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17.4 State Variables State variables describe the state of a system In the macroscopic approach to

thermodynamics, variables are used to describe the state of the system Pressure, temperature, volume, internal energy These are examples of state variables

The macroscopic state of an isolated system can be specified only if the system is in internal thermal equilibrium

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Transfer Variables When a process occurs in which energy is

transferred across the boundary of a system, the process can be described by some transfer variables.

A transfer of energy across the boundary represents a change in the system.

Transfer variables are not associated with any given state of the system, but with a change in the state of the system.

Heat and work are transfer variables Example of heat: we can only assign a value of

the heat if energy crosses the boundary by heat

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Work in Thermodynamics Work can be done on a

deformable system, such as a gas

Consider a cylinder with a moveable piston

A force is applied to slowly compress the gas

The compression is slow enough for all the system to remain essentially in thermal equilibrium

This is said to occur in a quasi-static process

A simplification model

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Heat vs. Work Work done on or by a system is the

amount of energy transferred between the system and its surroundings by mechanics

Heat is the energy transferred between a system and its surroundings due to a temperature difference

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Work, 2 The piston is pushed downward by a

force through a displacement:

A.dy is the change in volume of the gas, dV

Therefore, the work done on the gas is dW = -P dV

ˆext extdW d dy PAdy F r F j

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Work, 3 Interpreting dW = - P dV

If the gas is compressed, dV is negative and the work done on the gas is positive

If the gas expands, dV is positive and the work done on the gas is negative

If the volume remains constant, the work done is zero

The total work done is f

i

V

VW P dV

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PV Diagrams Used when the pressure

and volume are known at each step of the process

The state of the gas at each step can be plotted on a graph called a PV diagram

This allows us to visualize the process through which the gas is progressing

The curve is called the path

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PV Diagrams, cont The work done on a gas in a quasi-

static process that takes the gas from an initial state to a final state is the negative of the area under the curve on the PV diagram, evaluated between the initial and final states This is true whether or not the pressure

stays constant The work done does depend on the path

taken

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Work Done By Various Paths

Each of these processes have the same initial and final states

The work done differs in each process The work done depends on the path

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Work From A PV Diagram, Example 1 The volume of the

gas is first reduced from Vi to Vf at constant pressure Pi

Next, the pressure increases from Pi to Pf by heating at constant volume Vf

W = -Pi(Vf – Vi)

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Work From A PV Diagram, Example 2 The pressure of the

gas is increased from Pi to Pf at a constant volume

The volume is decreased from Vi to Vf

W = -Pf(Vf – Vi)

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Work From A PV Diagram, Example 3 The pressure and the

volume continually change

The work is some intermediate value between –Pf(Vf – Vi) and –Pi(Vf – Vi)

To evaluate the actual amount of work, the function P(V) must be known

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Isobaric Process An isobaric process is one that occurs

at a constant pressure The values of the heat and the work are

generally both nonzero The work done is W = P (Vf – Vi) where

P is the constant pressure

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Isovolumetric Process An isochoric process is one in which there is

no change in the volume Since the volume does not change, W = 0 From the First Law, Eint = Q If energy is added by heat to a system kept at

constant volume, all of the transferred energy remains in the system as an increase in its internal energy

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Isothermal Process for an Ideal Gas

This is a PV diagram of an isothermal expansion

The curve is a hyperbola

The curve is called an isotherm

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Isothermal Expansion, Details The curve of the PV diagram indicates

PV = constant The equation of a hyperbola

The work done on the ideal gas can be calculated by

ln f

i

VW nRT

V

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Heat Transfer, Example 1 The energy transfer, Q, into or

out of a system also depends on the process

The energy reservoir is a source of energy that is considered to be so great that a finite transfer of energy does not change its temperature

The piston is pushed upward, the gas is doing work on the piston

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Heat Transfer, Example 2 This gas has the same initial

volume, temperature and pressure as the previous example

The final states are also identical

No energy is transferred by heat through the insulating wall

No work is done by the gas expanding into the vacuum

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Energy Transfer, Summary Energy transfers by heat, like the

work done, depend on the initial, final, and intermediate states of the system

Both work and heat depend on the path taken

Both depend on the process followed between the initial and final states of the system

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17.5 The First Law of Thermodynamics

