Chapter 10

# To test the understanding of the lesson

## Question

• Give the differential expression of mass internal energy of an ideal gas.

• Give the differential expression of mass internal enthalpy of an ideal gas.

• What is the integral expression of the entropy of an ideal gas in terms of the and variables ?

### Solution

• Differential of mass internal energy of a perfect gas :

• Differential of the mass internal enthalpy of an ideal gas :

• Molar entropy of an ideal gas in terms of the variables then  :

## Question

What is the differential expression of the elementary work pressure forces received by a closed system in the general case and in the case of a reversible transformation ?

### Solution

In the general case, noting  :

If the transformation is reversible, and :

## Question

What is the work received by one mole of an ideal gas in a reversible isothermal transformation ?

### Solution

If the transformation is reversible isotherm :

And :

In the case of an expansion, and  : the external environment gets back work.

## Question

• Give the ideal gas equation of state in intensive variables , and (density).

• Give the equation of state of an ideal gas using the Boltzmann constant and particle density .

• What is the gas state equation of Van der Waals for moles, in variables , and  ?

### Solution

• Ideal gas equation of state in intensive variables , and  :

( is the molar mass)

• Equation of state of an ideal gas using the Boltzmann constant and particle density  :

• Gas state equation of Van der Waals for moles, in variables , and  :

## Question

What is an ideal gas ?

### Solution

• At the macroscopic scale : a perfect gas is a (real !) gas studied at low pressures (less than a few bars, is normal atmospheric pressure).

• At the microscopic scale : the particles of an ideal gas are punctual and do not interact with each other but only with the wall of the container they occupy during elastic collisions.

## Question

Give the value of the quadratic (RMS) speed of a perfect mono-atomic gas. What is its order of magnitude ?

### Solution

The root mean square velocity of a perfect mono-atomic gas :

Where is the mass of a particle, the molar mass, the Boltzmann constant and perfect gas constant.

Order of magnitude :

For Argon, at , .

For Helium, at , .

## Question

Express molars capacity and of an ideal gas in terms of (gas constant) and .

## Question

Give the differential formulation of the first law of thermodynamics for a closed system in its most general form.

### Solution

Noting the internal energy, macroscopic gravitational potential energy and the macroscopic kinetic energy :

Where and are the work and heat transfer received by the closed system.

## Question

Give the differential formulation of the second law of thermodynamics for a closed system (system macroscopically at rest).

With :

## Question

Give the first thermodynamic identity, in differential form ( relationship between , , , and ).

## Question

Leave the second thermodynamic identity, in differential form (relationship between , , , and ).

## Question

• How to calculate heat transfer during a transformation at a constant volume ?

• How to calculate heat transfer during a transformation at constant external pressure ?

### Solution

• A constant volume :

• A constant external pressure :

## Question

To state the theorem of Carnot on ditherme machines.

### Solution

Note and the temperatures of the cold source and the hot source.

For a dithermal machine operating irreversibly, the yield is still lower than that obtained during a reversible operation :

## Question

Give the relationship between and , and between and , and finally between and for isentropic evolution of an ideal gas of constant ratio .

### Solution

They write the laws of Laplace :

## Question

State the "first industrial valid law" for flowing fluids in industrial machines.

### Solution

Using mass quantities :

## Question

What is the statistical interpretation of entropy ?

### Solution

Entropy is related to the molecular disorder. The greater entropy of a system, the more disorder.

Thus, the entropy of a pure substance from a liquid to a gaseous state increases.

## Question

An electronic pulse generator emits pulses of energy , each at the frequency of .

The efficiency of the generator is equal to %.

How many liters of water per minute circulate in the cooling system of the generator so that the water temperature at the output does not increase by more than ?

The specific heat capacity of water is :

.

### Solution

In a period , energy given par the generator is .

For the water :

So :

.

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