pressure of 1.000 atm. Neglect the volume of the liquid compared with that of the vapor.
2.16 A sample of 2.000 mol of CO2gas is heated from 0.00◦C to 100.0◦C at a constant pressure of 1.000 atm. Findqandw for this process. State any assumptions or approximations.
a. Assume thatCP,mis constant and equal to its value in Table A.8 at 298.15 K. Assume that the gas is ideal.
b. Assume thatCP,mis given by the formula in Table A.6 with the parameters given there and that the gas is described by the truncated pressure virial equation of state:
PVmRT +A2P and use the fact thatA2B2.
2.3 Internal Energy: The First Law of Thermodynamics
Although Lavoisier discredited the phlogiston theory of combustion, which held that combustion was the loss of an “imponderable” fluid calledphlogiston, he was one of the principal promoters of the equally incorrectcaloric theory of heat espoused by Black, which asserted that heat was an imponderable fluid called “caloric.” The first experimental studies that discredited the caloric theory were done by Count Rumford.
Rumford was at one time in charge of manufacturing cannons for the Elector of Bavaria, the ruler who made him a count. Rumford noticed that when a cannon was bored with a dull boring tool, more heat was produced than when a sharp tool was used. He carried out a systematic set of experiments and was able to show by using a very dull tool that there was no apparent limit to the amount of heat that could be generated by friction. He immersed the cannon in water to transfer the heat from the cannon and even generated enough heat to have melted the cannon. Rumford’s results showed that “caloric” was not simply being extracted from the cannon, because if caloric existed only a definite amount could be stored in a cannon without melting it. Work must have been converted to heat.
Benjamin Thompson, Count Rumford, 1753–1814, was an American- British physicist who abandoned his family and left America after the American revolution because of his royalist sympathies. He pursued a checkered career in various countries, including Bavaria (where he ingratiated himself with the Elector of Bavaria), France (where he married Lavoisier’s widow), and England (where he founded the Royal Institution and hired Humphrey Davy as a lecturer).
Rumford calculated an approximate value for the “mechanical equivalent of heat,”
or the amount of heat to which a joule of work could be converted. Better values were obtained by Mayer in 1842 and Joule in 1847. Joule carried out experiments in which changes of temperature were produced either by doing work on a system or by heating it. His apparatus is schematically depicted in Figure 2.4. A falling mass turned a stirring paddle in a sample of water, doing work on the liquid. The rise in temperature of the water was measured and the amount of work done by the falling mass was compared with the amount of heat required to produce the same change in temperature. Joule found that the ratio of the work to the amount of heat was always the same, approximately 4.18 J of work for 1.00 cal of heat. The calorie is now defined to be exactly 4.184 J.
Julius Robert Mayer, 1814−1878, was a German physicist originally trained as a physician. He was apparently the first to assert that heat and work are two different means by which energy is transferred, and that energy can neither be created nor destroyed.
E X A M P L E 2.12
Calculate the rise in temperature of 100.0 g of water if the falling weight of Figure 2.4 has a mass of 5.00 kg and drops by 0.800 m. Neglect friction in the pulleys.
Solution
LetV be the potential energy of the mass.
∆V (5.00 kg)(9.80 ms−2)(−0.800 m) −39.2 J
Thermometer
Cord wound on drum
Pulley
Weight Stirrer
Figure 2.4 Joule’s Apparatus for Determining the Mechanical Equivalent of Heat (Schematic).
qC∆T (4.18 J K−1g−1)(100.0 g)∆T 418 J K−1 −∆V
∆T 39.2 J
418 J K−1 0.0937 K
Joule found that there was no detectable difference in the final state of the system whether its temperature was raised by doing work on it or by heating it. This shows that heat and work are two different means of changing a single property of the system.
