Entropy and the Second Law of Thermodynamics Practice Problems

During an isothermal and reversible process, 1 mole of helium gas, initially having a velocity distribution probability f(v), expands to occupy double its original volume. What is the change in the velocity distribution probability due to the isothermal expansion?

A jeweler heats an 8 g gold ring to 500 °C and places it on his workbench to cool down naturally to 30 °C. What is the entropy variation of the gold ring while it cools?

An ice cream cake at -18 °C left out in the kitchen melts within 30 minutes. The temperature in the kitchen is 30 °C. The kitchen-ice cream system is considered isolated, and the kitchen temperature is assumed constant. Without using calculations, state if this is a reversible or an irreversible process and identify the sign of the entropy variation.

Cryotherapy, a common method for treating athletes' injuries, works by filling a large tub with 100 kg of cold water at 10 °C and pouring in 1 kg of ice at 0 °C. The cold water is being cooled further by ice.  i) Is this a reversible or irreversible process, and what is the sign of the entropy variation (ΔS)? Do not use calculations.

What change in entropy (ΔS) occurs when a butter cube of mass 6.9 g at 35 °C is totally melted? Butter melts at 35 °C and has a latent heat of fusion of 60 J/g.

During an ice avalanche, 100 kg of ice at 0 °C falls into a large lake. The ice completely melts within a few hours. The lake temperature is assumed to be homogeneous and equal to 10 °C before and after the ice melts. Determine the total entropy change (ΔS) of the ice-water system.

A 150 g piece of solid wax at 40°C is placed on an 80°C metal plate. Determine the entropy changes of (i) the molten wax, (ii) the metal plate, and (iii) the universe, assuming that the temperature of the metal plate remains constant as the wax melts and subsequently reaches thermal equilibrium. Take the latent heat of fusion of wax to be 2.0 × 10⁵ J/kg, specific heat to be  2100 J/(kg.K), and its melting temperature to be 40°C.

A new diatomic gas called Zogon is discovered. Zogon has a molar mass of 25.0 g/mol and a boiling point of -171°C. A 350. mL sample of liquid Zogon, with a density of 980. kg/m³, changes to gas at constant pressure and reaches 45.0 °C. Zogon's latent heat of vaporization is 1.89 × 10⁵ J/kg. Find the entropy change for Zogon.

At an initial temperature of 300°C, a 50 g sample of a recently identified monoatomic gas, "Novagium," undergoes an isobaric transformation.  The molar mass of Novagium is 25 g/mol. During this process, the entropy of the Novagium increases by 40 J/K. Calculate the final temperature of the gas in this scenario in Kelvins (K).

Determine the equation for entropy change in the given scenario:  In a greenhouse with a total volume of V, two different gases, P and Q, are initially separated by a transparent curtain. Gases mix and disperse when the curtain is removed. At the same temperature, gas P, produced by plants, occupies a volume fraction of q_P and contains n_P moles, while gas Q, formed by decomposing organic matter, occupies a volume fraction q_Q and contains n_Q moles.

Determine the entropy change of (i) the gas and (ii) the whole system during a cooling process where a chef is trying to create a refreshing culinary experience. The chef brings a 3.0 mol sample of nitrogen gas in a 4.0 L rigid container down from 100°C to 25°C by putting it in a large basin of water at 25°C.

Calculate the change in entropy of 0.750 kg of ammonia at 200.0 °C when converted into steam at the same temperature through a reversible process. [Latent heat of vaporization of ammonia (L_vaporization) = 137 kJ/kg].

How will the entropy change if 300 g of ammonia gas is condensed from its boiling point of -33.3°C to form liquid ammonia at the same temperature? Consider that the latent heat of vaporization for ammonia is 1367 kJ/kg.

A reversible process is used in a laboratory experiment to transform 0.55 kg of ethanol at 78.5 °C into vapor. Find the change in entropy for ethanol, the surroundings, and the universe. If the process was irreversible, explain how these values would differ.

