Experiment 4 – Determination of the first excitation potential for mercury atom in Franck-Hertz experiment. Mohd Faiz Mohd Zin PHYS 341,West Virginia University
[email protected] Abstract We have measure measured d the excitation potentials of a mercury atom by performing Franck-Hertz experiment. The value measured in this experiment were were 4.911±0.087 4.911±0.087 V which is very close with the accepted value of 4.9 V and within error. We also observed the linearly increasing of acceleration voltage versus order of maxima and minima. The mean free path of an electron in mercury mercury vapor gas was measured to be 13.597 μm. The contact potential in this was found to be 2.879±0.097 V and the wavelength was also found at 251.684±4.373 nm which is in the ultraviolet spectrum.
II. Theory I. Introduction In 1913, Niels Bohr proposed his model of atom that explains the emission spectrum of hydrogen by quantization of energy levels of the electrons.[1] In his model, electrons require a discrete amount of energy to move up the orbitals while release a discrete amount of energy to move down of the orbitals. So, both Franck and Hertz conducted an experiment in 1914 which by observ obs erving ing ma maxim ximaa and min minima ima in tra transm nsmiss ission ion of electr ele ctrons ons thr throug ough h mer mercur cury y vap vapor or tha thatt suc succee ceeds ds to verifi ver ifies es th that at the ato atomi micc ele electr ctron on ene energ rgy y sta states tes are quantized.[5] Franck and Hertz verify that the atoms can only absorb a discrete amount of energy regardless how the energy was transferred to the atom. In this experiment, we will set-up the Franck-Hertz tube tu be wi with th dr drop opss of me merc rcur ury y al alon ong g th thee el elec ectr tric ical al connection and study the inelastic collision between electr ele ctrons ons and th thee mer mercur cury y ato atom m tha thatt ca can n res resul ults ts in quantized excitation excitation of the atom. We will measure the anode current, Ic by varying the acceleration voltage Va and det determ ermine ine the fir first st exc excita itatio tion n pot potent ential ial by observing its maxima and minima. [5]
A. Franck-Hertz Tube
FIG.1:: Sch FIG.1 Schema ematic tic of the sim simpli plifi fied ed Fra Franck nck-He -Hertz rtz circuit. Franck-Hertz tube is a triode with a cathode (K), a grid and an anode (A).[4] The grid and the cathod are separated in a large distance so that the shorter mean free path of the electrons will have high probability for collision between the electrons and the mercury gas molecule mole cules. s. Short distance distance betw between een the grid and the anode is to decrease the collisions beyond the grid. There is an acceleration voltage Va across cathode and the grid and a deceleration voltage Vd across andode and grid where Vd: 1.5 V. So, electrons that arrive at the grid need to have an energy greater than eVd to reach the anode A. [4]
B. Kinetic gas theory and mean free path
According to ideal gas law, the state of an amount of gas can be determined by equation: p = nk B T (1) where n is the number of atoms density and k B is the Boltzmann constant. A drop of mercury in the Franck-Hertz tube will partially evaporates with increasing temperature. So, a two-phase of liquid and gas formed and can be shown by the Clausius-Clape Clausius-Clapeyron yron equation: [6]
dp λ = dT ( Vv−Vl ) T
III. Experimental Design
(2)
where p is the pressure, T is temperature in K, λ is the vaporisation energy and Vv and Vt is the volumina of the vapor and liquid phase respectively. By means of integration and inserting number into the equation (2):
lg p=10.55 −
3333 0,86 ·l ·lg g T (3) T
p is pressure in Torr ( 1 Torr = 133.33 Pa). In this experiment, we will use high temperature which o
is above 170 C to ensure a short distance for mean free path of electrons. l of an electron moves from the l Mean free path, ̄ cathode K to the anode A is given by [5]
̄l l =
1
√ 2 πn Ro
2
(4)
Where Ro is the radius of a mercury atom which is equal to 1.5×10 equation will yield:
−10
m Further development of this
̄l l =
k B T
√ 2 πp R o 2
(5)
C. Inelastic Electron Scattering The first excited state of mercury is 4.9 eV above the ground state. This means that the mercury atoms can absorb 4.9 eV energy if the accelerated electrons contai con tain n thi thiss muc much h of ene energ rgy y. How Howeve everr, whe when n the accelerat acce lerated ed elec electrons trons have less energy than 4.9 eV which means Va is less than 4.9 V, the collisions form between the electrons and the molecules are elastic and since electron mass is very small compared with the
mas asss
of a
merc rcur ury y
ato tom, m,
( ∆ E
∝
the electrons will pass through the grid and reach the anode to be measured as current. If the accelerated electrons contain energy equals to 4.9 eV eVa, a, the ele electr ctrons ons wi will ll hav havee eno enough ugh ene energ rgy y to collide inelastically with the mercury atoms as they reach toward the grid. For this type of collision, the mercury atoms will absorb completely 4.9 eV energy of the electrons. Thus, they have insufficient energy to overcome the deceleration voltage Vd across the grid to the anode and so they fall back to the grid. At this point the anode current Ic will register a minimum amount of current.
