# 6.E: The Second Law (Exercises)

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- 199211

- Contributed by Patrick Fleming
- Assistant Professor (Chemistry and Biochemistry) at California State University East Bay

## Q5.1

What is the minimum amount of work needed to remove 10.0 J of energy from a freezer at -10.0 °C, depositing the energy into a room that is 22.4 °C?

## Q5.2

Consider the isothermal, reversible expansion of 1.00 mol of a monatomic ideal gas (C_{V} = 3/2 R) from 10.0 L to 25.0 L at 298 K. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.

## Q5.3

Consider the isobaric, reversible expansion of 1.00 mol of a monatomic ideal gas (C_{p} = 5/2 R) from 10.0 L to 25.0 L at 1.00 atm. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.

## Q5.4

Consider the isochoric, reversible temperature increase of 1.00 mol of a monatomic ideal gas (C_{V} = 3/2 R) °Ccupying 25.0 L from 298 K to 345 K. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for this process.

## Q5.5

Consider the adiabatic expansion of 1.00 mol of a monatomic ideal gas (C_{V} = 3/2 R) from 10.0 L at 273 K to a final volume of 45.0 L. Calculate \(\Delta T\), \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.

## Q5.6

15.0 g of ice (\(\Delta H_{fus} = 6.009\, kJ/mol\)) at 0 °C sits in a room that is at 21 °C. The ice melts to form liquid at 0 °C. Calculate the entropy change for the ice, the room, and the universe. Which has the largest magnitude?

## Q5.7

15.0 g of liquid water (C_{p} = 75.38 J mol^{-1} °C^{-1}) at 0 °C sits in a room that is at 21 °C. The liquid warms from 0 °C to 21 °C. Calculate the entropy change for the liquid, the room, and the universe. Which has the largest magnitude?

## Q5.8

Calculate the entropy change for taking 12.0 g of H_{2}O from the solid phase (C_{p} = 36.9 J mol^{-1} K^{-1}) at -12.0 °C to liquid (C_{p} = 75.2 J mol^{-1} K^{-1}) at 13.0 °C. The enthalpy of fusion for water is \(\Delta H_{fus} = 6.009 \,kJ/mol\).

## Q5.9

Using Table T1, calculate the standard reaction entropies (\(\Delta S^o\)) for the following reactions at 298 K.

- \(CH_3CH_2OH(l) + 3 O_2(g) \rightarrow 2 CO_2(g) + 3 H_2O(l)\)
- \(C_{12}H_{22}O_{11}(s) + 12 O_2 \rightarrow 12 CO_2(g) + 11 H_2O(l)\)
- \(2 POCl_3(l) \rightarrow 2 PCl_3(l) + O_2(g)\)
- \(2 KBr(s) + Cl2(g) \rightarrow 2 KCl(s) + Br_2(l)\)
- \(SiH_4(g) + 2 Cl(g) \rightarrow SiCl_4(l) + 2 H_2(g)\)

## Q5.10

1.00 mole of an ideal gas is taken through a cyclic process involving three steps:

- Isothermal expansion from V
_{1}to V_{2}at T_{1} - Isochoric heating from, T
_{1}to T_{2}at V_{2} - Adiabatic compression from V
_{2}to V_{1}

- Graph the process on a V-T diagram.
- Find \(q\), \(w\), \(\Delta U\), and \(\Delta S\) for each leg. (If you want, you can find \(\Delta H\) too!)
- Use the fact that \(\Delta S\) for the entire cycle must be zero (entropy being a state function and all …), determine the relationship between V
_{1}and V_{2}in terms of C_{v}, T_{1}and T_{2}.

## Q5.11

2.00 moles of a monatomic ideal gas (C_{V} = 3/2 R) initially exert a pressure of 1.00 atm at 300.0 K. The gas undergoes the following three steps, all of which are reversible:

- isothermal compression to a final pressure of 2.00 atm,
- Isobaric temperature increase to a final temperature of 400.0 K, and
- A return to the initial state along a pathway in which

\[p = a+bT\]

where \(a\) and \(b\) are constants. Sketch the cycle on a pressure-temperature plot, and calculate \(\Delta U\) and \(\Delta S\) for each of the legs. Are \(\Delta U\) and \(\Delta S\) zero for the sum of the three legs?

## Q5.12

A 10.0 g piece of iron (C = 0.443 J/g °C) initially at 97.6 °C is placed in 50.0 g of water (C = 4.184 J/g °C) initially at 22.3 °C in an insulated container. The system is then allowed to come to thermal equilibrium. Assuming no heat flow to or from the surroundings, calculate

- the final temperature of the metal and water
- the change in entropy for the metal
- the change in entropy for the water
- the change in entropy for the universe

## Q5.13

Considers a crystal of \(CHFClBr\) as having four energetically equivalent orientations for each molecule. What is the expected residual entropy at 0 K for 2.50 mol of the substance?

## Q5.14

A sample of a certain solid is measured to have a constant pressure heat capacity of 0.436 J mol^{-1} K^{-1} at 10.0 K. Assuming the Debeye extrapolation model

\[ C_p(T) = aT^3\]

holds at low temperatures, calculate the molar entropy of the substance at 12.0 K.