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Week 12: Thermodynamics

\(\Delta\) ← if you see the word "Delta" instead of a triangle, please refresh the page! Thanks :)

NOTE: this page was updated Friday, April 17, ~12pm to add more problems :)

 

Overview⚓︎

I recommend reviewing the following concepts, they will probably come up in recitation:

  • The first law of thermodynamics

    • \(\Delta U = Q - W\)
    • taking \(W\) to mean work done by the gas
    • keeping track of signs carefully

  • Internal energy and heat capacities for a monatomic ideal gas

    • \(U = \tfrac32 nRT\)
    • so \(\Delta U = \tfrac32 nR\Delta T\)
    • \(C_V = \tfrac32 R\) and \(C_P = \tfrac52 R\)
    • internal energy depends only on temperature

  • Special thermodynamic processes

    • isothermal: \(\Delta T = 0\), so \(\Delta U = 0\)
    • constant volume: \(W = 0\)
    • adiabatic: \(Q = 0\)
    • constant pressure: \(Q = nC_P\Delta T\)

  • Work from a \(PV\) diagram

    • area under the curve
    • positive for expansion
    • negative for compression
    • for straight-line segments, using trapezoid areas

  • Using \(PV=nRT\) to compare states on a diagram

    • if \(n\) is fixed, then \(T \propto PV\)
    • using endpoint states to find temperature changes
    • for a monatomic gas, \(\Delta U = \tfrac32 (P_fV_f - P_iV_i)\)

  • Heat engines and thermal efficiency

    • \(e = \dfrac{W}{Q_H}\)
    • \(W = Q_H - Q_C\)
    • power is work per time
    • keeping track of whether quantities are per cycle or per second

  • Carnot efficiency

    • \(e_{\mathrm{Carnot}} = 1 - \dfrac{T_C}{T_H}\)
    • reservoir temperatures must be in Kelvin
    • real engines must have efficiency below Carnot

 

 

Practice Problems⚓︎

  • If you're comfortable with the "★★★" problems, you should do great during recitation.
  • To print these questions, simply press Ctrl + P while on this page, and it should come out formatted nicely -- just make sure to refresh first so all the math renders properly.

Difficulty key:

  • ★☆☆ = beginner
  • ★★☆ = standard
  • ★★★ = challenging / multi-step

Thermodynamics Constants / Reminders

  • \(R = 8.31 \ \mathrm{J/(mol \cdot K)}\)
  • For a monatomic ideal gas: \(U = \tfrac32 nRT\)
  • \(C_V = \tfrac32 R\)
  • \(C_P = \tfrac52 R\)
  • First law: \(\Delta U = Q - W\)
  • Isothermal ideal-gas process: \(\Delta U = 0\)
  • Constant-volume process: \(W = 0\)
  • Adiabatic process: \(Q = 0\)
  • Heat-engine efficiency: \(e = \dfrac{W}{Q_H}\)
  • Carnot efficiency: \(e_{\mathrm{Carnot}} = 1 - \dfrac{T_C}{T_H}\)
  • Useful conversion: \(1\,\mathrm{kPa \cdot L} = 1\,\mathrm{J}\)
  • Latent heat of vaporization of water: \(L_v = 2.26 \times 10^6\,\mathrm{J/kg}\)
  • Power: \(1\,\mathrm{W} = 1\,\mathrm{J/s}\)

 

[1] First Law, Internal Energy, and Special Processes⚓︎

  1. ★☆☆
    A rigid container holds \(0.200 \ \mathrm{mol}\) of a monatomic ideal gas. The gas is heated from \(310 \ \mathrm{K}\) to \(370 \ \mathrm{K}\).

    • (a) Find the change in internal energy.
    • (b) Find the work done by the gas.
    • (c) Find the heat added to the gas.

  2. ★☆☆
    During an isothermal expansion of an ideal gas, the gas does \(680 \ \mathrm{J}\) of work on its surroundings.

    • (a) What is the change in internal energy of the gas?
    • (b) How much heat flows into or out of the gas?

  3. ★★☆
    A sample of \(0.350 \ \mathrm{mol}\) of a monatomic ideal gas is compressed isothermally at \(300 \ \mathrm{K}\) from \(3.60 \ \mathrm{L}\) to \(1.20 \ \mathrm{L}\).

    • (a) Find the change in internal energy.
    • (b) Find the work done by the gas.
    • (c) Find the heat flow into or out of the gas.

  4. ★★☆
    A closed rigid container holds \(0.240 \ \mathrm{mol}\) of a monatomic ideal gas at \(290 \ \mathrm{K}\). The gas is heated until its pressure is \(2.50\) times its initial value.

    • (a) What is the final temperature?
    • (b) What is the change in internal energy?
    • (c) How much heat is added to the gas?

  5. ★★★
    In an adiabatic process, the temperature of \(3.20 \ \mathrm{mol}\) of a monatomic ideal gas drops from \(500^\circ\mathrm{C}\) to \(140^\circ\mathrm{C}\).

    • (a) How much heat is exchanged with the surroundings?
    • (b) What work does the gas do?
    • (c) What is the change in internal energy of the gas?

  6. ★★★
    The temperature of a \(2.80 \ \mathrm{mol}\) sample of a monatomic ideal gas is initially \(320 \ \mathrm{K}\). Its internal energy is doubled by the addition of heat.

    • (a) How much heat is needed if the gas is heated at constant volume?
    • (b) How much heat is needed if the gas is heated at constant pressure?