4
mc m Hg
)
electrons doesn't lose any energy to the gas in that collision but only change in its direction.[1] Therefore the electrons will arrive at the grid with kinetic energy equal to eV eVa. a. The presence of deceleration potential Vd across the grid to the anode will oppose electrons that have energy lesser than eVd. So if Va Va is larger than Vd
The apparatus is set-up and fundamentally similar as in FIG 1. The filament is heated at midrange which is about 5.5 V and the Amplifier Gain is in maximum whil wh ilee Am Ampl plif ifie ierr Ze Zero ro is in th thee mi midr dran ange ge.. An acceleration voltage Va Va is applied between cathode and the gri grid d whi while le dec decele elera ratio tion n vol voltag tagee Vd is app applie lied d between grid and the anode A.[4] The Franck-Hertz tube contains drop of mercury and so it is hea heated ted to vap vapori orized zed the me mercu rcury ry.. We o
carefully set the temperature T at 170 C as it is a suitable temperature for excitation as it gives the mean free path of electrons in the right distance which must be very small compared compared with the dista distance nce betw between een cath ca thod odee an and d th thee an anod odee wh whic ich h in ou ourr tu tube be ha have ve a distance of 8mm.[4] This distance is large compared with wi th th thee me mean an fr free ee pa path th of th thee el elec ectr tron onss at th this is temperature for mercury gas.[3] In thi thiss exp experi erimen ment, t, we wer weree exp expect ected ed to see multi mul tiple pless of exc excita itatio tion n ene energ rgy y for mer mercur cury y ato atoms ms which is the result of inelastic collisions that occur between between electrons electrons and atoms at cert certain ain energy that excites mercury energy level from ground state to the first state. We also measured the anode current Ic by varying the acceleration voltage Va by using picoammeter and control contr ol unit respectivel respectively y. At every peaks, we try to meas me asur uree th thee an anod odee cu curr rren entt Ic at th thee su surr rrou ound ndin ing g accelerat acce leration ion volt voltage age Va to incre increase ase the accu accuracy racy in determine the peaks.
IV. Result and analysis A. Analysis of the Franck-Hertz curve About 75 data points were taken to obtain FIG 2 and they were curve-fit by using summation of Lorentzian function for each of the peaks. The Franck-Hertz curve shown sho wn in FIG 1 for forme med d the expecte expected d seq sequen uence ce of maxima and minima. We know that the deceleration volt vo ltag agee Vd is 1. 1.5 5 V an and d so a cu curr rren entt sh shou ould ld be
observable if Va Va is exceeding 1.5V. 1.5V. Our data shows that the currents start to increase slightly at 1.663 V. As we can see from FIG 2, the current continue to increase due to the sufficient energy of the electrons to overc ove rcome ome the dec decele elerat ration ion vol voltag tage. e. How Howeve ever, r, the current starts to drops dramatically at the first peak which is at 7.79 V. The sudden drops of the anode current Ic indicates that the electrons have gain enough kinetic energy to collide inelastically with the mercury atoms just as they reach the grid. The mercury atoms will gain energy while electrons lose its energy and fall out before reaching the anode and therefore the anode curr cu rren entt Ic de decr crea easi sing ng sh shar arpl ply y un unti till be beco come mess a minimum. However, at certain point, the current starts to inc increa rease se aga again. in. So th this is me means ans tha thatt the col colli lisio sion n between electrons and mercury atoms become elastic again. From FIG 2, the amplitude of a single maximum increases with higher acceleration voltage Va. This is due to a multiple inelastic collisions occured between the accelerated electrons and the mercury atoms. So ∆V V a , the currents will whenever Va is multiple of ∆ start sta rt to dro drop p aga again. in. The There refor foree hig higher her acc accele elera ratio tion n voltage Va will gives more electrons sufficient energy to ov over erco come me de dece cele lera rati tion on vo volt ltag agee Vd un unle less ss th thee elec el ectr tron onss ga gain in en enou ough gh ki kine neti ticc en ener ergy gy to co coll llid idee inelea ine least stica ically lly wi with th the mer mercur cury y ato atoms. ms. Due to the stat st atis isti tica call fl fluc uctu tuat atio ions ns an and d th thee ki kine neti ticc en ener ergy gy distribution of the electrons, the anode current in the minima remains larger than zero.
