Page 65 - Physical chemistry eng

42 CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics

P2.2 The temperature of 1.75 moles of an ideal gas the right to result in a final pressure P = 4.00 bar in the
f
increases from 10.2°C to 48.6°C as the gas is compressed adi- right part. Consider the compression of the gas in the right
abatically. Calculate q, w, ¢U , and ¢H for this process, part to be a reversible process.
assuming that C V,m = 3R>2 . a. Calculate the work done on the right part in this process
P2.3 A 2.50 mole sample of an ideal gas, for which and the final temperature in the right part.
C V,m = 3R>2 , is subjected to two successive changes in state: b. Calculate the final temperature in the left part and the
3
(1) From 25.0°C and 125 * 10 Pa , the gas is expanded amount of heat that flowed into this part.
3
isothermally against a constant pressure of 15.2 * 10 Pa to P2.12 In the reversible adiabatic expansion of 1.75 mol of
twice the initial volume. (2) At the end of the previous an ideal gas from an initial temperature of 27.0°C, the work
process, the gas is cooled at constant volume from 25.0°C to done on the surroundings is 1300. J. If C V,m = 3R>2 , calcu-
–29.0°C. Calculate q, w, ¢U , and ¢H for each of the stages. late q, w, ¢U , and ¢H .
Also calculate q, w, ¢U , and ¢H for the complete process.
P2.13 A system consisting of 82.5 g of liquid water at 300. K
P2.4 A hiker caught in a thunderstorm loses heat when her is heated using an immersion heater at a constant pressure of
clothing becomes wet. She is packing emergency rations that if 1.00 bar. If a current of 1.75 A passes through the 25.0 ohm
completely metabolized will release 35 kJ of heat per gram of resistor for 100. s, what is the final temperature of the water?
rations consumed. How much rations must the hiker consume to
P2.14 A 1.25 mole sample of an ideal gas is expanded
avoid a reduction in body temperature of 2.5 K as a result of heat
from 320. K and an initial pressure of 3.10 bar to a final
loss? Assume the heat capacity of the body equals that of water
pressure of 1.00 bar, and C P,m = 5R>2 . Calculate w for the
and that the hiker weighs 51 kg. State any additional assumptions.
following two cases:
P2.5 Count Rumford observed that using cannon boring
a. The expansion is isothermal and reversible.
machinery a single horse could heat 11.6 kg of ice water
b. The expansion is adiabatic and reversible.
(T = 273 K) to T = 355 K in 2.5 hours. Assuming the same
rate of work, how high could a horse raise a 225 kg weight in Without resorting to equations, explain why the result to part
–1 –1
2.5 minutes? Assume the heat capacity of water is 4.18 J K g . (b) is greater than or less than the result to part (a).
P2.6 A 1.50 mole sample of an ideal gas at 28.5°C expands P2.15 A bottle at 325 K contains an ideal gas at a pressure of
3
3
isothermally from an initial volume of 22.5 dm to a final vol- 162.5 * 10 Pa . The rubber stopper closing the bottle is
3
ume of 75.5 dm . Calculate w for this process (a) for expan- removed. The gas expands adiabatically against P external =
3
5
sion against a constant external pressure of 1.00 * 10 Pa , 120.0 * 10 Pa , and some gas is expelled from the bottle in the
and (b) for a reversible expansion. process. When P = P external , the stopper is quickly replaced.
The gas remaining in the bottle slowly warms up to 325 K. What
P2.7 Calculate q, w, ¢U , and ¢H if 2.25 mol of an ideal
is the final pressure in the bottle for a monatomic gas, for which
gas with C V,m = 3R>2 undergoes a reversible adiabatic
expansion from an initial volume V = 5.50 m 3 to a final vol- C V,m = 3R>2 , and a diatomic gas, for which C V,m = 5R>2 ?
i
ume V f = 25.0 m 3 . The initial temperature is 275 K. P2.16 A 2.25 mole sample of an ideal gas with C V,m = 3R>2
5
initially at 310. K and 1.25 * 10 Pa undergoes a reversible
P2.8 Calculate w for the adiabatic expansion of 2.50 mol of an
adiabatic compression. At the end of the process, the pressure
ideal gas at an initial pressure of 2.25 bar from an initial temper- 6
is 3.10 * 10 Pa . Calculate the final temperature of the gas.
ature of 450. K to a final temperature of 300. K. Write an
Calculate q, w, ¢U , and ¢H for this process.
expression for the work done in the isothermal reversible expan-
sion of the gas at 300. K from an initial pressure of 2.25 bar. P2.17 A vessel containing 1.50 mol of an ideal gas with
What value of the final pressure would give the same value of w P = 1.00 bar and C P,m = 5R>2 is in thermal contact with a
i
as the first part of this problem? Assume that C P,m = 5R>2 . water bath. Treat the vessel, gas, and water bath as being in
thermal equilibrium, initially at 298 K, and as separated by
P2.9 At 298 K and 1 bar pressure, the density of water is
–3
-1
0.9970 g cm , and C P,m = 75.3 J K mol -1 . The change in adiabatic walls from the rest of the universe. The vessel, gas,
volume with temperature is given by ¢V = V initial b¢T and water bath have an average heat capacity of
-1
b
P
where , the coefficient of thermal expansion, is C = 2450. J K . The gas is compressed reversibly to
2.07 * 10 -4 K -1 . If the temperature of 325 g of water is P = 20.5 bar . What is the temperature of the system after
f
thermal equilibrium has been established?
increased by 25.5 K, calculate w, q, ¢H , and ¢U .
P2.18 An ideal gas undergoes an expansion from the initial state
P2.10 A muscle fiber contracts by 3.5 cm and in doing so
described by P , V , T to a final state described by P , V , T in (a) a
lifts a weight. Calculate the work performed by the fiber. i i f f
process at the constant external pressure P , and (b) in a reversible
Assume the muscle fiber obeys Hooke’s law F =-k x with f
–1
a force constant k of 750. N m . process. Derive expressions for the largest mass that can be lifted
through a height h in the surroundings in these processes.
P2.11 A cylindrical vessel with rigid adiabatic walls is sep-
P2.19 An ideal gas described by T = 275 K , P = 1.10 bar ,
arated into two parts by a frictionless adiabatic piston. Each i i
and V = 10.0 L is heated at constant volume until P =
part contains 45.0 L of an ideal monatomic gas with i
5
C V,m = 3R>2 . Initially, T = 300. K and P = 1.75 * 10 Pa 10.0 bar . It then undergoes a reversible isothermal expansion
i
i
in each part. Heat is slowly introduced into the left part using until P = 1.10 bar . It is then restored to its original state by the
extraction of heat at constant pressure. Depict this closed-cycle
an electrical heater until the piston has moved sufficiently to