60
2 Work, Heat, and Energy: The First Law of Thermodynamics
P R O B L E M S
Section 2.3: Internal Energy: The First Law of
Thermodynamics
2.17 According to special relativity, the total energy of a system
is given by
E mc
2
where m is the mass of the system and c is the speed of
light.
a. Assuming a substance with a molar mass of
0.100 kg mol −1 , calculate the amount of energy
necessary to change the molar mass from its rest-mass
value to a value larger by 1.0 ppm (part per
million).
b. Compare this energy with a typical chemical bond
energy, roughly 400 kJ mol −1 .
c. If the energy in part a were all kinetic energy of the
center of mass of the system, calculate the speed of the
center of mass of the system of 1.000 mol.
2.18 a. Calculate q, w, and ∆U if 2.000 mol of neon (assumed
ideal) is heated at a constant pressure of 1.000 atm
from a temperature of 0.00 ◦ C to a temperature of
250.00 ◦ C.
b. Calculate q, w, and ∆U if the same sample of neon is
heated at a constant volume from the same initial state
to 250.00 ◦ C and is then expanded isothermally to the
same final volume as in part a.
2.19 Calculate q, w, and ∆U for melting 100.0 g of ice at 0.0 ◦ C
and a constant pressure of 1.000 atm. The density of ice is
0.916 g mL −1 .
2.20 Calculate q, w, and ∆U for vaporizing 2.000 mol of liquid
water at 100.0 ◦ C to steam at 100.0 ◦ C at a constant pressure
of 1.000 atm.
2.21 Consider the following three processes: (1) A sample of
2.000 mol of helium gas is isothermally and reversibly
expanded from a volume of 10.00 L and a temperature of
400.0 K to a volume of 40.00 L. (2) The same sample is
reversibly cooled at a constant volume of 10.00 L from
400.0 K to a temperature of 300.0 K, then expanded
reversibly and isothermally to a volume of 40.00 L, and
then heated reversibly from 300.0 K to 400.0 K at a
constant volume of 40.00 L. (3) The same sample is
expanded irreversibly and isothermally at a temperature of
400.0 K from a volume of 10.00 L to a volume of 40.00 L
with a constant external pressure of 1.000 atm. Calculate
∆U, q, and w for each process.
2.22 1.000 kg of water is pressurized isothermally at 298.15 K
from a pressure of 1.000 atm to a pressure of 10.00 atm.
Calculate w for this process. State any
assumptions.
2.4
Calculation of Amounts of Heat and Energy
Changes
Like work, heat is not a state function. The amount of heat put into a system can depend
on the path taken from the initial to the final state, as was the case with work.
Exercise 2.7
Show that if
dU dw + dq
and if dU is an exact differential and if dw is an inexact differential then dq must be an inexact
differential.
2 Work, Heat, and Energy: The First Law of Thermodynamics
P R O B L E M S
Section 2.3: Internal Energy: The First Law of
Thermodynamics
2.17 According to special relativity, the total energy of a system
is given by
E mc
2
where m is the mass of the system and c is the speed of
light.
a. Assuming a substance with a molar mass of
0.100 kg mol −1 , calculate the amount of energy
necessary to change the molar mass from its rest-mass
value to a value larger by 1.0 ppm (part per
million).
b. Compare this energy with a typical chemical bond
energy, roughly 400 kJ mol −1 .
c. If the energy in part a were all kinetic energy of the
center of mass of the system, calculate the speed of the
center of mass of the system of 1.000 mol.
2.18 a. Calculate q, w, and ∆U if 2.000 mol of neon (assumed
ideal) is heated at a constant pressure of 1.000 atm
from a temperature of 0.00 ◦ C to a temperature of
250.00 ◦ C.
b. Calculate q, w, and ∆U if the same sample of neon is
heated at a constant volume from the same initial state
to 250.00 ◦ C and is then expanded isothermally to the
same final volume as in part a.
2.19 Calculate q, w, and ∆U for melting 100.0 g of ice at 0.0 ◦ C
and a constant pressure of 1.000 atm. The density of ice is
0.916 g mL −1 .
2.20 Calculate q, w, and ∆U for vaporizing 2.000 mol of liquid
water at 100.0 ◦ C to steam at 100.0 ◦ C at a constant pressure
of 1.000 atm.
2.21 Consider the following three processes: (1) A sample of
2.000 mol of helium gas is isothermally and reversibly
expanded from a volume of 10.00 L and a temperature of
400.0 K to a volume of 40.00 L. (2) The same sample is
reversibly cooled at a constant volume of 10.00 L from
400.0 K to a temperature of 300.0 K, then expanded
reversibly and isothermally to a volume of 40.00 L, and
then heated reversibly from 300.0 K to 400.0 K at a
constant volume of 40.00 L. (3) The same sample is
expanded irreversibly and isothermally at a temperature of
400.0 K from a volume of 10.00 L to a volume of 40.00 L
with a constant external pressure of 1.000 atm. Calculate
∆U, q, and w for each process.
2.22 1.000 kg of water is pressurized isothermally at 298.15 K
from a pressure of 1.000 atm to a pressure of 10.00 atm.
Calculate w for this process. State any
assumptions.
2.4
Calculation of Amounts of Heat and Energy
Changes
Like work, heat is not a state function. The amount of heat put into a system can depend
on the path taken from the initial to the final state, as was the case with work.
Exercise 2.7
Show that if
dU dw + dq
and if dU is an exact differential and if dw is an inexact differential then dq must be an inexact
differential.