# Irodov Solutions → Thermodynamics and Molecular Physics → The First Law of Thermodynamics. Heat Capacity

 2.26. Demonstrate that the interval energy U of the air in a room is independent of temperature provided the outside pressure p is constant. Calculate U, if p is equal to the normal atmospheric pressure and the room's volume is equal to V = 40 m3. Free solution >> 2.28. Two thermally insulated vessels 1 and 2 are filled with air and connected by a short tube equipped with a valve. The volumes of the vessels, the pressures and temperatures of air in them are known (V1, p1, T1 and V2, p2, T2). Find the air temperature and pressure established after the opening of the valve. Free solution >> 2.29. Gaseous hydrogen contained initially under standard conditions in a sealed vessel of volume V = 5.0 l was cooled by ΔT = 55 K. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas. Free solution >> 2.30. What amount of heat is to be transferred to nitrogen in the isobaric heating process for that gas to perform the work A = 2.0 J? Free solution >> 2.31. As a result of the isobaric heating by ΔT = 72 K one mole of a certain ideal gas obtains an amount of heat Q = 1.60 kJ. Find the work performed by the gas, the increment of its internal energy, and the value of γ = Cp/CV. Free solution >> 2.32. Two moles of a certain ideal gas at a temperature T0 = 300 K were cooled isochorically so that the gas pressure reduced n = 2.0 times. Then, as a result of the isobaric process, the gas expanded till its temperature got back to the initial value. Find the total amount of heat absorbed by the gas in this process. Free solution >> 2.33. Calculate the value of γ = Cp/CV for a gaseous mixture consisting of ν1 = 2.0 moles of oxygen and ν2 = 3.0 moles of carbon dioxide. The gases are assumed to be ideal. Free solution >> 2.34. Find the specific heat capacities cv and cp for a gaseous mixture consisting of 7.0 g of nitrogen and 20 g of argon. The gases are assumed to be ideal. Free solution >> 2.35. One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a temperature T. The space over the piston opens into the atmosphere. What work has to be performed in order to increase isothermally the gas volume under the piston n times by slowly raising the piston? The friction of the piston against the cylinder walls is negligibly small. Free solution >> 2.36. A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volume V0, in which an ideal gas is contained under the same pressure p0 and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas η times compared to that of the other by slowly moving the piston? Free solution >> 2.37. Three moles of an ideal gas being initially at a temperature T0 = 273 K were isothermally expanded n = 5.0 times its initial volume and then isochorically heated so that the pressure in the final state became equal to that in the initial state. The total amount of heat transferred to the gas during the process equals Q = 80 kJ. Find the ratio γ = Cp/CV for this gas. Free solution >> 2.39. One mole of oxygen being initially at a temperature T0 = 290 K is adiabatically compressed to increase its pressure η = 10.0 times. Find: (a) the gas temperature after the compression; (b) the work that has been performed on the gas. Free solution >> 2.40. A certain mass of nitrogen was compressed η = 5.0 times (in terms of volume), first adiabatically, and then isothermally. In both cases the initial state of the gas was the same. Find the ratio of the respective works expended in each compression. Free solution >> 2.41. A heat-conducting piston can freely move inside a closed thermally insulated cylinder with an ideal gas. In equilibrium the piston divides the cylinder into two equal parts, the gas temperature being equal to T0. The piston is slowly displaced. Find the gas temperature as a function of the ratio η of the volumes of the greater and smaller sections. The adiabatic exponent of the gas is equal to γ. Free solution >> 2.42. Find the rate v with which helium flows out of a thermally insulated vessel into vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of helium in the vessel is T = 1,000 K. Free solution >> 2.43. The volume of one mole of an ideal gas with the adiabatic exponent γ is varied according to the law V = a/T, where a is a constant. Find the amount of heat obtained by the gas in this process if the gas temperature increased by ΔT. Free solution >> 2.44. Demonstrate that the process in which the work performed by an ideal gas is proportional to the corresponding increment of its internal energy is described by the equation pVn = const, where n is a constant. Free solution >> 2.45. Find the molar heat capacity of an ideal gas in a polytropic process pVn = const if the adiabatic exponent of the gas is equal to γ. At what values of the polytropic constant n will the heat capacity of the gas be negative? Free solution >> 2.46. In a certain polytropic process the volume of argon was increased α = 4.0 times. Simultaneously, the pressure decreased β = 8.0 times. Find the molar heat capacity of argon in this process, assuming the gas to be ideal. Free solution >> 2.47. One mole of argon is expanded polytropically, the polytropic constant being n = 1.50. In the process, the gas temperature changes by ΔT = -26 K. Find: (a) the amount of heat obtained by the gas; (b) the work performed by the gas. Free solution >> 2.48. An ideal gas whose adiabatic exponent equals γ is expanded according to the law p = αV, where α is constant. The initial volume of the gas is equal to V0. As a result of expansion the volume increases η times. Find: (a) the increment of the internal energy of the gas; (b) the work performed by the gas; (c) the molar heat capacity of the gas in the process. Free solution >> 2.49. An ideal gas whose adiabatic exponent equals γ is expanded so that the amount of heat transferred to the gas is equal to the decrease of its internal energy. Find: (a) the molar heat capacity of the gas in this process; (b) the equation of the process in the variables T, V; (c) the work performed by one mole of the gas when its volume increases η times if the initial temperature of the gas is T0. Free solution >> 2.51. An ideal gas with the adiabatic exponent γ undergoes a process in which its internal energy relates to the volume as U = aVα, where a and α are constants. Find: (a) the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by ΔU; (b) the molar heat capacity of the gas in this process. Free solution >> 2.52. An ideal gas has a molar heat capacity Cv at constant volume. Find the molar heat capacity of this gas as a function of its volume V, if the gas undergoes the following process: (a) T = T0eαV; (b) p = p0eαV, where T0, p0 and α are constants. Free solution >> 2.53. One mole of an ideal gas whose adiabatic exponent equals γ undergoes a process p = p0 + α/V, where p0 and α are positive constants. Find: (a) heat capacity of the gas as a function of its volume; (b) the internal energy increment of the gas, the work performed by it, and the amount of heat transferred to the gas, if its volume increased from V1 to V2. Free solution >> 2.54. One mole of an ideal gas with heat capacity at constant pressure Cp undergoes the process T = T0 + αV, where T0 and α are constants. Find: (a) heat capacity of the gas as a function of its volume; (b) the amount of heat transferred to the gas, if its volume increased from V1 to V2. Free solution >> 2.55. For the case of an ideal gas find the equation of the process (in the variables T, V) in which the molar heat capacity varies as: (a) C = CV + αT; (b) C = CV + βV; (c) C = CV + ap, where α, β and a are constants. Free solution >> 2.56. An ideal gas has an adiabatic exponent γ. In some process its molar heat capacity varies as C = α/T, where α is a constant. Find: (a) the work performed by one mole of the gas during its heating from the temperature T0 to the temperature η times higher; (b) the equation of the process in the variables p, V. Free solution >> 2.57. Find the work performed by one mole of a Van der Waals gas during its isothermal expansion from the volume V1 to V2 at a temperature T. Free solution >> 2.60. Two thermally insulated vessels are interconnected by a tube equipped with a valve. One vessel of volume V1 = 10 l contains ν = 2.5 moles of carbon dioxide. The other vessel of volume V2 = 100 l is evacuated. The valve having been opened, the gas adiabatically expanded. Assuming the gas to obey the Van der Waals equation, find its temperature change accompanying the expansion. Free solution >>