of the fit for five peaks. Most of the data were fitted well except for several outliers at the minima. B. Excitation potential energy Method of least-square fit was performed between acceleration voltage Va Va and the number of peaks as shown in FIG 3. We obtained a good linearity of our data points and the slope of this graph is the average differences of acceleration potential between two peaks. The equation of the graph is:
y =( 4.911 ±0.087 ) x +2.579 ±0.289
(4.911±0.087 )V and ∆V V a is ∆V ∆ V a=( 4.911±0.087 ) V where ∆
The slope of this graph is
so the acceleration voltage difference between adjacent maxima. The error obtained is random error in which results from statistical calculations. Therefore, by linear regression to the series of maxima, we have measured first excitation potential for mercury atoms which is 4.911 ±0.087V . We We have measured a good value in comparison with theoretical value which is 4.9 V and is inside the error boundaries.
FIG 3: Acceleration voltage Va increasing linearly with number of peak. Method least-square fit was used and standard error obtained is 3.474 V. Next we try to observe the series of minima and perform linear regression with the number of order for minima as in FIG 4. Acceleration voltage is linearly proportional to the order of minima and with equation y =( 5.239±0.072 ) x +3.287 ±0.072 . Just like the maxima, the slope of this graph is FIG 2: Anode current current Ic versus Acceleration voltage Va which has been fitted with Lorentzian. It is the sum
∆V ∆ V a
which is 5.239± 0.072V .The theoretical value for this is slightly deviate outside the error boundaries
which might be estimated too optimisticall optimistically y when taking the measurement.
In this experiment we regulate the temperature at 443K. With With equation (3) we can obtain the pressure the gas whch is 797.595 Pa and we can use both temperature T and pressure obtained in equation (5) to obtain 13.597μm. 13.597μm. So the mean free path for the electrons is very small compared with distance from the cathode to anode due to the high temperature set in this experiment. Therefore we know that the electrons will undergo a sufficient number of collisions before reaching the grid and it will lose energy faster before it can contains higher energy values for higher excitation state mercure to occur. With this mean free path we can assume that higher energy levels than the first excitation state can be neglected.
V. Data and error analysis In FIG 1,w 1,wee use used d Lor Lorent entzia zian n fun functi ction on whi which ch is given giv en as FIG 4: Acceleration voltage Va increasing linearly with order of minima. Method least-square fit was used and standard error obtained is 3.706 V. As we have measured the excitation energy of the mercury atom, we found the spectral frequency corresponding to this energy by equation
f =
E 4.911eV = h 4.133×10−15 eV 15
= 1.188±0.021 ×10 Hz From frequency we can find wavelength:
λ =
c = 2.99×108 ÷1.188×1015 f
= 251.684 ± 4.373 nm The wavelength is corresponds to a strong line in the ultraviolet emission spectrum of mercury. C. Contact potential difference Contact potential difference is the difference between the work functions of the two plates of cathode and anode. This is due to the change of the amount of energy in the electrons as it lose some energy when being emitted by the cathode and gaine some when absorbed into the plate. We can substrace the average peak to peak voltage which is ∆ ∆V V a=( 4.911±0.087 ) V from the
acceleration voltage for the first peak which is 7.79V and the contact potential difference is 2.879 ±0.097 V . D. Mean free path of electrons
y = y 0+( 2
A w ' ). where π 4∣ x − x c 2∣+ w 2
