QUADRATURE OSCILLATORS
3.1. Quadrature Oscillator Topologies
An ideal gas is one that would obey the gas laws and the ideal gas equation under all conditions – so ideal gases cannot be liquefied. In 1863, the Irish physician and chemist Thomas Andrews succeeded in plotting a series of p–V curves for carbon dioxide;
these curves deviated from the Boyle’s law curves at high pressures and low temperatures. Until this time it had been believed that certain gases could never be liquefied. Andrews showed that there was a critical temperature above which the gas could not be liquefied by simply increasing the pressure.
He demonstrated that for carbon dioxide the critical temperature is approximately 31 °C.
The difference between ideal gases and real gases
The 19th century experimenters showed that ideal gases are just that – ideal – and that real gases do not behave as ideal gases. Under high pressures and low temperatures all gases can be liquefied and thus become almost incompressible. This can be seen if we plot a graph of ___ pVRT against p for one mole of a real gas – we would expect this plot to give a horizontal straight line of value of 1.00 for
an ideal gas. Figure 10 is such a plot for nitrogen.
The pressure axis is in atmospheres – this is a non-SI unit that is appropriate for high pressures and quite commonly used by experimenters (e.g.
900 atm converts to 900 × 1.01 × 105 Pa or 9.1 × 107 Pa). The ideal gas line is a better fit to the real gas line at 1000 K and better still at low pressure.
Figure 9 Maxwell–Boltzmann distribution for the speed of gas molecules.
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
200 400 600 800 1000 1200 1400
relative frequency
speed/m s−1
Maxwell−Boltzmann speed distribution
0
273 K 373 K 473 K
Figure 10 Deviation of a real gas from an ideal gas situation.
300
200 K 500 K
1000 K ideal gas
0 1 2 3
600 900
P (atm) PV
RT
3 . 2 M O D E L L I N G A G A S
Thus we can see that the best approximation of a real gas to an ideal gas is at high temperature and low pressures. In deriving the ideal gas equation we made assumptions that are not true for a real gas. In particular we said that “there are no intermolecular forces between the molecules in between collisions.” This is not true for real gases on two accounts:
● Short-range repulsive forces act between gas molecules when they approach each other – thereby reducing the effective distance in which they can move freely (and so actually reducing the useful gas volume below the value of V that we considered).
● At slightly greater distances the molecules will exert an attractive force on each other – this causes the formation of small groups
of loosely attached molecules. This in turn reduces the effective number of particles in the gas. With the groups of molecules at the same temperature as the rest of the gas, they have the same translational kinetic energy and momentum as other groups of molecules or individual molecules. The overall effect of intermolecular attraction is to reduce the pressure slightly.
The Dutch physicist Johannes van der Waals was awarded the 1910 Nobel Physics Prize for his work in developing a real gas equation, which was a modification of the ideal gas equation. In this equation two additional constants were included and each of these differed from gas to gas. In the ensuing years many have tried to formulate a single simple equation of state which applies to all gases, no one has been successful to date.
TOK
Empirical versus theoretical models
Theoretical models such as the kinetic theory of gas are based on certain assumptions. If these assumptions are not met in the real world, the model may fail to predict a correct outcome as is the case with real gases at high pressure and low temperature.
Despite the best efforts of van der Waals and others, there is no theorectical model that gives the right answer for all gases in all states. This, however, does not mean that we cannot make correct predictions about a particular gas – we can use an empirical model that is based on how the gas has behaved in these circumstances previously. We can sensibly expect that if the conditions are repeated then the gas will behave in the same way again. In the 17th century, with no understanding of the nature of gas molecules, a theoretical model of gases was not possible and so Boyle, Charles, and others could only formulate empirical laws that were repeatable with the precision that their apparatus allowed. In the 21st century we still rely on empirical methods when gases are not behaving ideally; even when the van der Waals modification of the equation of state for an ideal gas is used, the constants that need to be included are different from gas to gas and determined experimentally. So even though it is, perhaps, aesthetically pleasing to have a “one size fits all” equation, the reality is that theoretical models rarely account for all circumstances. Even Newton’s laws of motion must be
modified when objects are moving at speeds approaching that of light. Theoretical models are not inherently better than empirical models, or vice versa. The best model, whether it is theoretical or empirical, is the model that gives the best predictions for a particular set of circumstances.
r When a set of experimental results are not in line with theory does this mean that the theory must be abandoned?
r In March 2012 the following report appeared on the BBC website:
“The head of an experiment that appeared to show subatomic particles travelling faster than the speed of light has resigned from his post…..
Earlier in March, a repeat experiment found that the particles, known as neutrinos, did not exceed light speed. When the results from the Opera group at the Gran Sasso underground laboratory in Italy were first published last year, they shocked the world, threatening to up-end a century of physics as well as relativity theory – which holds the speed of light to be the Universe’s absolute speed limit.”
