Entropy Equations for Special Processes Practice Problems

A universal law suggests that in any process, the quantity of energy that cannot be used for useful work equates T_L∆S, where T_L represents minimum accessible temperature and ∆S signifies total alteration in entropy during the process. Confirm if this law holds true in case a ball rolling down an inclined plane comes to a halt.

An underlying theory suggests that in any given process, the quantity of energy that becomes inaccessible (or lost) for doing useful work equals $T_L\Delta S$, where $T_L$ signifies the lowest accessible temperature and $\Delta S$ denotes the total change in entropy throughout the process. Determine the expression for the lost energy E_lost for the free adiabatic expansion that occurs inside of an idealized gas.

Find out how much maximum work can be obtained from cooling down a 5.0 kg iron block that is initially heated up to a temperature of 800 K and is later brought to ambient conditions with an average temperature of around 295 K. Note that the specific heat capacity of iron is 450 J/kg.K.

Imagine you have 0.25 mol of sodium. At very low temperatures, its specific heat is described by C_V = a ⋅T + b ⋅T³ and the values for these variables are a = 2.5 mJ/mol·K² and b = 3.0 mJ/mol·K⁴. Now, suppose you increase the temperature of this sodium sample from 2.0 K to 6.0 K. Calculate the entropy change when the temperature changes.

A certain crystalline solid "Y" has a specific heat value that obeys the law C_V = 4$\times$10³ (T/T_D)³ J·mol⁻¹·K⁻¹ at low temperatures where T_D represents its Debye temperature which is equal to 1500K. If one mole of this solid "Y" is heated from 300 K to its Debye Temperature, what would be the resulting change in entropy?