First Law of Thermodynamics for a Closed System

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This is a topic from Thermodynamics:

Contents

Statement

The 1st Law of Thermodynamics for a closed system states that:

Ein - Eout = ΔESystem

It is a basic implication of the law of conservation of energy, and as such requires no derivation.

  • The right hand side:
ΔESystem = ΔEInternal + ΔEKinetic + ΔEPotential
=m(u2 - u1) + m((v2)2 - (v1)2)/2 + mg(z2 - z1)

where z represents height (we don't use 'h' here, because 'h' represents enthalpy)
u represents specific internal energy.

  • The left hand side:
  • Taking into account heat transfer only:
Ein - Eout = Q12
  • Taking into account work transfer only:
Ein - Eout = -W12
  • Thus taking both heat and work into account:
Ein - Eout = Q12 - W12
  • Therefore the law can be rewritten in full as:
Q12 - W12 = m(u2 - u1) + m((v2)2 - (v1)2)/2 + mg(z2 - z1)

Stationary Systems

It is very common for a problem to involve a system which is stationary. For a stationary system the change in the kinetic energy and potential energy are zero, and so:

Q12 - W12 = m(u2 - u1) and
dQ - dW = dU

This is the most common form of the first law in this course.
In order to apply the first law of thermodynamics to problems we must first explore the concept of internal energy, as well as the related concept, enthalpy.

Internal Energy

  • Internal energy is a property, the unit of which is the joule.
  • It is a measure of the energy associated with the random, molecular motion of a substance due to its temperature. Therefore the internal energy is the sum of the microscopic internal kinetic and potential energy of the molecules which make up a system, as opposed to the macroscopic kinetic and potential energy of the system itself.
  • It follows that the internal energy of a system is linearly proportional to the amount of matter in the system - its mass- as well as the temperature of the system:
U α mT
U/m α T
u α T
The symbol α is the Greek letter alpha and means 'directly proportional to'
  • This constant of proportionality is known as the specific heat capacity

Specific Heat Capacity CV

  • Specific heat capacity is the energy required to increase the temperature of one kilogram of some substance by one degree.
  • The magnitude of the specific heat capacity is always based on one kilogram, not the extent of the system, and so specific heat capacity is an intensive property (as are all 'specific' properties).
  • Specific heat capacity for a heat transfer which occurs at constant volume is represented by CV
  • If heat is transferred to or from a substance at a constant volume, then:
CV = du/dT
du = CV dT
  • Integrating both sides:
Screen Shot 2013-03-14 at 4.28.19 PM.png
  • Multiplying both sides by mass changes specific internal energy to internal energy:
Screen Shot 2013-03-14 at 4.31.31 PM.png
  • This change in internal energy is irrespective of the type of process which caused the temperature change.

Enthalpy

  • Enthalpy is a property, defined in relation to other existing properties, including internal energy:
Enthalpy = Internal Energy + Pressure x Volume
H = U + PV
  • The unit of enthalpy is the joule, because:
Screen Shot 2013-03-19 at 7.07.25 PM.png
and decomposing with S.I units:
Screen Shot 2013-03-19 at 7.07.33 PM.png

and so enthalpy is the sum of two energy properties.

  • Thus, even though enthalpy's definition is based on other properties, it can be defined in its own right as a measure of the total energy of a stationary thermodynamic system, including the internal energy of a substance and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.

Specific Heat Capacity CP

  • The definition of specific heat capacity for a constant pressure process arises from an examination of the definition of enthalpy.
  • Dividing both sides of the definition for enthalpy by mass gives:
H/m = U/m + PV/m
h = u + Pv

where h, u and v are specific heat, specific internal energy and specific volume respectively.

  • By the ideal gas law, Pv = RT, and so:
h = u + RT
  • Since internal energy is a function of temperature, enthalpy is the sum of two functions of temperature, and so:
h α T
  • Having established this proportionality the same method can be applied to derive CP as with CV.
  • Thus if heat is transferred to or from a substance at constant pressure:
CP = dh/dT
dh = CPdT
  • Integrating both sides:
Screen Shot 2013-03-19 at 8.04.47 PM.png
  • Multiplying both sides by mass changes specific internal energy to internal energy:
Screen Shot 2013-03-19 at 8.04.52 PM.png

Relating CV and CP

Relation with the Gas Constant

  • The two constants can be related using the definitions of specific enthalpy and the ideal gas law:
(1) h = u + Pv
(2) Pv = RT
  • Writing these definitions in differential form:
(1) dh =du + Pdv
(2) Pdv = RdT
  • Substituting (1) into (2):
dh = du + RdT
  • Dividing both sides by dT:
dh/dT = du/dT + R
  • Recalling the respective definitions of CP and CV:
CP = CV + R

The Isentropic Index

The isentropic index or ratio of specific heat capacities is represented by k or γ:

k = γ = CP/CV

Combining Relations

Combining these two relations leads to some useful expressions.

  • Rearranging the isentropic index:
CVk = CP
CV = CP/k
  • Substituting into the gas constant relation:
CP = CP/k + R
CP(1 - 1/k) = R
CP(k-1)/k = R
  • Therefore:
CP = kR/(k-1)
  • And since CV = CP/k:
CV = R/(k-1)

Common Application of Enthalpy in Problem Solving

Considering an isobaric process (constant pressure) for a stationary system, find the heat transfer, Q.

Q12 - P(V2 - V1) = (U2 - U1)
  • Rearranging gives:
Q12 = (U2 + PV2) - (U1 + PV1)
Q12 = H2 - H1
= ΔH
Q12 = mCP(T2 - T1)
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