The First Law of Thermodynamics is a special case of the Law of Conservation of Energy It takes into account changes in internal energy

and energy transfers by heat and work The First Law of Thermodynamics states that

Eint = Q + W A special case of the continuity equation All quantities must have the same units of measure

of energy

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The First Law of Thermodynamics, cont For infinitesimal changes in a system

dEint = dQ + dW No practical distinction exists between

the results of heat and work on a microscopic scale Each can produce a change in the internal

energy of the system Once a process or path is defined, Q and

W can be calculated or measured

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Isovolumetric Process An isochoric process is one in which there is

no change in the volume Since the volume does not change, W = 0 From the First Law, Eint = Q If energy is added by heat to a system kept at

constant volume, all of the transferred energy remains in the system as an increase in its internal energy

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Isothermal Process for an Ideal Gas

An isothermal process is one that occurs at a constant temperature

Since there is no change in temperature, Eint = 0

Therefore, Q = - W Any energy that enters the system by

heat must leave the system by work

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Adiabatic Process An adiabatic process is one

during which no energy enters or leaves the system by heat Q = 0 This is achieved by

Thermally insulating the walls of the system

Having the process proceed so quickly that no heat can be exchanged

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Adiabatic Process, cont

Since Q = 0, Eint = W If the gas is compressed adiabatically,

W is positive so Eint is positive and the temperature of the gas increases Work is done on the gas

If the gas is expands adiabatically the temperature of the gas decreases

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Adiabatic Processes, Examples Some important examples of adiabatic

processes related to engineering are The expansion of hot gases in an internal

combustion engine The liquefaction of gases in a cooling

system The compression stroke in a diesel engine

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Special Processes, Summary Adiabatic

No heat exchanged Q = 0 and Eint = W

Isobaric Constant pressure W = P (Vf – Vi) and Eint = Q + W

Isothermal Constant temperature Eint = 0 and Q = - W

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Adiabatic Free Expansion This is an example of adiabatic free

expansion The process is adiabatic because it

takes place in an insulated container

Because the gas expands into a vacuum, it does not apply a force on a piston and W = 0

Since Q = 0 and W = 0, Eint = 0 and the initial and final states are the same

No change in temperature is expected

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Cyclic Processes A cyclic process is one that originates and ends at

the same state On a PV diagram, a cyclic process appears as a

closed curve The internal energy must be zero since it is a state

variable If Eint = 0, Q = -W In a cyclic process, the net work done on the

system per cycle equals the area enclosed by the path representing the process on a PV diagram

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17.7 Molar Specific Heat Several processes can

change the temperature of an ideal gas

Since T is the same for each process, Eint is also the same

The heat is different for the different paths

The heat associated with a particular change in temperature is not unique

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Molar Specific Heat, 2 We define specific heats for two

processes that frequently occur Changes with constant pressure Changes with constant volume

Using the number of moles, n, we can define molar specific heats for these processes

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Molar Specific Heat, 3 Molar specific heats:

Q = n Cv T for constant volume processes Q = n Cp T for constant pressure processes

Q (in a constant pressure process) must account for both the increase in internal energy and the transfer of energy out of the system by work

Q(constant P) > Q(constant V) for given values of n and T

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Ideal Monatomic Gas A monatomic gas contains one atom

per molecule When energy is added to a monatomic

gas in a container with a fixed volume, all of the energy goes into increasing the translational kinetic energy of gas There is no other way to store energy in

such a gas

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Ideal Monatomic Gas, cont Therefore, Eint = 3/2 n R T

E is a function of T only In general, the internal energy of an

ideal gas is a function of T only The exact relationship depends on the type

of gas At constant volume, Q = Eint = n Cv T

This applies to all ideal gases, not just monatomic ones

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Monatomic Gases, final Solving for Cv gives Cv = 3/2 R = 12.5 J/mol . K

For all monatomic gases This is in good agreement with experimental results

for monatomic gases

In a constant pressure process, Eint = Q + W and Cp – Cv = R This also applies to any ideal gas Cp = 5/2 R = 20.8 J/mol. K

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Ratio of Molar Specific Heats We can also define

Theoretical values of CV, CP, and are in excellent agreement for monatomic gases

67.12/R32/R5

CC

V

P

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Sample Values of Molar Specific Heats

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