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Home , Monatomic gas

Course 1 Laboratory Second Semester

Experiment: Specific Heats Ratio

1 Ratio of Specific Heats C p/C v for

1 Some Theory

The ratio of the specific heat at constant pressure to that at constant volume, C p/C v = γ, is an important quantity in thermal . The magnitude of γ for a , on the basis of classical theory, reveals information about the complexity of the . So a measurement of C p/C v reveals the molecular nature of the gas, e.g. monatomic, diatomic, etc… This relationship may be obtained from the gas law assuming a random distribution of velocities of the molecules 1 1 PV = R T = Mv2 = N m v 2 , (1) 3 3 where M is the total mass of gas in a volume V, N denotes the total number of gas particles in the mass M, and their average effective velocity is v. It is convenient to take the mass M to be the mass of a mole of the gas and to introduce the average kinetic energy of the gas particles ( KE ) 1 2 1 2 RT = Nm v2 = N m v2 = ( KE ). (2) 3 A 3 A 2 3 This gives 1 3 R mv2 = T (3) 2 2 NA showing that the average kinetic energy of a is proportional to the y of gas. The rate R/N A = k is the , R = molar , and N A = the Avogadro number. For a monatomic gas ( KE ) in (2) denotes the total , since only translational x motion in three dimensions is possible. It is z customary to associate three degrees of freedom ( f = 3) with the kinetic energy. The The three possible directions with which a monatomic molecule may translate. energy per degree of freedom is therefore 1 1 (KE ) = R T . (4) 3 2 In general, for f degrees of freedom the total energy U by the equipartition principle is f (5). U = R T 2

The molecular specific heat at constant volume (C v), i.e. the rate of the change of energy U with temperature T becomes

2 ∂U  R   Cv =   = f . (6) ∂T v 2

Since C p − Cv = R,  f  C = C + R =  + 1 R . (7) p v  2  Dividing equation (7) by equation (6) gives the expression for γ 2 γ = C / C = 1 + . (8) p v f For monatomic gases such as the noble gases , , , and , as well as for metallic vapours, only translational motion is possible so that f = 3. This gives γ = 1.66, a result in agreement with experiment. The value of γ observed for diatomic gases is usually 1.4, indicating a value of f equal to 5.

Rotational degrees of freedom of a . Note there is no rotation around the z-axis as on the quantum scale such rotations do not change the energy state of the molecule.

This can be explained by assuming that a diatomic molecule is of a form of a dumbbell. The additional degrees of freedom are considered as due to the possibility of rotation about two axes mutually at right angles to each other and to a line through the centre of the . For many triatomic molecules the value of γ is 1.33 or less, indicating the existence of additional degrees of freedom. It is also found that at higher the value of γ becomes smaller. This can be explained by the introduction of vibrational degrees of freedom of the molecule. The method of measurement of the specific heat ratio γ relies on the assumption that the changes of volume and pressure of gas in the closed tube containing a vibrating piston are reversible adiabatic (i.e. heat is not lost from the system), and for a given mass of gas they are described by the equation γ pV = constant (9) The isothermal change is described by pV = constant and the gradient of the isothermal curve at any point is obtained by differentiation dp p = - . dV V

3 The gradient of the adiabatic curve at any point is obtained by differentiation of (9) dp γp = - . (10) dV V 2 Apparatus The apparatus consists of a precision bore glass tube containing a freely moving light aluminium alloy piston which carries a small cylindrical bar magnet. The cylinder is kept in the tube at a Annular Bar Piston Magnet height of the coil external to the tube by means of a magnet attached to the coil. The piston is set Coils into oscillation by the magnet/coil system energised by a suitable frequency current from a Magnet signal generator through an amplifier. The Glass apparatus drawn not to scale is shown in Figure 1, Tube and the circuit connections are shown in Figure 2.

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Coil Figure 1. The C /C apparatus (Not to Signal Frequency p v Generator Meter Amplifier scale). Figure 2. Circuit Connections The glass tube is clamped in a vertical position in a laboratory stand, not shown in the figure. The glass tube is positioned in the centre of the coil. The annular magnet in the coil assembly supports the piston at the height of the coil. The ac. current from the signal generator is passed through the coils causing the piston to oscillate, the amplitude of oscillation being greatest when the frequency of the generator output coincides with frequency of the natural mechanical resonance of the system.