[2] \(PV\) Diagrams, Work, and Heat⚓︎

  1. ★☆☆
    A gas expands at constant pressure \(P = 180 \ \mathrm{kPa}\) from \(1.40 \ \mathrm{L}\) to \(4.10 \ \mathrm{L}\). What is the work done by the gas?

  2. ★☆☆
    A fixed amount of an ideal gas goes from state \(A\) to state \(B\).

    \[A: (P=150 \ \mathrm{kPa},\ V=2.00 \ \mathrm{L},\ T=300 \ \mathrm{K})\]
    \[B: (P=250 \ \mathrm{kPa},\ V=3.00 \ \mathrm{L})\]

    What is the temperature at point \(B\)?

  3. ★★☆
    A monatomic ideal gas goes from state \(A\) to state \(C\) in two steps on a \(PV\) diagram:

    • \(A \to B\): constant volume from \((1.50 \ \mathrm{L},\ 200 \ \mathrm{kPa})\) to \((1.50 \ \mathrm{L},\ 350 \ \mathrm{kPa})\)
    • \(B \to C\): constant pressure from \((1.50 \ \mathrm{L},\ 350 \ \mathrm{kPa})\) to \((4.50 \ \mathrm{L},\ 350 \ \mathrm{kPa})\)

    If the temperature at point \(A\) is \(200 \ \mathrm{K}\), find:

    • (a) the temperature at point \(C\)
    • (b) the total work done by the gas
    • (c) the change in internal energy
    • (d) the total heat added to the gas

  4. ★★☆
    A monatomic ideal gas moves from state \(A\) to state \(B\) along a straight-line path on a \(PV\) diagram.

    \[A: (V=2.00 \ \mathrm{L},\ P=120 \ \mathrm{kPa},\ T=280 \ \mathrm{K})\]
    \[B: (V=5.00 \ \mathrm{L},\ P=300 \ \mathrm{kPa})\]

    Find:

    • (a) the temperature at point \(B\)
    • (b) the work done by the gas
    • (c) the change in internal energy
    • (d) the heat added to the gas

  5. ★★★
    A monatomic ideal gas moves along the following four-segment path on a \(PV\) diagram:

    • from \(A=(2.00 \ \mathrm{L},\ 100 \ \mathrm{kPa})\) horizontally to \((3.00 \ \mathrm{L},\ 100 \ \mathrm{kPa})\)
    • then linearly to \((5.00 \ \mathrm{L},\ 250 \ \mathrm{kPa})\)
    • then horizontally to \((7.00 \ \mathrm{L},\ 250 \ \mathrm{kPa})\)
    • then linearly to \(B=(9.00 \ \mathrm{L},\ 150 \ \mathrm{kPa})\)

    If the temperature at point \(A\) is \(240 \ \mathrm{K}\), find:

    • (a) the temperature at point \(B\)
    • (b) the total work done by the gas
    • (c) the change in internal energy
    • (d) the total heat added to the gas

  6. ★★★
    A monatomic ideal gas is taken from state \(A\) to state \(C\), where

    \[A: (V=2.00 \ \mathrm{L},\ P=150 \ \mathrm{kPa},\ T=300 \ \mathrm{K})\]
    \[C: (V=6.00 \ \mathrm{L},\ P=300 \ \mathrm{kPa}).\]

    Compare the following two paths between the same endpoints:

    • Path 1: a straight-line path from \(A\) directly to \(C\)
    • Path 2: first go vertically to \((2.00 \ \mathrm{L},\ 300 \ \mathrm{kPa})\), then horizontally to \(C\)

    For each path, find:

    • (a) the work done by the gas
    • (b) the heat added to the gas

    Also find:

    • (c) the change in internal energy between \(A\) and \(C\)
    • (d) which path requires more heat input

[3] Heat Engines and Carnot Efficiency⚓︎

  1. ★☆☆
    A heat engine absorbs \(2.20 \times 10^3 \ \mathrm{J}\) of heat each cycle and exhausts \(1.50 \times 10^3 \ \mathrm{J}\) each cycle.

    • (a) How much work is done per cycle?
    • (b) What is the engine's efficiency?

  2. ★☆☆
    A heat engine does \(480 \ \mathrm{J}\) of work in each cycle, and each cycle lasts \(0.250 \ \mathrm{s}\). What is the power output of the engine?

  3. ★★☆
    A heat engine operates at three-quarters of the efficiency of a Carnot engine working between a hot reservoir at \(525^\circ\mathrm{C}\) and a cold reservoir at \(75^\circ\mathrm{C}\).

    What is the operating efficiency of the engine?

  4. ★★☆
    The efficiency of a heat engine is \(55.0\%\) of the efficiency of a Carnot engine operating between \(45^\circ\mathrm{C}\) and \(245^\circ\mathrm{C}\). If the engine absorbs heat at a rate of \(20.0 \ \mathrm{kW}\), at what rate is heat exhausted?

  5. ★★★
    A heat engine has a thermal efficiency of \(34.0\%\). Its heat input each cycle is supplied by the condensation of \(4.20 \ \mathrm{kg}\) of steam at \(100^\circ\mathrm{C}\).

    • (a) How much heat is absorbed each cycle?
    • (b) How much work is done each cycle?
    • (c) How much heat is lost to the surroundings each cycle?

  6. ★★★
    A proposed ocean-thermal power plant would operate between warm surface water at \(24^\circ\mathrm{C}\) and deep water at \(7^\circ\mathrm{C}\).

    • (a) What is the maximum possible efficiency of such a plant?
    • (b) If the electrical power output is \(80.0 \ \mathrm{MW}\), how much heat must be absorbed from the warm reservoir each hour, assuming the plant operates at that maximum efficiency?