HERMODYNAMICS (SPA519) Detailed s to Coursework 1 ISSUE: Septeber 6 th 017 HAND-IN: October rd 017 QUESION 1: (5 arks) he siple kinetic theory arguent sketched in the lectures and in Feynan's lecture notes shows that the ideal gas teperature is a easure of the ean kinetic energy of a olecule of ass, 1 v kb Conversely, we define the root ean square speed of a typical gas olecule by vrs v kb a) A olecule of sulphur dioxide (SO, olecular weight 64 au) in the Venusian atosphere has a root ean square speed of 55 s -1 i) What is the teperature of that atosphere? [4 ks] he kinetic teperature is defined by the equilibriu relation 1 1 kb v v rs so we calculate v rs giving kb 7 641.6610 55 1.810 74.5 ii) What is the ean kinetic energy in ev of a olecule of nitrogen (N, olecular weight 8 au) in that sae atosphere? [4 ks] In equilibriu all olecules have the sae average kinetic energy deterined solely by the teperature as in the relation at the top above. We can find the ean kinetic energy of the SO and that of the N ust be the sae; 1
7 1 641.6610 55 0 v 1.510 0 0 1 1.510 1.510 v 0. 095eV e -19 1.6 10 C iii) What is the ratio of the root ean square speeds of a sulphur dioxide olecule and a nitrogen olecule in the Venusian atosphere? [4 ks] Since vrs kb / the ratio of olecular speeds at the sae is v v N rs SO rs N SO 8 64 0.66 b) In the outer atosphere of a star like the Sun soe un-ionised atos exist but with the electrons excited to higher energy states as a result of collisions between rapidly oving atos. he typical energy needed to excite an electron fro one energy level in an ato to another is about 1 ev. Estiate how hot such a gas would have to be for a typical nitrogen ato to possess enough kinetic energy to excite an electron when it collides with another olecule? What would be the root ean square speed of such an ato? [4 ks] o excite an ato, an incident particle ust have a ean kinetic energy of the order of 1 ev. For this energy we find the corresponding teperature fro k 1 ev 11.6 10 19 B 19 1.6 10 1.810 1 770 he corresponding root ean square velocity is vrs k B 1.810 770 7 141.6610 7 1.810,714 1 s vrs = :71 10 s -1
c) In the centre of a star, atter exists as a gas of copletely ionised nuclei and electrons or plasa. he root ean square speed of an electron in this plasa is 0.c where c is the speed of light in vacuu. What is the teperature of the plasa? [4 ks] If an electron has vrs = 0.c, then v rs kb 1 9.110 0. 1.810 8 10 8 1.7810 d) he 001 Nobel Prize in physics was awarded for the production of an exotic state of atter called a Bose-Einstein Condensate (BEC). he first such successful experient produced a BEC containing about 000 rubidiu atos (Rb olecular ass 87 au) reduced to a teperature of about 0n (nanoelvin!). Although the properties of the atos in the BEC are deterined by quantu physics, there are other Rb atos present which are not in the BEC but which behave as an ordinary gas also at this extreely low teperature. Estiate the ean kinetic energy and the root ean square speed of an ato of ordinary" Rb at this teperature. For rubidiu at 0 n we find the ean kinetic energy to be [5 ks] 1 8 1 1 v kb 1.51.810 10 4.1410.6 10 ev and the root ean square speed to be vrs kb / 8 1.810 10 7 1.6610 87.4 10 1 s.4 /s
QUESION : (Practice) he resistance of a wire is given by; R R 0 1 where is the teperature in degrees Celsius easured on the ideal gas scale and thus R 0 is the resistance at the ice point. he constants and are.8 10-0 C -1 and -.0 10-6 0 C - respectively. At a teperature of 70 0 C on the Celsius scale calculate; a) he ratio of the resistance of the theroeter to the value R 0. R R 0 0 6 70 R0 1 (70 C) (70 C) R0 1.8 10 70.0 10 70 70 R 1 0.66 1.47 10 1. R R 70 R0 1.5 0 5 b) he epirical teperature R on the resistance scale. 0 R 70 R 7.16 7.161.5 41.45 R0 c) he teperature Gas on the gas scale. Gas 7.15 70 4.15 Note that as an epirical teperature R has no units, this is not the elvin scale. Only if we have calibrated the resistance theroeter against an ideal gas theroeter can we convert the resistance into elvin. he teperature on the elvin scale is 7.15+70 = 4.15. Note the slight discrepancy between this gas teperature and the epirical teperature. QUESION (Practice he table below lists the observed values of the pressure P of a gas in a constantvolue gas theroeter at an unknown teperature, and at the triple point of water, as the ass of gas used in the theroeter is reduced. P ( Hg) 1000.0 750.0 500.0 50.0 PP ( Hg) 155. 1151.6 767.8 8.95 4
Fro this data; a) Write down an equation relating these quantities to the ideal gas teperature. he ideal gas teperature is defined as the liit Gas li 7.16 PP 0 P b) Using the definition of the ideal gas teperature and the above data either graphically or otherwise estiate the ideal gas teperature to two decial places. P P c) Fro the graph in a) estiate the ideal gas teperature to two decial places. We find for the liiting value by extrapolation of the value of P to P P 0 P P P 7.16 P P P P 0 7.16 0.6511 177.85 5