Calore specifico dei gas perfetti

Il primo principio della termodinamica permette di capire perché, a parità di massa e di aumento di temperatura non è la stessa cosa scaldare un gas a volume costante a pressione costante. Infatti nel primo caso tutto il calore va ad incrementare l’energia interna del gas; nel secondo una parte del calore serve a far compiere al gas un lavoro esterno e perciò ne occorre di più.

Vogliamo mostrare che esiste una relazione tra l’energia interna e i calori specifici dei gas perfetti.

Sappiamo che il calore specifico di una sostanza è definito come:

Nel caso dei gas questa equazione è modificata riferendo il calore specifico non più all’unità di massa della sostanza ma a 1 mole di gas. Quindi:  

Quando un gas scambia calore, la quantità di calore scambiata è diversa a seconda del tipo di trasformazione termodinamica alla quale il gas è sottoposto; in particolare, risultano interessanti i due casi in cui lo scambio di calore avviene rispettivamente a pressione costante e a volume costante. Definiamo:

Per calcolare c_p e c_v per un gas perfetto  dobbiamo scrivere il primo principio della termodinamica che dice che  Q = DU+L, tenendo conto che l’energia interna di un gas monoatomico può essere scritta nel seguente modo: U=Ec=3/2n RT      e quindi:    ΔU = 3/2n RΔT

Uso questa espressione nel primo principio:  Q =3/2nR DT + L

Tenendo presente che quando V=cost si ha L = 0, si ricava: c_v=3/2 R

Tale relazione mostra che il calore specifico a volume costante di un gas perfetto non solo è indipendente dalla temperatura, ma è lo stesso per tutti i gas. Ricordiamo che il valore riportato vale per i gas monoatomici (He, Ne…) per i quali l’energia interna è solo energia cinetica traslazionale. Tenendo conto della cosiddetta relazione di Mayer c_p-c_v = R  si ricava che c_p = 5/2R. Infine otteniamo che:

DU = n c_v DT

Questa è l’espressione generalmente usata per il calcolo della variazione dell’energia interna di un gas perfetto. Allora possiamo scrivere per un gas perfetto il primo principio della termodinamica nella  forma: Q = n c_v DT + L

Se avessimo scelto un gas biatomico invece di uno monoatomico, essendo in questo caso:

avremmo ottenuto per il calore specifico a volume costante il valore c_v = 5/2R  e c_p = 7/2 R. 

Motore di Carnot

Vai al link per vedere l'animazione del funzionamento del motote
http://www.galileo.fr.it/marc/termologia_e_termodinamica/carnot/Carnot_Engine.htm
applet originale © .Wan Ching Hui

Cool microscope feels the heat

"Physicists in Germany have invented a new kind of microscope that uses a gas of extremely cold atoms to map the surface of nanoscale structures. The researchers say that their device is complimentary to atomic-force microscopes (AFMs) and that they ultimately hope to create a probe with precision that is limited only by fundamental quantum uncertainties."

Cool microscope feels the heat - physicsworld.com

Il birraio di Salford

James Prescott Joule FRS (1818 – 1889) was an English physicist and brewer, born in Salford, Lancashire. Joule studied the nature ofheat, and discovered its relationship to mechanical work (see energy). This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics. The SI derived unit of energy, the joule, is named after him. He worked with Lord Kelvin to develop the absolute scale of temperature, made observations onmagnetostriction, and found the relationship between the current through a resistance and the heat dissipated, now called Joule's law.
http://en.wikipedia.org/wiki/James_Prescott_Joule

On the Relation between Heat and the Mechanical Power.

On the Existence of an Equivalent Relation between Heat and the ordinary Forms of Mechanical Power.

By James P. Joule, Esq.

