# Irodov Solutions → Thermodynamics and Molecular Physics → The Second Law of Thermodynamics. Entropy

 2.113. In which case will the efficiency of a Carnot cycle be higher: when the hot body temperature is increased by ΔT, or when the cold body temperature is decreased by the same magnitude? Free solution >> 2.114. Hydrogen is used in a Carnot cycle as a working substance. Find the efficiency of the cycle, if as a result of an adiabatic expansion (a) the gas volume increases n = 2.0 times; (b) the pressure decreases n = 2.0 times. Free solution >> 2.115. A heat engine employing a Carnot cycle with an efficiency of η = 10% is used as a refrigerating machine, the thermal reservoirs being the same. Find its refrigerating efficiency ε. Free solution >> 2.116. An ideal gas goes through a cycle consisting of alternate isothermal and adiabatic curves (Fig. 2.2). The isothermal processes proceed at the temperatures T1, T2, and T3. Find the efficiency of such a cycle, if in each isothermal expansion the gas volume increases in the same proportion. Free solution >> 2.117. Find the efficiency of a cycle consisting of two isochoric and two adiabatic lines, if the volume of the ideal gas changes n = 10 times within the cycle. The working substance is nitrogen. Free solution >> 2.120. An ideal gas goes through a cycle consisting of (a) isochoric, adiabatic, and isothermal lines; (b) isobaric, adiabatic, and isothermal lines, with the isothermal process proceeding at the minimum temperature of the whole cycle. Find the efficiency of each cycle if the absolute temperature varies n-fold within the cycle. Free solution >> 2.121. The conditions are the same as in the foregoing problem with the exception that the isothermal process proceeds at the maximum temperature of the whole cycle. Free solution >> 2.124. Calculate the efficiency of a cycle consisting of isothermal, isobaric, and isochoric lines, if in the isothermal process the volume of the ideal gas with the adiabatic exponent γ (a) increases n-fold; (b) decreases n-fold. Free solution >> 2.128. Making use of the Clausius inequality, demonstrate that all cycles having the same maximum temperature Tmax and the same minimum temperature Tmin are less efficient compared to the Garnot cycle with the same Tmax and Tmin. Free solution >> 2.131. The entropy of ν = 4.0 moles of an ideal gas increases by ΔS = 23 J/K due to the isothermal expansion. How many times should the volume ν = 4.0 moles of the gas be increased? Free solution >> 2.132. Two moles of an ideal gas are cooled isochorically and then expanded isobarically to lower the gas temperature back to the initial value. Find the entropy increment of the gas if in this process the gas pressure changed n = 3.3 times. Free solution >> 2.133. Helium of mass m = 1.7 g is expanded adiabatically n = 3.0 times and then compressed isobarically down to the initial volume. Find the entropy increment of the gas in this process. Free solution >> 2.134. Find the entropy increment of ν = 2.0 moles of an ideal gas whose adiabatic exponent γ = 1.30 if, as a result of a certain process, the gas volume increased α = 2.0 times while the pressure dropped β = 3.0 times. Free solution >> 2.135. Vessels 1 and 2 contain ν = 1.2 moles of gaseous helium. The ratio of the vessels' volumes V2/V1 = α = 2.0, and the ratio of the absolute temperatures of helium in them T1/T2 = β = 1.5. Assuming the gas to be ideal, find the difference of gas entropies in these vessels, S2 - S1. Free solution >> 2.136. One mole of an ideal gas with the adiabatic exponent γ goes through a polytropic process as a result of which the absolute temperature of the gas increases τ-fold. The polytropic constant equals n. Find the entropy increment of the gas in this process. Free solution >> 2.137. The expansion process of ν = 2.0 moles of argon proceeds so that the gas pressure increases in direct proportion to its volume. Find the entropy increment of the gas in this process provided its volume increases α = 2.0 times. Free solution >> 2.138. An ideal gas with the adiabatic exponent γ goes through a process p = p0 - αV, where p0 and α are positive constants, and V is the volume. At what volume will the gas entropy have the maximum value? Free solution >> 2.139. One mole of an ideal gas goes through a process in which the entropy of the gas changes with temperature T as S = aT + Cv ln T, where a is a positive constant, Cv is the molar heat capacity of this gas at constant volume. Find the volume dependence of the gas temperature in this process if T = T0 at V = V0. Free solution >> 2.140. Find the entropy increment of one mole of a Van der Waals gas due to the isothermal variation of volume from V1 to V2. The Van der Waals corrections are assumed to be known. Free solution >> 2.141. One mole of a Van der Waals gas which had initially the volume V1 and the temperature T1 was transferred to the state with the volume V2 and the temperature T2. Find the corresponding entropy increment of the gas, assuming its molar heat capacity Cv to be known. Free solution >> 2.143. Find the entropy increment of an aluminum bar of mass m = 3.0 kg on its heating from the temperature T1 = 300 K up to T2 = 600 K if in this temperature interval the specific heat capacity of aluminum varies as c = a + bT, where a = 0.77 J/(g*K), b = 0.46 mJ/(g*K2). Free solution >> 2.144. In some process the temperature of a substance depends on its entropy S as T = aSn, where a and n are constants. Find the corresponding heat capacity C of the substance as a function of S. At what condition is C < 0? Free solution >> 2.145. Find the temperature T as a function of the entropy S of a substance for a polytropic process in which the heat capacity of the substance equals C. The entropy of the substance is known to be equal to S0 at the temperature T0. Draw the approximate plots T(S) for C > 0 and C < 0. Free solution >> 2.146. One mole of an ideal gas with heat capacity Cv goes through a process in which its entropy S depends on T as S = α/T, where α is a constant. The gas temperature varies from T1 to T2. Find: (a) the molar heat capacity of the gas as a function of its temperature; (b) the amount of heat transferred to the gas; (c) the work performed by the gas. Free solution >> 2.148. One of the two thermally insulated vessels interconnected by a tube with a valve contains ν = 2.2 moles of an ideal gas. The other vessel is evacuated. The valve having been opened, the gas increased its volume n = 3.0 times. Find the entropy increment of the gas. Free solution >> 2.149. A weightless piston divides a thermally insulated cylinder into two equal parts. One part contains one mole of an ideal gas with adiabatic exponent γ, the other is evacuated. The initial gas temperature is T0. The piston is released and the gas fills the whole volume of the cylinder. Then the piston is slowly displaced back to the initial position. Find the increment of the internal energy and the entropy of the gas resulting from these two processes. Free solution >> 2.152. A piece of copper of mass m1 = 300 g with initial temperature t1 = 97 °C is placed into a calorimeter in which the water of mass m2 = 100 g is at a temperature t2 = 7 °C. Find the entropy increment of the system by the moment the temperatures equalize. The heat capacity of the calorimeter itself is negligibly small. Free solution >> 2.154. N atoms of gaseous helium are enclosed in a cubic vessel of volume 1.0 cm3 at room temperature. Find: (a) the probability of atoms gathering in one half of the vessel; (b) the approximate numerical value of N ensuring the occurrence of this event within the time interval t ≈ 1010 years (the age of the Universe). Free solution >> 2.157. A vessel of volume V0 contains N molecules of an ideal gas. Find the probability of n molecules getting into a certain separated part of the vessel of volume V. Examine, in particular, the case V = V0/2. Free solution >>