## Saturday, 6 September 2014

### ISOTHERMAL AND ADIABATIC CHANGES PROBLEMS

1. An ideal gas undergoes isothermal expansion at 0 oC from 0.010 m3 to 0.200 m3. For 5 moles of gas, cal­culate
(a) The work done,
(c) the change in internal energy.

2. A gas cylinder contains 6.4 kg of oxygen at a pressure of 5 atmospheres. An exactly similar cylinder contains 4.2 kg of nitrogen at the same temperature. What is the pressure of the nitrogen? (Relative molecular masses: oxygen = 32, nitrogen = 28. Assume that each behaves as a perfect gas.)

3. Air at 20 oC is allowed to expand under adiabatic conditions until its pressure has fallen to one-third of its original value. What is the final temperature of the air?
(Ratio of the principal specific heats of air = 1.4)

4. A certain volume of helium at 15 oC is expanded adiabatically until its volume is trebled. Calculate the temperature of the gas immediately after the expansion has taken place. (Ratio of the principal specific heat capacities of helium = 1.67)

5. Some nitrogen at 20 oC is expanded isothermally to twice its original volume. It is then cooled at constant volume to –10 oC and finally compressed adiabati­cally to its original pressure and volume,
(a) Draw a PV diagram to show this change.
(b)  Calculate the pressure after the isothermal expansion.
(c)   Calculate the pressure after the cooling at constant volume.

6. The density of neon at 27 oC and a pressure of 1.00x105 Pa is 0.90 kg m-3. Calculate the mass of neon in a neon discharge tube of volume 250 cm3 if the pressure inside it is 1.00x105 Pa at a temp­erature of 150 oC.

7. Two identical glass bulbs are joined with a thin glass tube and filled with air which is initially at 20 oC. What will the pressure in the apparatus become if one bulb is immersed in steam and the other in melting ice?

8.  When a gas expands adiabatically it does work on its surroundings although there is no heat input to the gas. Where does this energy come from?

9.  A mass of 1 g of hydrogen at 20 oC and 105 Pa has its volume halved by an adiabatic change.
Calculate the change in the internal energy of the gas.
(Ideal gas constant = 8.31 J mol-1 ratio of the principal specific heats for hydrogen = 1.40)

10. A mass of air with an initial volume of 5 litres at a temperature of 0 oC and a pressure of 105 Pa is compressed isothermally to 2.5 litres, and then allowed to expand adiabatically to 12.5 litres.
(a) Show the process on a PV diagram and calculate the final pressure and temperature of the air.
(b) At volume was the pressure momentarily atmos­pheric? (Assume  for air = 1.40)

11.     State the first law of thermodynamics. How is the equation modified for:
(b) an input of heat at constant volume.
12.    Suppose that you wanted to increase the efficiency of a Carnot engine. Would it be better to increase the input temperature T1 or to decrease the output temperature T2 by an equal amount? Explain your answer.

13. A Carnot engine operates between two reservoirs at 300 K and 400 K. What is the efficiency of the cycle?

14. A Carnot engine whose low-temperature reservoir is at 280 K has an efficiency of 40 per cent. By how much must the temperature of the high-temperature reservoir be raised if the efficiency is to be increased to 50 per cent, the temperature of the low-tempera­ture reservoir being unchanged?

15.  A hot air balloon when inflated contains 1 500 m3 of hot air at atmospheric pressure (10-5 Pa). The tem­perature outside the envelope is 300 K and the mass of the balloon (excluding the hot air) and its passen­gers is 250 kg. Assuming that the air may be treated as an ideal gas of molar mass 29x10-3 kg mol-1 find the mean temperature to which the air in the balloon must be heated if the balloon is just to rise.
(Take the molar gas constant R to be 8.3 J mol-1 K-1.)
16. An ideal gas is trapped in a cylinder by a movable piston. Initially it occupies a volume of 8x10-3 m3 and exerts a pressure of 108 kPa.
(a) It first undergoes an isothermal expansion until its volume is 27 x 10-3m3. Calculate the new pressure.
(b) It is then compressed adiabatically back to a volume of 8x10-3 m3. Calculate the final pressure.
(c)   State what single process will bring the gas back to its initial condition.
(d)  Sketch and label the three stages on a PV diagram. (Ratio of the principal molar heat capacities of the gas = 5/3)

17. Two grams of helium gas at 27 oC are contained in a chamber by a frictionless piston at atmospheric pressure 1.0x 105 Pa. Calculate
(a) the number of molecules per unit volume in the chamber,
(b) their mean kinetic energy, and
(c) the internal energy of the gas.

Ten grams of neon gas are pumped into the chamber, which is maintained at constant volume. Calculate the final pressure in the chamber when both gases are present and their temperature is 27 0C. How are the values of the quantities in (a) to (c) changed?
The chamber, which has negligible heat capacity, is then thermally insulated and the gas is allowed to return to atmospheric pressure slowly and adia­batically. Treating the gas as ideal, explain qualitatively whether the internal energy and temperature of the gas will rise, fall or stay the same.

Why is such a cumbersome device as the constant volume gas thermometer used as the standard ther­mometer over a range of temperatures? Why are several different pressures of gas used in the ther­mometer to ensure a correct temperature measure­ment?
(R= 8.3 J mol-1 K-1; L = 6.0x1023 mol-1  MHe = 0.004 kg mol-1; MNe = 0.020 kg mol-1)

18. Air under a pressure of 106 Pa in a closed vessel is suddenly released by opening and closing a tap.
(a) If the external pressure is 1 atmosphere, what fraction of the original air remains in the vessel?
If the initial temperature is 300 K, find
(b) the temperature just after the tap is closed,
(c) the pressure some time after the tap is closed.