ABSOLUTE ZERO

Transcription

1 ABSOLUE ZEO INODUCION As you may know, for an ideal gas at constant volume, the relation between its ressure and its temerature t, measured in 0 C, is given by: t m b () hus a lot of t versus will be a straight line with sloe m and intercet b. his relation, called Charles Law, is aroximately true for all gases rovided the ressure is reasonably small. Different gases will have different values for the sloe. However, the extraolated value of the temerature when the ressure is zero, the intercet, turns out to be the same for all gases. Since the ressure cannot have a value less than zero, this temerature is the minimum value any gas (or any object) may have, and is called Absolute Zero. his in turn defines an absolute temerature scale, the Kelvin. Kelvin t( 0 C) Absolute Zero () In this exeriment you will find the value of Absolute Zero by taking temerature, ressure data for a fixed quantity of gas held at constant volume in a Pyrex glass bulb. In everything that follows, temeratures measured in Kelvin are given the symbol, while temeratures in Celsius are indicated by t. HE EXPEIMEN You will measure the ressure and temerature of the gas in the bulb for temeratures from 0 0 C to about 00 0 C. Note that the volume of the gas is at least aroximately constant; the constancy of the volume is exlored further in the ANALYSIS section below. he bulb is laced in a bath of water, whose temerature is controlled by adding ice to the bath or heating it with the hot late. he temerature of the bath is measured with the sulied thermometer. ABSOLUE ZEO

2 he thermometer is a Partial Immersion tye, which means that it will give accurate readings only if it is immersed in the bath to the line indicated on the stem near the bulb, with the stem remaining at room temerature. Pay articular attention to the measured temeratures when the bath is a mixture of water and melting ice and when the water in the bath is boiling. he manometer you will use to measure the ressure of the gas in the glass bulb has a secified accuracy of ± 0.5% of the reading ± 5 digits. he last art of the secification means that the last digit read is uncertain by ± 5. For examle, if the manometer reads: mm of mercury (orr) then 0.5% of the reading is.8 orr. hus the total error of accuracy is orr, and the ressure is: ±.3 orr he most difficult art of the exerimental rocedure is maniulating the hot late controls so that the temerature of the bath remains constant long enough for the temerature of the gas in the bulb to equal the temerature of the bath as measured by the thermometer. When equilibrium is achieved the ressure will be constant in time. he difficulty of achieving this condition will be one factor in determining the errors you should assign to your measurements. here are three data oints where thermal equilibrium is easy to achieve. One is when the bath is at room temerature, a second is when the bath is a mixture of water and melting ice, and the third is when the water in the bath is boiling. Measure atmosheric ressure with the barometer in MP5 or MP6 when you begin to take data and again when you are finished. Note the room temerature with the thermometer beside the barometer for both measurements. ANALYSIS he first, most simle analysis of the data will involve fitting the temerature versus ressure data to a straight line. he intercet of the fit is your exerimental value of Absolute Zero. As a first term Core exeriment you should also calibrate the thermometer as described below. As a second term exeriment, the other calibrations described below may also be alied to your data. ABSOLUE ZEO

3 Calibration of the hermometer You may calibrate the thermometer using the measured temeratures for boiling water and for a mixture of water and melting ice. he boiling oint of water goes down as atmosheric ressure decreases. hus boiling water on a mountainto has a lower temerature than boiling water at sea level. he boiling ointt B, in Celsius, at a given atmosheric ressure H is given by: t ( H 760) (3) B H C is measured in mm of mercury (orr) at 0 Celsius. his is the reading you obtained from the barometer adjusted to 0 Celsius using the table asted alongside the barometer. he factor is an exerimentally determined number for the change in the boiling oint of water er orr difference in ressure between the observed ressure and the Standard Pressure. You may assume that the actual temerature of the water and melting ice is 0 Celsius rovided equilibrium has been achieved. Is there an areciable difference between the exected values of the boiling oint and the freezing oint of water, and the values you actually read on the thermometer? If so, you may need to correct your temerature values. It is reasonable to assume a linear correction and you should devise one that gives you the correct values at both calibration oints. C C Volume Measurement In the exeriment you will have noticed that not all of the traed air is heated when you heat the water around the bulb. he air in the tubes connecting the bulb to the manometer resumably stays close to room temerature. o take this effect into account we do the following calculation. Suose there are n moles of traed air, n being in the heated region itself and n being in the connecting tube, most of which is aroximately at room temerature. As the air is heated, it becomes less dense in the bulb relative to the tube, and therefore n decreases and n increases, but n remains constant: n n n (4) For the bulb, with volume V : V = n (5) ABSOLUE ZEO