A is area, w is width, xc is center and y0 is y-offset. From this function we obtain curve for each peak and adding up all together to curve-fit the Franck-Hertz curve. Thee ran Th range ge for our mea measur sured ed val value ue for the fir first st excitation potential energy of mercury atom is from 4.824V 4.824 V to 4.998 4.998V V whic which h incl includes udes theo theoreti retical cal value inside ins ide the err error or bou bounda ndarie ries. s. How Howeve everr for ser series ies of minima we measured the range of 5.167V to 5.311V which whi ch doe doesn' sn'tt inc includ ludee to the theore oretic tical al val value ue of 4.9 4.9V V inside ins ide it itss er error ror bou bounda ndarie ries. s. Our guess is tha thatt the devi de viat atio ion n is ca caus used ed by ou ourr la lack ck of da data ta po poin ints ts surrounding our minima and the reason for our maxima is mu much ch be bett tter er th than an mi mini nima ma is be beca caus usee we on only ly collectin coll ecting g many data points surr surroundi ounding ng the peak but neglecting the surrounding minima. As we have shown the linearly increasing between anode ano de pot potent ential ial Va Va for bot both h ser series ies of max maxim imaa and minima, we can assume that both maxima and minima have equal acceleration potential difference and thus able to obtain accurate measurement for first excitation potential by using either one of them. Thee wa Th wave vele leng ngth th em emit itte ted d is in th thee ra rang ngee of 247.31 247 .311nm 1nm to 256 256.05 .057 7 nm whi which ch is loc locate ated d in the middle ultraviolet spectrum. This experiment could be improved by taking much more data points especially surrounding the maxima and minima and if we could determine whether the successive minima or maxima is equally spaced.
VI. Conclusion We have found an agreement with the accepted value of the excitation potential of mercury atom. For series of maxima we have measured the potential to be 4.911 ±0.087V an and d fo forr se seri ries es of mi mini nima ma is
5.239± 0.072V
. From FIG 3 and FIG 4, we havee sho hav shown wn tha thatt the acc accele elerat ration ion vol voltag tagee li linea nearly rly increasing with the order for both maxima and minima. Then Th en we ca can n as assu sume me th that at th thee di dist stan ance ce be betw twee een n successive maxima is equal with the distance between succes suc cessiv sivee mi minim nima. a. The Theref refore ore the fir first st exc excita itatio tion n potential can be measured by observing the series of maxima and minima. In this experiment, the regular spacings between maxima maxi ma in th thee pl plot ot of an anod odee cu curr rren entt Ic ag agai ains nstt accelerating voltage Va from FIG 1 have verifiedthe quanti qua ntizat zation ion of ato atomic mic ele electr ctron on ene energ rgy y sta states tes of mercury merc ury atom atomss that demonstrate demonstrate the tran transmis smission sion of energy by electron to atoms is in multple discrete. We also have measured the contact potential to be 2.879 ±0.097 V and observed that the electrons lose some of energy when it leaves cathode and gain some of energy when it absorbs anode. The mean free path was measured to be 13.587 μm which is very small sma ll val value ue in re relat lative ive wit with h the dis distan tance ce bet betwee ween n cathod cat hodee and ano anode de and we can neglect neglect the higher higher excitation state for mercury atoms. Other than that, we have verify the emission spectrum of mercury to be in middle ultraviolet spectrum by
measuring
its
wavelength which is 251.684 ±4.373 nm . So the spectrum of mercury should include a line whose wavelength is at that measured value.
7. References Modern n Physi Physics cs,, 2n [1] Tay aylor lor,, Za Zafir firato atoss & Dub Dubson son,, Moder 2nd d Edition, 2004: Section 5.06 & 5.10 [2] Melissinos A., Experiments A., Experiments in Modern Physics , Academic Press, May 1966.
Experiment , Description folder. [3] Franck-Hertz Experiment , [3] Franck-Hertz [4] Instruction Manual and Experiment Guide for the PASCO PASCO scientific Franck-Hertz Experiment. PASCO scientific. [5] Preston, Dietz., The Art of Experimental Physics, John Wiley & Sons, 1991. "Clapeyron on and [6]Salzman, [6] Salzman, William William R. (2001-08-21). "Clapeyr Clausius–ClapeyronEquations" . Chemical Universi rsity ty of Arizon Arizona. a. Archi Archived ved Thermodynamics. Thermodynamics. Unive from the original on 2007-07-0 2007-07-07. 7. Retrieved Retrieved 2007-102007-1011.
