TUTORIAL 3
Thermodynamics — Questions
3.6. Express the volume expansivity and the isothermal compressibility as functions of density ρ and its partial
derivatives. For water at 50°C and 1 bar, κ = 44.18 × 10−6 bar−1. To what pressure must water be
compressed at 50°C to change its density by 1%? Assume that κ is independent of P.
3.9. For liquid water the isothermal compressibility is given by κ = c/[V(P + b)], where c and b are functions of
temperature only. If 1 kg of water is compressed isothermally and reversibly from 1 to 500 bar at 60°C, how
much work is required? At 60°C, b = 2700 bar and c = 0.125 cm³ g−1.
3.17. One mole of an ideal gas with C = (7/2)R and C = (5/2)R expands from P = 8 bar and T = 600 K to
P
V
1
1
P = 1 bar by each of the following paths: (a) Constant volume; (b) Constant temperature; (c) Adiabatically.
2
Assuming mechanical reversibility, calculate W, Q, ∆U, and ∆H for each process. Sketch each path on a
single P–V diagram.
3.40. Derive an equation for the work of mechanically reversible, isothermal compression of 1 mol of a gas
from an initial pressure P1 to a final pressure P2 when the equation of state is the virial expansion truncated to
Z = 1 + B′P. How does the result compare with the corresponding equation for an ideal gas?
3.42. For methyl chloride at 100°C the second and third virial coefficients are B = −242.5 cm³ mol−1 and C =
25,200 cm6 mol−2. Calculate the work of mechanically reversible, isothermal compression of 1 mol of methyl
chloride from 1 bar to 55 bar at 100°C. Base calculations on: (a) Z = 1 + B/V + C/V²; (b) Z = 1 + B′P + C′P²,
where B′ = B/(RT) and C′ = (C − B²)/(RT)². Why don’t both equations give exactly the same result?
Tutorial 3 | Page 1
TUTORIAL- 2
5.3. The following heat engines produce power of 95,000 kW. Determine in each case the rates at
which heat is absorbed from the hot reservoir and discarded to the cold reservoir. (a) A Carnot engine
operates between heat reservoirs at 750 K and 300 K. (b) A practical engine operates between the
same heat reservoirs but with a thermal efficiency η = 0.35.
5.4. A particular power plant operates with a heat-source reservoir at 350°C and a heat sink reservoir
at 30°C. It has a thermal efficiency equal to 55% of the Carnot-engine thermal efficiency for the same
temperatures. (a) What is the thermal efficiency of the plant? (b) To what temperature must the heatsource reservoir be raised to increase the thermal efficiency of the plant to 35%? Again, η is 55% of
the Carnot-engine value.
5.8. With respect to 1 kg of liquid water: (a) Initially at 0°C, it is heated to 100°C by contact with a heat
reservoir at 100°C. What is the entropy change of the water, of the heat reservoir? What is ΔStotal? (b)
Initially at 0°C, it is first heated to 50°C by contact with a heat reservoir at 50°C and then to 100°C by
contact with a reservoir at 100°C. What is ΔStotal? (c) Explain how the water might be heated from 0°C
to 100°C so that ΔStotal = 0.
5.9. A rigid vessel of 0.06 m3 volume contains an ideal gas, CV = (5/2) R, at 500 K and 1 bar. (a) If heat
in the amount of 15,000 J is transferred to the gas, determine its entropy change. (b) If the vessel is
fitted with a stirrer that is rotated by a shaft so that work in the amount of 15,000 J is done on the gas,
what is the entropy change of the gas if the process is adiabatic? What is ΔStotal? What is the
irreversible feature of the process?
5.11. For an ideal gas with constant heat capacities, show that: (a) For a temperature change from T1
to T2, ΔS of the gas is greater when the change occurs at constant pressure than when it occurs at
constant volume. (b) For a pressure change from P1 to P2, the sign of ΔS for an isothermal change is
opposite that for a constant-volume change.
5.21. One mole of an ideal gas, CP = (7/2) R and CV = (5/2) R, is compressed adiabatically in a
piston/cylinder device from 2 bar and 25°C to 7 bars. The process is irreversible and requires 35%
more work than a reversible, adiabatic compression from the same initial state to the same final
pressure. What does the entropy change of the gas?