If a block of ice weighing 0.75 kg at a temperature of 0°C melts to form water at the same temperature through a reversible process, calculate the change in entropy for its environment.

Determine the change in entropy of the universe (water + surroundings) as a whole if you have a pot with 0.85 kg of boiling water at 100°C and allow it to evaporate on your stove top completely through a reversible process.

Two separate insulated vessels contain argon (Ar) and neon (Ne) gases respectively, with each vessel containing 2.0 moles stored at the same temperature. When they are allowed to mix by connecting both vessels to each other, how will their system's entropy value change?

Two cylinders, both containing argon gas at the same initial temperature and pressure "P", expand from Y liters to 5Y liters. Note that one cylinder undergoes an isothermal expansion, while the other cylinder undergoes an adiabatic expansion. Which cylinder will end up having a higher final pressure value?

An insulated steel container weighing 300 g at an initial temperature of -10°C is filled with hot coffee weighing around half a kilogram and has an initial temperature of +70°C. After some time, they reach thermal equilibrium. What will be the final temperature? [Hint: Specific heat capacity of steel (c_steel) = 0.49 J/g°C, the specific heat capacity of coffee (c_coffee) = 4.18 J/g°C.]

Suppose a student is heating 2.00 kg of ethanol from 0°C to 75°C in a university laboratory. Calculate the change in entropy (Δ𝑆) during this process. Given the specific heat capacity of ethanol is 2440 J/kg·K. 

A 2.00 kg aluminum rod is heated from 20°C to 200°C. Calculate the change in entropy of the rod using the integral: ∆S = ∫dQ/T. Note that the specific heat capacity of aluminum is equal to 900 J/kg$\cdot$K.

Determine the total entropy change of a reversible isothermal process and justify your answer.

A sample of hydrogen gas with a quantity of 0.75 mol undergoes heating at constant volume from an initial temperature of -10°C to a final temperature of +20°C. Using ∆S = ∫ dQ / T, calculate its change in entropy.

A steel cup with a mass of 200 g initially at room temperature (20°C) is filled with hot tea weighing 0.60 kg. The tea is initially at its boiling point temperature (100°C). After some time, thermal equilibrium is reached at a temperature of 97°C. Using ∆S = ∫ dQ / T, compute the entropy change due to this process. Assume that the specific heat capacities for steel and water are 490 J/kg·K and 4180 J/kg·K, respectively.

A physicist takes two pistons that initially contain the same amount of hydrogen gas at the same temperature and pressure. To examine the behavior of the hydrogen, he compresses them from volume V to volume V/5 (reversibly). He compresses one of the pistons isothermally, and the other is compressed adiabatically. Using integration, calculate the change in entropy for each process.

Two spaceships weighing 3000 kg are speeding towards each other in outer space at 50 km/h. They collide head-on and come to a complete stop because of the impact. Calculate how much change in the entropy of the universe due to this event, assuming the surrounding temperature is 30°C.

Two copper vessels, one containing 500 g of mercury that is initially at a temperature of −10°C and another containing an equal amount of mercury that is initially at a temperature of +40°C, are brought into thermal contact with each other until they reach thermal equilibrium without any heat exchange occurring with their surroundings. Calculate the final equilibrium temperature and total entropy change. Use ∆S = ∫ dQ / T [Hint: The specific heat capacity of mercury is 140 J/kg$\cdot$°C]

Consider a Carnot refrigeration cycle operating between two reservoirs. The high-temperature reservoir is at 𝑇_𝐻 and the low-temperature reservoir is at 𝑇_𝐶. Calculate the total entropy change for the entire Carnot cycle.

A geothermal power plant operates between a hot reservoir at 900 K and a cold reservoir at 600 K. Compare the efficiency of this real power plant to that of an ideal Carnot engine if the plant produces 600 J of work per cycle for a heat input of 3000 J.