Home Physics Thermodynamics Mix The figure below shows an apparatus for the …
Physics Thermodynamics Mix MCQ (Single Correct)

The figure below shows an apparatus for the determination of C p /C v for a gas, according to the method of Clement and Desormes. A bottle G, of reasonable capacity (say a few liters), is fitted with a tap H, and a manometer M. The difference in pressure between the inside and the outside can thus be determined by observation of the difference h in heights of the two columns in the manometer. Thue bottle is filled with the gas to be investigated, at a very slight excess pressure over the outside atmospheric pressure. The bottle is left in peace (with the tap closed) until the temperature of the gas in the bottle is the same as the same the outside temperature in the room. Let the reading of the manometer be h 1 . The tap H is then opened for a very short time, just sufficient for the internal pressure to become equal to the atmospheric pressure (in which case the manometer reads h = 0). With the tap closed the bottle is left in peace for a while, until the inside temperature has become equal to the outside temperature. Let the final reading of the manometer be h. From the values of h i and h f it is possible to find C p /C v .

A
Derive an expression for C p /C v in terms of h i and h f in the above experiment.
B
Suppose that the gas in question is oxygen. What is your theoretical prediction for C p /C v at 20ºC, within the framework of statistical mechanics?

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Sol. The equation of state of ideal gas is pV = nkT. Since the initial and final T, V of the gas in the bottle are the same, we have p f /p i = n f /n i .

Meanwhile, n f /n i = V/V', where V' is the volume when the initial gas in the bottle expands adiabatically to pressure p 0 . Therefore

= . = ,

γ = = .

Since h i /h 0 < 1 and h f /h 0 < 1, we have γ = h i /(h i – h f ).

Oxygen consists of diatomic molecules. When t = 20ºC, only the translational and rotational motions of the molecules contribute to the specific heat. Therefore

C v = , C p = , γ = .

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