HW-1
2.5. An electric hand mixer draws 1.5 amperes at 110 volts. It is used to mix 1 kg of cookie dough for 5 minutes.
After mixing, the temperature of the cookie dough is found to have increased by 5°C. If the heat capacity of the
dough is 4.2 kJ⋅kg−1⋅K−1, what fraction of the electrical energy used by the mixer is converted to internal energy
of the dough? Discuss the fate of the remainder of the energy.
2.6. One mole of gas in a closed system undergoes a four-step thermodynamic cycle. Use the data given in the
following table to determine numerical values for the missing quantities indicated by question marks.
2.8. A tank containing 20 kg of water at 20°C is fitted with a stirrer that delivers work to the water at the rate of
0.25 kW. How long does it take for the temperature of the water to rise to 30°C if no heat is lost from the water?
For water, CP = 4.18 kJ⋅kg−1⋅°C−1.
2.10. A steel casting weighing 2 kg has an initial temperature of 500°C; 40 kg of water ini tially at 25°C is
contained in a perfectly insulated steel tank weighing 5 kg. The cast ing is immersed in the water and the
system is allowed to come to equilibrium. What is its final temperature? Ignore the effects of expansion or
contraction, and assume constant specific heats of 4.18 kJ⋅kg−1⋅K−1 for water and 0.50 kJ⋅kg−1⋅K−1 for steel.
2.23. An incompressible (ρ = constant) liquid flows steadily through a conduit of circular cross-section and
increasing diameter. At location 1, the diameter is 2.5 cm and the velocity is 2 m⋅s−1; at location 2, the
diameter is 5 cm.
(a) What is the velocity at location 2?
(b) What is the kinetic-energy change (J⋅kg−1) of the fluid between locations 1 and 2?
2.26. Water at 28°C flows in a straight horizontal pipe in which there is no exchange of either heat or work with
the surroundings. Its velocity is 14 m⋅s−1 in a pipe with an internal diameter of 2.5 cm until it flows into a
section where the pipe diameter abruptly increases. What is the temperature change of the water if the
downstream diameter is 3.8 cm? If it is 7.5 cm? What is the maximum temperature change for an enlargement
in the pipe?
2.31. Steam flows at steady state through a converging, insulated nozzle, 25 cm long and with an inlet diameter
of 5 cm. At the nozzle entrance (state 1), the temperature and pressure are 325°C and 700 kPa and the velocity
is 30 m⋅s−1. At the nozzle exit (state 2), the steam temperature and pressure are 240°C and 350 kPa. Property
values are:
H1 = 3112.5 kJ⋅kg−1
H2 = 2945.7 kJ⋅kg−1
V1 = 388.61 cm3⋅g−1
V2 = 667.75 cm3⋅g−1
What is the velocity of the steam at the nozzle exit, and what is the exit diameter?
Tutorial 6
6.2. (a) Making use of the fact that Eq. (6.21)
is an exact differential expression, show that:
What is the result of application of this equation to an ideal gas?
(b) Heat capacities CV and CP are defined as temperature derivatives respectively of U and H. Because these
properties are related, one expects the heat capacities also to be related. Show that the general expression
connecting CP to CV is:
Show that Eq. (B) of Ex. 6.2 is another form of this expression.
6.3. If U is considered a function of T and P, the “natural” heat capacity is neither CV nor CP, but rather the
derivative (∂ U / ∂ T) P . Develop the following connections between
To what do these equations reduce for an ideal gas? For an incompressible liquid?
6.4. The PVT behavior of a certain gas is described by the equation of state: P ( V − b ) = RT where b is a
constant. If in addition CV is constant, show that:
(a) U is a function of T only.
(b) γ = const.
(c) For a mechanically reversible process,
6.6. A pure fluid, described by the canonical equation of state: G = F(T) + KP, where F(T) is a substance-specific
function of temperature and K is a substance-specific constant. Determine for such a fluid expressions for V, S,
H, U, CP, and CV. These results are consistent with those for an important model of liquid-phase behavior. What
is the model?
6.7. Estimate the change in enthalpy and entropy when liquid ammonia at 270 K is compressed from its
saturation pressure of 381 kPa to 1200 kPa. For saturated liquid ammonia at 270,
6.8. Liquid isobutane is throttled through a valve from an initial state of 360 K and 4000 kPa to a final pressure of
2000 kPa. Estimate the temperature change and the entropy change of the isobutane. The specific heat of
liquid isobutane at 360 K is 2.78 J·g−1·°C−1. Estimates of V and β may be found from Eq. (3.68).
Determine expressions for GR, HR, and SR implied by the Dieterici equation:
Here, parameters a and b are functions of composition only.