Since energy is defined as the capacity to do work, the work done on the sample of water must have increased its energy. Therefore the heat transferred must also have increased its energy. Joule’s sample of water could have kinetic energy if its center of mass were moving, and it could have gravitational potential energy. The work and heat that Joule added to his system did not change either of these forms of energy, so there must be an additional form of energy, which we call theinternal energyor the thermodynamic energyand denote byU. If the system is close to the surface of the earth, the total energy of the system is
Etotal 1
2mv2c+Mgzc+U (2.3-1)
wheremis the mass of the system,vcis the speed of its center of mass,zcis the vertical coordinate of its center of mass, andgis the acceleration due to gravity.
Based on the experiments of Rumford, Mayer, Helmholtz, Joule, and many others since the time of Joule, we now state thefirst law of thermodynamicsas it applies to a system whose kinetic and potential energy do not change:For a closed system and for any process that begins and ends with equilibrium states
∆Uq+w (2.3-2)
whereqis the amount of heat transferred to the system andwis the work done on the system and where∆Uis the change in the value of U,the internal energy, which is a state function:
∆UU(final)−U(initial) (2.3-3) In spite of the work of Rumford, Mayer, and Joule the credit for announcing the first law of thermodynamics went to Helmholtz.
Hermann Ludwig von Helmholtz, 1821−1894, was a German physicist and physiologist who studied the energy of muscle contraction and who was one of the first to propose that the energy for all processes on the earth ultimately came from solar radiation.
We accept the first law of thermodynamics as an experimentally established law and accept the internal energy as a state function. This law is a version of the law of conservation of energy, which is a general law of physics to which there are no known exceptions. Apparent violations of energy conservation led particle physicists to search for previously unknown particles that could be transferring energy to or from a system, leading to the discovery of theneutrino.2Occasionally an unknown inventor in search of gullible investors announces a machine that will produce more energy than it takes in, violating the first law of thermodynamics. Such nonexistent machines are known asperpetual motion machines of the first kind.
It is the total energy of a system that is governed by conservation of energy. The first law of thermodynamics as stated in Eq. (2.3-2) applies to a closed system whose center of mass is not accelerated and whose gravitational potential energy does not change.
If work is done to change the kinetic or potential energy of the system as a whole this amount of work must be subtracted from the total work done to obtain the amount of work that changes the internal energy.
The internal energy includes the rest-mass energy of the system. We always deal only with energy changes or measure energy relative to some specified reference state in order to avoid including the rest-mass energy in our calculations.
E X A M P L E 2.13
Find the value of the rest-mass energy of 1.000 mol of argon gas, using Einstein’s equation, Emc2.
Solution
Erest-mass(0.039948 kg)(2.9979×108m s−1)23.5903×1015J
2E. Fermi,Z. Physik,88, 161 (1934).
Exercise 2.6
a. Assume that argon is an ideal gas withCV,m12.47 J K−1mol−1. Find∆Uif 1.000 mol of argon gas is heated at constant volume from 298.15 K to 500.0 K. Find the ratio of this energy difference to the rest-mass energy of the system. Find the difference between the observed mass of the system at 298.15 K and at 500.0 K.
b. Explain why it would be difficult to use values of total energies for chemical purposes if the rest-mass energy were included.
For a one-phase simple system containing one component the equilibrium state is specified by the values of three variables, at least one of which must be extensive. Since the internal energy is a state function we can write
UU(T,V,n) (2.3-4)
or
UU(T,P,n) (2.3-5)
The internal energy is a state function, but heat and work are not state functions.
Because heat and work are both means of changing the value of the internal energy, they do not maintain separate identities after a transfer of energy is finished. The following analogy has been used.3Heat transferred to the system is analogous to rain falling on a pond, work done on the system is analogous to the influx of a stream into the pond, and energy is analogous to water in the pond. Evaporation (counted as negative rainfall) is analogous to heat flow to the surroundings, and efflux from the pond into a second stream is analogous to work done on the surroundings. Once rain falls into the pond, it is no longer identifiable as rain, but only as water. Once stream flow is in the pond, it also is identifiable only as water, and not as stream flow. The amount of water in the pond is a well-defined quantity (a state function), but one cannot separately state how much rain and how much stream flow are in the pond. Similarly, there is no such thing as the heat content of a system in a given state and no such thing as the work content of a system in a given state.