[In the letter to the Editors of the 'Philosophical Magazine.']
series 3, vol. xxvii, p. 205Gentlemen,
The principal part of this letter was brought under the notice of the British association at its last meeting at Cambridge. I have hitherto hesitated to give it further publication, not because I was in any degree doubtful of the conclusions at which I had arrived, but because I intended to make a slight alteration in the apparatus calculated to give still greater precision to the experiments. Being unable, however, just at present to spare time necessary to fulfil this design, and being at the same time most anxious to convince the scientific world of the truth of the positions I have maintained, I hope you will do me the favour of publishing this letter in your excellent Magazine.
The apparatus exhibited before the Association consisted of a brass paddle-wheel working horizontally in a can of water. Motion could be communicated to this paddle by means of weights, pulleys, &c., exactly in the matter described in a previous paper.*
The paddle moved with great resistence in the can of water, so that the weights (each of four pounds) descended at the slow rate of about one foot per second. The height of the pulleys from the ground was twelve yards, and consequently, when the weights had descended through that distance, they had to be wound up again in order to renew the motion of the paddle. After this operation had been repeated sixteen times, the increase of the temperature of the water was ascertained by means of a very sensible and accurate thermometer.
A series of nine experiments was performed in the above manner, and nine experiments were made in order to eliminate the cooling or heating effects of the atmosphere. After reducing the result to the capacity for heat of a pound of water, it appeared that for each degree of heat evolved by the friction of water a mechanical power equal to that which can raise a weight of 890 lb. to the height of one foot had been expended.
The equivalents I have already obtained are; -- 1st, 823 lb., derived from magneto-electrical experiments (Phil. Mag. ser. 3 vol. xxiii. pp. 263, 347); 2nd, 795 lb., deduced from the cold produced by the rarefaction of air (Ibid. May 1845, p. 369); and 3rd, 774 lb. from experiments (hitherto unpublished) on the motion of water through narrow tubes. This last class of experiments being similar to that with the paddle wheel, we may take the mean of 774 and 890, or 832 lb., as the equivalent derived from the friction of water. In such delicate experiments, where one hardly ever collects more than one another than that above exhibited could hardly have been expected. I may therefore conclude that the existence of an equivalent relation between heat and the ordinary froms of mechanical power is proved; and assume 817 lb., the mean of the results of three distinct classes of experiments, as the equivalent, until more accurate experiments shall have been made.
Any of your readers who are so fortunate as to reside amid the romantic scenery of Wales or Scotland could, I doubt not, confirm my experiments by trying the temperature of the water at the top and at the bottom of a cascade. If my views be correct, a fall of 817 feet will course generate one degree of heat, and the temperature of the river Niagra will be raised about one fifth of a degree by its fall of 160 feet.
Admitting the correctness of the equivalent I have named, it is obvious that the vis viva of the particles of a pound water at (say) 51° is equal to the vis viva possessed by a pound of water at 50° plus the vis viva which would be acquired by a weight of 817 lb. after falling through the perpendicular height of one foot.
Assuming that the expansion of elastic fluids on the removal of pressure is owing to the centrifugal force of revolving atmospheres of electricity, we can easily estimate the absoute quantity of heat in matter. For in an elastic fluid the pressure will be proportional to the square of the velocity of the revolving atmosphere, and the vis viva of the atmospheres will also be proportional to the square of their velocity; consequently the pressure of elastic fluids at the temperatures 32° and 33° is 480 : 481; consequently the zero of temperature must be 480° below the freezing-point of water.
We see then what an enormous quantity of vis viva exists in matter. A single pound of water at 60° must possess 480° + 28° = 508° of heat; in other words, it must possess a vis viva equal to that acquired bt a weight of 415036 lb. after falling through the perpendicular height of one foot. The velocity with which the atmosphere of electricity must revolve in order to present this enormous amount of vis viva must of course be prodigious, and equal probably to the velocity of light in the planetary space, or to that of an electric discharge as determined by the experiments of Wheatstone.
* Phil. Mag. ser. 3, vol. xxiii, p. 436. The paddle-wheel used by Rennie in his experiments on the friction of water (Phil. Trans. 1831, plate xi, fig, 1) was somewhat similar to mine. I have employed, however, a greater number of "floats," and also a corresponding number of stationary floats, in order to prevent the rotatory motion of the can.
I remain, Gentlemen,
Yours Respectfully,
James P Joule.

