries and instead has a single or more
openings through which there is a possibility of mass flow besides transfer of energy
between the surrounding and the system.
 Adiabatic system is a thermally isolated system at the boundaries hence not allowing heat
energy exchange
 Homogenous system encompasses a system having a medium that is homogenous
throughout with regard to the chemical and physical make up.
 Heterogeneous system defines a system composed of at least two various phases of
working of chemical or physical medium for example water and oil or water and steam.
 Isolated system refers to a system that is not under the influence of the surrounding in
whatever way meaning not even the mass or energy crosses the system boundary and
keeps an isolated situation from the environment. Such a system is entirely enclosed and
has insulated boundaries.
Thermodynamic processes
Isobaric process is a process where the pressure is kept constant. This is often attained through
enabling expansion or contraction of the volume in such a way that any change in pressure is
neutralized which would be as a result of transfer of heat. In an isobaric process, there are
basically internal changes in energy and there is work that is done by the system as well as

transfer of heat taking place. This means that there is no single quantity in the first law of
thermodynamics which is readily reducing to zero.
Isochoric process defines a process where the volume remains constant. Because volume is kept
constant, there is no work done by the system and hence W=0. This could be termed as most
likely the simplest way through which the variables of thermodynamics are controlled as it is
obtainable through placing the system is a confined container that does not expand or contract.
Isothermal process is a process where the temperatrure is kept constant. Generally, an
isothermal process refers to a process in which changes occurs in the work, internal energy as
well as heat energy with no changes in the temperature. This can be best illustrated by the Carnot
Cycle which offers a description of the working of heat engine through the supply of heat to a
certain gas. The result is an expansion of the gas in the cylinder which then pushes the piston
resulting in some work done. The expanded gas or heat then has to be push off the cylinder or
otherwise dumped to allow the subsequent heat or expansion cycle occur. The internal energy of
any ideal gas completely depends on the temperature hence any deviations in the internal energy
in the process of an isothermal process for any ideal gas is zero.
Adiabatic process takes place excluding the transfer of heat or mass of matter between the
thermodynamic system and the environment using a material that is strongly insulated otherwise
by conducting the process so fast there time for any significant transfer of heat is not allowed to
take place. Deploying the first law of thermodynamics in a process as an adiabatic process, delta-
is obtained since delta-U refers to the deviation in the internal energy while W refers to the work
done by the system.

Positive work is done by a system which undergoes expansion when subjected to adiabatic
conditions and hence there is a decrease in the internal energy while there is negative work done
by a system which undergoes contraction under the same conditions. The internal energy
therefore increases. The concept of adiabatic process is best explained by the compression and
expansion strokes that are experienced in the internal combustion engine in which the little
amount of heat transfers to outside the systems are negligible and technically all the changes of
energy are transferred to the moving piston.
Equations of state occur in the form pV=… that encompass the energy units
Going through the same systematically, from the first law:
dU=TdS-pdV
Expressing dU in terms of V and T
dU =( ∂ U
∂ V )TdV +( ∂ U
∂T )VdT
The second partial derivative defines the constant volume heat capacity
To examine the first partial derivative, Maxwell relation from the initial law is used
( ∂T
∂ V )S
=−( ∂ p
∂ S )V
That enables a deviation of
dU =(T ( ∂ p
∂ T )V
− p )+dV +Cv dT

pV =kT2
In which k is a constant. Using the same equation just obtained;
( ∂ U
∂V )T
=kT 2 /V
Hence
dU =kT 2 dV
V +Cv dT
Properties of thermodynamic systems
A property of the thermodynamic systems defines any features of the system that is directly or
indirectly measurable even as the system remains in equilibrium. The properties of
thermodynamic systems may be classified into two:
Intensive or Intrinsic properties refer to the features that do not depend on the mass of the
system. Such properties are inherent and are not determined by the system mass for example
temperature, viscosity, pressure and specific volume (Parrondo, Horowitz and Sagawa, 2015).
Extrinsic or Extensive properties refer to the features that depend on the system mass. The worth
of such properties is directly pegged on the mass of the system in which an increase in the
extrinsic properties goes along with an increase in the mass of the system for example weight,
total or net volume and energy.
The First Law of Thermodynamics
The First Law of Thermodynamics states that energy can neither be created nor destroyed but
converted from a form to another or into work.