3 The Bangor method of C p/C v measurements The method has been developed by Clark and Kate in 1940 at the University College of North Wales, Bangor. It is based on the fact that the velocity of sound in a gas depends on the specific heats ratio γ. The rapid compressions and rarefactions in a train of sound waves are considered to be adiabatic. As an extension of this idea Clark and Kate assumed that the resonant frequency in a system described in preceding section is also dependent upon γ as similar adiabatic process takes place. Three modes of resonance are possible in the system: (a) with both ends of the tube closed, (b) with one end open and (c) with both ends open. Consider arrangement (a) with both ends of the tube closed by rubber bungs, and the system oscillating in a simple harmonic motion. The restoring force on the piston will be at any instant:

− (Ks + 2 K)y (11) where Ks is the force constant of the supporting magnet field

4 K is the force constant of the gas on one side of the piston y is the displacement of the piston from its resting position at that instant. Equation (10) in the preceding section can be written dp dV = - γ . p V Ky Also dp = and dV = - Ay A where A is the cross-sectional area of the tube and V is the volume of gas on one side of the piston Combining above equations gives γpA 2 K = (12) V The resonance frequency of the system is 1 KK+ 2 f = s (13) 1 2π M where M is the mass of the piston. For the arrangement (b) with one bung in place, 2 K is reduced to K as the restoring force of the gas now acts on one side of the piston and the new, lower resonance frequency is 1 KK+ f = s (14) 2 2π M If both bungs are removed or the tube evacuated, arrangement (c), the restoring force is only due to the fixed fields of the magnets and the resonance frequency is 1 K f = s . (15) 3 2π M Combining equations (12), (13) and (14) gives 4π 2 MV γ = ()f2 − f 2 (i) pA 2 1 2 or using equations (12), (13) and (15) 2π 2 MV γ = ()f2 − f 2 (ii) pA 2 1 3 Either of the above can be used to obtain γ, but for practical reasons it is found that equation (i) is best suited for investigating air at atmospheric pressure. Method (ii) may be used for any gas.

5 4 Experimental Procedure • Measure the internal diameter of the glass tube using Vernier callipers and also its length. Weigh the piston: do not use the digital scales as it has a ferromagnetic pan which will affect the weight of the piston. • Place the glass tube in the clamp stand such that it is vertical and passes through the centre of the coil. Make sure that the centre mark is just above the coil. Clamp the tube in position but not too firmly. You should be able to slide the tube up and down for vertical position adjustment. • Place your hand over the bottom of the tube and drop the piston, open end down, into the tube. Allow the piston to come to rest at the magnet. Line the centre mark of the tube with the half volume mark on the piston. This ensures that the volume of air on either side of the piston is equal. • Fit the solid bung in the bottom of the tube and the bored bung at the top. Both bungs should be tight. Close the bored bung with the small bung. This helps to avoid compression of the air in the tube. • Connect up the circuit shown in figure 2. Ensure that all earths (colour coded black) are connected to the same point. The oscillator should be set to produce a sinusoidal signal. Set the range on the oscillator so that frequencies of 1-50 Hz can be scanned in a single sweep with greatest sensitivity. • Start the oscillator at approximately 40Hz and set the signal voltage so that the piston is just moving. Ensure that the piston is free to oscillate, slight adjustment of the vertical orientation of the tube may be necessary. Adjust the frequency until a resonance is found. Once on resonance, you may increase the signal voltage to produce movement of approximately 4mm. If the piston leaves the confines of the coil then the signal is too large. Monitor the amplifier output signal on the CRO to see if it remains sinusoidal; if not then lower the signal voltage. Make measurements of the oscillation amplitude as a function of frequency. Perform a fit to your data using the Mathcad peak fitting spreadsheet in order to find the resonant frequency.

• Locate the lower frequency resonances with the top bung removed ( f2) then both removed ( f3). You will need to reduce the signal voltage in each case. • Calculate γ using equations 15 i and ii including errors. Comment on the values obtained and also identify any potential sources of error. • If the closed tube were filled with Helium at atmospheric pressure and room temperature, what would be the resonant frequency?

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