Dal sito


http://www.chemteam.info/Chem-History/Joule-Heat-1845.html

Fermi Telescope and the dark matter

"New results from NASA's Fermi Gamma-Ray Space Telescope appear to confirm a larger-than-expected rate of high-energy positrons reaching the Earth from outer space. This anomaly in the cosmic-ray flux was first observed by the Italian-led PAMELA spacecraft in 2008 and suggests the existence of annihilating dark-matter particles. Physicists believe that about 80% of the mass in the universe is in the form of a mysterious substance known as dark matter. ... researchers are attempting to find direct evidence of it on Earth using either heavily shielded underground detectors or with particle accelerators. But they also have a third, less direct, option – using satellites or balloon-based instruments to detect the particles that some theories predict are created in space when two dark-matter particles collide and annihilate."

Has Fermi glimpsed dark matter? - physicsworld.com

Telescope optics set to aid gravitational detection

"A British team is designing the optics for a telescope that will be able to detect the gravitational effects of violent cosmic events, such as when two black holes collide.
The €790m (£688m) Einstein Telescope should be completed by 2025, by which time it will be capable of detecting gravitational waves around 100 orders of magnitude fainter than current devices can." Telescope optics set to aid gravitational detection News The Engineer

Asta e fune

One end of a uniform beam weighing 30N and 1 m long is attached to a wall with a hinge. The other end is supported by a wire. Find the tension of the rope. What is the action on the wall?

A+T+W=0  (somma vettoriale)

r×W+2r×T=0 (polo in O)

Momento del peso = L m g sin 60°/2

Momento tensione fune = LT sin 30°

  L m g sqrt(3) / 4 = L T /2

   T = 2mg/sqrt(3)

A_x= T cos 60° = mg  ;  A_y= mg-T sin60°=mg-mg/sqrt(3)

n.10 - disco e asta

Un perno P passante per il centro del disco (vedi la figura) permette al disco di ruotare liberamente nel piano della figura che è un piano verticale. Il disco ha raggio b e la sua massa  è m₁ . Una sbarra omogenea e di lunghezza L è saldata al bordo del disco. La sbarra ha la direzione della lunghezza perpendicolare al bordo del disco e si distende solo nel piano del disco. La sua massa è m₂.

Trovare il momento d’inerzia del sistema (disco e sbarra) rispetto all’asse del perno, ossia l’asse perpendicolare al disegno e passante per il centro del disco.

Se il sistema ruota, che direzione ha il momento angolare?

Calcolate l’accelerazione angolare del sistema, quando viene rilasciato dalla posizione mostra in figura.

Calcoliamo il momento d’inerzia ricordando che esso è una quantità additiva. Dato che conosciamo il momento d’inerzia del disco: ½ m₁b², a questo basta aggiungere il momento d’inerzia dell’asta che calcoliamo nel seguente modo. Prendiamo un asse x perpendicolare al perno e diretto lungo l’asta. Supponiamo una piccola massa lunga dx, dm=ρdx, dove ρ è la densità dell’asta pari a m₂/L. Quindi

Discutiamo ora il momento angolare con ω avente la direzione dell’asse del perno, poiché l’asse del perno è quello di rotazione, come ci dice il problema. Facciamo sempre riferimento alla figura usata per calcolare il momento di'inerzia. L'asse di rotazione è l'asse P e usiamo per il calcolo il polo O (il centro del disco) in figura.

Si applica quindi la relazione Iα=τ al sistema.  Il sistema ruota attorno al punto fisso P. Le forze esterne sono l’azione del  sostegno del perno e il peso del disco e dell’asta. Poiché il peso del disco è applicato nel centro del disco , se prendiamo questo centro come polo per il calcolo dei momenti, il peso del disco non ha momento, come l’azione del supporto del perno. L’unica forza che ha momento è il peso dell’asta.

Il momento meccanico è dovuto solo  al peso della sbarra,  m₂g, applicata al CM_sbarra

Unbound planets could abound in the universe

"Ten planets that appear to be drifting in interstellar space have been spotted by an international team of astronomers. The planets are so far from any host stars that they may not orbit a star at all, and could be drifting unbound through space. The team believes that such rogue planets could outnumber normal stars almost 2:1 and their existence could confirm computer simulations of solar-system formation."

Unbound planets could abound in the universe - physicsworld.com