Suppose a system is in a state I (the initial state) and has an internal energy, U1, it converts into
state II (last state) having internal energy U2 through the supply of q heat amount. During the
process, work W is carried out by system and heat is being absorbed that is utilized in enhancing
the internal energy of the system as we; as doing certain mechanical wok W.I (Guryanova,
Popescu, Short, Silva and Skrzypczyk, 2016)
How equations of states are derived from first principles
The change in internal energy of system is determined as sum of the heat distributed and the
work done.
Hence,
Last internal energy=U2=U1+q+W
(U2-U1)=q+w
∆ U =q+W
In work type involving pressure and volume, W=P ∆ V
Therefore, ∆ U =q+ P ∆V
How equations of states are applied to various systems
Fist Law of Thermodynamics can be applied in various processes:
Isothermal processes: Internal energy tends to be a function of changes in temperature and since
temperature does not change, internal energy is as well constant. This results in no change in the
internal energy

∆ U =0
Using first law of thermodynamics,
∆ U =q+W
0=q+W
q=-W otherwise W=-q
Hence in isothermal process, the heat taken in is fully used in performing work to environment
Adiabatic Process: In this process heat exchange does not occur
q=0
Using first law of thermodynamics,
∆ U =q+W
∆ U =0+W
Hence, W=q
Hence an enhancement in the internal energy of the system results from work performed by
surrounding on very system otherwise work performed by system on environment is as a result
of the expense in the system internal energy.
Isochoric Process: In this system changes in the volume do not occur
∆ V =0
Hence the work done, W=PDV=0

∆ U =q+W
∆ U =q+0
∆ U =q
Hence an increase in the internal energy is as a result of absorption of heat from environment
otherwise reduction in internal energy results from the heat release from the system to the
environment.
How equations of states are derived from the system constants
An ideal gas has the equation
pV =μRT
Where p is gas pressure, V volume and μ number of moles of the gas. R defines the universal gas
constant which is 8.3144 j/⁰K moles while T is the absolute temperature. Following the first law
of thermodynamics regarding energy conservation, it may be expressed in the differential form
dq=du+pdV
In which dq is the input of thermal energy du the change in the gas internal energy and pdV work
done by gas when expanding through volume change (Rathakrishnan, 2019).
Constant Volume Process
Should V be constant, dV=0 dq=du meaning all thermal input gets to the internal energy of the
gas to the gas. An increase in temperature is expected and dq=μ CV dT

Constant Pressure Process
Should p=constant, dp=0 and using the equation pdV=μRdT , there is direct proportionality
between the work done by expansion of the gas a via the differential volume dV and the change
in temperature dT. dq= μ Cp dT suppose the specific heat of the gas is at constant pressure, Cp
μ Cp dT =μ Cv dT +μRdT
Upon simplification, a significant constitutive relationship between the variables Cv, R and Cp is
found as shown:
Cp=CV+R
Constant Temperature Process
Suppose T=constant, dp=0 and from the equation d (pV) =0 meaning that there is inverse
proportionality between pressure and volume. Still, dp=pdV meaning no change in the internal
energy occurs (du=0) and every thermal input to the gas gets into expansion work.
(b) Index of compression in Polytropic processes
From the equation p2
p 1 = (V 1
V 2 )
P1V1=nRT1; P2V2=nRT2
We obtain
p2
p 1 = ( nRT1 / p1
nRT2 / p2 )
n
= ( T 1 P2
T 2 P1 ) n
=
( P2
P1 )
n
( T 1
T 2
)
n

Otherwise
( T1
T2 )=( p2
p1
)
1−n
n Or T1
T2
=
( p2
p1 )n −1
n
Still,
T2
T1
=
( V 1
V 2 )n−1
i) ( P1 V 1 )n= ( P2 V 2 )n
P2
P1
= ( V 1
V 2 )
n
=101.325/10= (0.001/20*10-6) n
10.1325=50n; n=log 10.1325/log 50
n=0.59
ii) ( P1 V 1 )n= ( P2 V 2 )n
P2
P1
= ( V 1
V 2 )n
=0.1/1.8= (8/1) n
0.0555=8n; n=log 8/log 0.0555
n=0.72
Assumptions
 The ideal gas laws hold
 The expansion of the containers is as a result of the gas they contain only

References
Gaskell, D.R. and Laughlin, D.E., 2017. Introduction to the Thermodynamics of Materials. CRC
press
Guryanova, Y., Popescu, S., Short, A.J., Silva, R. and Skrzypczyk, P., 2016. Thermodynamics of
quantum systems with multiple conserved quantities. Nature communications, 7, p.12049
Parrondo, J.M., Horowitz, J.M. and Sagawa, T., 2015. Thermodynamics of information. Nature
physics, 11(2), p.131
Rathakrishnan, E., 2019. Applied gas dynamics. Wiley