4 whereas for the connecting volume V : Here V n (6) is the room temerature in Kelvin. We eliminate n and n : V V n (7) his looks like the normal ideal gas law excet for the second term on the left hand side, which should be smaller than the other two terms. earranging, we obtain: n V (8) V V However we must consider one more correction. Exansion of the Pyrex Glass Bulb Given the volume V at room temerature, the coefficient of volume exansion gives the volume V at temerature according to: V V ( )] (9) [ Using this to correct for the volume of the bulbv, Equation 8 becomes n V V (0) V ( ) V ( ) hus a lot of / versus / will be a straight line if the ideal gas law holds over the temerature range exlored in this exeriment. Further, the general form Equation 0 is: () where: n (a) V ( ) ABSOLUE ZEO

5 V V (b) V ( ) his may be rewritten by rearranging and using a binomial exansion to get: ( ) / ( ) 3 (3) If this becomes: (4) Since, the absolute temerature, is the temerature t in Celsius minus the value of Absolute Zero, this equation can be written as: or: t b (5a) b t ( ) (5b) where b is the value of Absolute Zero. Fitting t versus to a second order olynomial and using Equation 5a to find Absolute Zero is unlikely to give a good result. his is in art because the fitter will fit to values of / and /, although they are not free arameters but instead are numbers that you may calculate. hus, you should use Equation 5b to determine the value of b by: Estimating V, V, and n and calculating and. Note that this will use your first, simle, determination for Absolute Zero in order to find. You should be very careful about units for the ressure, the gas constant, etc. Calculating b for each of your data oints. Averaging the result of each calculation of b to give a final value of Absolute Zero. CONSANS Density of mercury at 0 O C = g/cm Coefficient of volume exansion of Pyrex glass deg mole of any gas occuies.4 liters at SP (Standard emerature and Pressure). Universal gas constant = 8.3 J/(mole K) 3 mole = gram molecular weight = 0 kg molecular weight ABSOLUE ZEO

6 - 4 - PEPAAOY QUESIONS. Above we refer to ressures in units of orr, which is the ressure exerted by a cylinder of mercury, as shown to the right, of height L = mm. Assume the area A of the cylinder is square meter. What is that ressure in units of Pascals = Newton/meter exerted by the column of mercury?. Does your answer to Question deend on the area A of the column? Why? Does it deend on the column being circular? Why? 3. At what deth below the surface of a lake will you exerience a ressure twice the ressure you exerience on the surface? 4. If you are on to of a mountain where atmosheric ressure is 740 orr, what is the boiling oint of water? he answer to this question exlains why you can t make a good cu of tea on to of a mountain. 5. he temerature of a gas is a measure of the average kinetic energy of the molecules of the gas. What is the average kinetic energy of the gas molecules when the temerature of the gas is 0 Kelvin? 6. Is it meaningful to talk about the temerature of a erfect vacuum? his is otentially a somewhat subtle question. EFEENCE Almost any first year university level hysics textbook. AUHOS his Guide Sheet was written by David M. Harrison, June 00. he art on calibrations and corrections is based on material written by Derek A.L. in 990. Milton From and ony Key contributed to revious versions of the Guide Sheet in 995 and 996, 998 resectively. ABSOLUE ZEO