PY2104 - Introduction to Thermodynamics and Statistical Physics ...

PY2104 - Introduction to Thermodynamics andStatistical PhysicsProblem Set 11) A lead block is dropped from a height of 10 meters to the ground. As itimpacts the ground, 40% of its kinetic energy is converted into heat. By howmuch does its temperature increases ?(Note: the specific heat capacity of lead is C = 130J/K/kg).2) A person living in a 4m x 5m x 5m room turns on a 100-W fan before heleaves the warm room at 100 kPa, 30°C, hoping that the room will be coolerwhen he comes back after 5 hours. Disregarding any heat transfer determinethe temperature he discovers when he comes back.(Assume the air to be an ideal gas with molar heat capacity C v = 3 2 R)3) Consider 1 mole of perfect gas at volume V 1 and pressure P 1 performing aquasistatic cycle which consists of the following four successive stages: (i)expansion to volume V 2 at constant pressure. (ii) Pressure drop from P 1 toP 2 at constant volume (V 2 ). (iii) Compression back to volume V 1 while thepressure (P 2 ) is kept constant; and (iv) increase of the pressure from P 2 toP 1 , with constant volume V 1 .Find: (i) the work done on the gas in the cycle, and (ii) the heat absorbedby the gas during this cycle.Hint: plot the states of the gas in a P −V diagram.4) Solveasimilarproblemtoproblem(3)above, butwiththefollowing4stages:(i) temperature increase from T 1 to T 2 at constant volume, V 1 ; (ii) isothermalexpansion from (T 2 , V 1 ) to (T 2 , V 2 ); (iii) cooling at a constant volume (V 2 )from T 2 to T 1 ; and (iv) isothermal compression back to (T 1 , V 1 ).5) A Carnot cycle is defined as the following cycle: (i) isothermal expansionfrom volume V 1 to volume V 2 at constant temperature (T 1 ); (ii) adiabaticexpansion from volume V 2 to volume V 3 (during which the temperature decreasesto T 2 ); (iii) isothermal compression from volume V 3 to volume V 4at constant temperature, T 2 ; and (iv) adiabatic compression back to theoriginal state of the system (T 1 , V 1 ).What is the work done on the gas during each of the steps, and during theentire cycle ?6) Consider a gas in a jar. (i) The particles that constitute the gas are inconstant motion: their position and velocity are not fixed in time, yet thepressure they excerpt on the vessel is. Is this gas in equilibrium? (ii) Thesame jar is sitting in the train to Dublin. During the journey, all particlesin the jar have a net velocity in the general direction of Dublin. Is the gasin equilibrium?7) Systems A, B, and C are gases with pressures and volumes P A ,V A ; P B ,V B ;and P C ,V C . When A and C are in thermal equilibrium, one finds empiricallythat the equationP A V A −P C V C = nP A

is satisfied. When B and C are in equilibrium, the relationholds. Here, n and n ′ are constants.P B V B −P C V C = − n′ P C V CV B(i) What are the three functions of state which are equal to one another atthermal equilibrium? Argue that these functions can be called “the empiricaltemperature”, T . Is any of these gases ideal?(ii) What is the relation expressing thermal equilibrium between gases A andB ?