**ISOTHERMAL AND ADIABATIC CHANGES**

**1. An ideal gas undergoes isothermal expansion at 0**

^{o}C from 0.010 m^{3 }to 0.200 m^{3}. For 5 moles of gas, calculate**(a) The work done,**

**(b) the heat added, and**

**(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**

^{o}C 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**

^{o}C 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**

^{o}C is expanded isothermally to twice its original volume. It is then cooled at constant volume to –10^{o}C 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**

^{o}C and a pressure of 1.00x10^{5}Pa is 0.90 kg m^{-3}. Calculate the mass of neon in a neon discharge tube of volume 250 cm^{3}if the pressure inside it is 1.00x10^{5 }Pa at a temperature of 150^{o}C.**7. Two identical glass bulbs are joined with a thin glass tube and filled with air which is initially at 20**

^{o}C. 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**

^{o}C and 10^{5 }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**

^{o}C and a pressure of 10^{5}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:**

**(a) an adiabatic change,**

**(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 T**

_{1 }or to decrease the output temperature T_{2}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 m**

^{3 }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}m^{3}and exerts a pressure of 108 kPa.**(a) It first undergoes an isothermal expansion until its volume is 27 x 10**

^{-3}m^{3}. Calculate the new pressure.**(b) It is then compressed adiabatically back to a volume of 8x10**

^{-3}m^{3}. 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**

^{o}C are contained in a chamber by a frictionless piston at atmospheric pressure 1.0x 10^{5 }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**

^{0}C. 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 = 6.0x10^{23}mol^{-1 }M_{He}= 0.004 kg mol^{-1}; M_{Ne}= 0.020 kg mol^{-1})**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.**