The media were very quick to applaud the experiment when the initial results were published, but the scientific community was less quick to abandon a well-established “law”. Which of the two groups was showing bias in this case?
Questions
1 Two objects are in thermal contact. State and explain which of the following quantities will determine the direction of the transfer of energy between these objects?
a) The mass of each object.
b) The area of contact between the objects.
c) The specific heat capacity of each object.
d) The temperature of each object. (4 marks)
2 Two objects are at the same temperature.
Explain why they must have the same internal
energy. (2 marks)
3 The internal energy of a piece of copper is increased by heating.
a) Explain what is meant, in this context, by internal energy and heating.
b) The piece of copper has mass 0.25 kg. The increase in internal energy of the copper is 1.2 × 103 J and its increase in temperature is 20 K. Estimate the specific heat capacity
of copper. (4 marks)
4 Calculate the amount of energy needed to raise the temperature of 3.0 kg of steel from 20 °C to 120 °C. The specific heat capacity of steel is
490 J kg–1 K–1. (2 marks)
5 Calculate the energy supplied to 0.070 kg of water contained in a copper cup of mass 0.080 kg. The temperature of the water and the cup increases from 17 °C to 25 °C.
Specific heat capacity of water = 4200 J kg–1K–1 Specific heat capacity of copper = 390Jkg–1 K–1
(2 marks)
6 The temperature difference between the inlet and the outlet of an air-cooled engine is 30.0 K. The engine generates 7.0 kW of waste power that the air extracts from the engine.
Calculate the rate of flow of air (in kg s–1) needed to extract this power.
Specific heat capacity of air (at constant pressure)
= 1.01 × 103 J kg–1 K–1 (3 marks)
7 2.0 kg of water at 0 °C is to be changed into ice at this temperature. The same mass of water now at 100 °C is to be changed into steam at this temperature.
Specific latent heat of fusion of water = 3.34 × 105 J kg–1
Specific latent heat of vaporization of water = 2.26 × 106 J kg–1
a) Calculate the amount of energy needed to be removed from the water to freeze it.
b) Calculate the amount of energy required by the water to vaporize it.
c) Explain the difference between the values calculated in a) and b). (6 marks)
8 (IB) A container holds 20 g of neon and also 8 g of helium. The molar mass of neon is 20 g and that of helium is 4 g.
Calculate the ratio of the number of atoms of neon to the number of atoms of helium.
(2 marks)
9 A fixed mass of an ideal gas is heated at constant volume. Sketch a graph to show the variation with Celsius temperature t with pressure p of the gas? (3 marks)
10 Under what conditions does the equation of state for an ideal gas, pV = nRT, apply to a
real gas? (2 marks)
11 a) (i) Explain the difference between an ideal gas and a real gas.
(ii) Explain why the internal energy of an ideal gas comprises of kinetic energy only.
b) A fixed mass of an ideal gas has a volume of 870 cm3 at a pressure of 1.00 × 105 Pa and a temperature of 20.0 °C. The gas is heated at constant pressure to a temperature of 21.0 °C.
Calculate the change in volume of the gas.
(6 marks)
Q U E S T I O N S
a) The pressure p of a fixed mass of an ideal gas is directly proportional to the kelvin temperature T of the gas, that is, p ∝ T.
(i) State the relation between the pressure p and the volume V for a change at constant temperature.
(ii) State the relation between the volume V and kelvin temperature T for a change at a constant pressure.
b) The ideal gas is held in a cylinder by a moveable piston. The pressure of the gas is p
1, its volume is V
1 and its kelvin temperature is T
1.
The pressure, volume and temperature are changed to p
2, V
2 and T
2 respectively. The change is brought about as illustrated below.
p1, V1, T1 p2, V1, T' p2, V2, T2
heated at constant volume to pressure p2 and temperature T'
heated at constant pressure to volume V2 and temperature T2
(i) p1, p2, T1 and T ′ (ii) V1, V2, T′ and T2
c) Use your answers to b) to deduce, that for an ideal gas
pV = KT,
where K is a constant. (6 marks)
13 A helium balloon has a volume of 0.25 m3 when it is released at ground level. The temperature is 30 °C and the pressure
1.01 ×105 Pa. The balloon reaches a height such that its temperature has fallen to –10 °C and its pressure to 0.65 × 105 Pa.
a) Calculate the new volume of the balloon.
b) State two assumptions that must be made about the helium in the balloon.
c) Calculate the number of moles of helium in
the balloon. (6 marks)
4 O S C I L L AT I O N S A N D WAV E S
Introduction
All motion is either periodic or non-periodic. In periodic motion an object repeats its pattern of motion at fixed intervals of time: it is regular and repeated. Wave motion is also periodic and
there are many similarities between oscillations and waves; in this topic we will consider the common features but also see that there are differences.