Work in t/d processes

Elementary work:

, so

The work on total pass

depends on the actual pass.

For the process

Work diagram.

The work done by the system is equal the square under the curve in work diagram.

The first law of T/D can be written now:

Heat capacity

The heat capacity of a body by the definition is:

and

The specific heat is:

Molar heat capacity is:

Heat capacity is a function of process:

Let’s consider the process :

and so

From practice we know that heat capacity of ideal gas doesn’t depend on temperature therefore:

and

Finally we can write for internal energy of ideal gas:

Let’s consider the process :

This formula is valid for all process.

If then

Then we obtain the Meier’s equation:

The adiabatic constant and internal energy

By the definition we can define specific heat ratio:

and

For internal energy:

and

Polytropic process

Polytropic process – it is a process with constant heat capacity:

;

The equation of state

then

these are the equations for polytropic process.

Heat capacity for polytropic process:

;

Work in polytropic process

There are two ways to calculate the work:

1. from the definition: (do it at home)

2. from the first law:

When (adiabatic process)

as it is expected.

When (isothermal process)

Iso-processes with ideal gas

1) Isobaric process:

The equation for isobaric process is:

and

Work:

Heat:

2) Isochoric process:

The equation for isochoric process is:

and

Work:

Heat:

3) Isothermal process:

The equation for isothermal process is:

and

Work:

Heat:

4) Adiabatic (isentropic) process:

The equation for adiabatic process is:

and

Work:

Heat:

Adiabatic process

By the definition of adiabatic process: .

Then

For internal energy there is an expression:

Then

We have

It means:

,

the solution of this equation is:

And finally:

The table for iso-processes

	Process	Capacity	n	Equation
1	Isobaric		0
2	Isochoric
3	Isothermal		1
4	Adiabatic	0
5	Polytropic

The classical theory of heat capacity

The Equipartition of Energy.

The average energy of a molecule:

Where i – is the number of degrees of freedom.

Monatomic molecule (as a material point) has only translational motion therefore .

Diatomic molecule at low temperature can move translational ( ) and rotational ( ) without vibration and has .

Diatomic molecule at high temperature can move translational ( ), rotational ( ) and oscillate. For this molecule:

.

Factor 2 in oscillation motion is because of the oscillator has the kinetic and potential energy.

So, the average energy of a molecule in thermal equilibrium is:

for monatomic molecule;

for diatomic molecule at low temperature;

for diatomic molecule at high temperature.

The air is an example.