The First Law of Thermodynamics and Some Simple Processes Quiz

PDF page 688: A process by which a gas does work on a piston at constant pressure is called an isobaric process. Since the pressure is constant, the force exerted is constant and the work done is given as 15.10

PDF page 689: diagram for an isobaric process. You can see in the figure that the work done is the area under the graph. This property of diagrams is very useful and broadly applicable: the work done on or by a system in going from one state to another equals the area under the curve on a diagram.

PDF page 690: the change in volume, or the area under the curve over that interval. Thus the total area under the curve equals the total work done. (b) Work must be done on the system to follow the reverse path. This is interpreted as a negative area under the curve. We can see where this leads by considering Figure 15.11(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in Figure 15.11(b), then work is done on the system. The area under the curve in that case is negative, because is negative. diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in Figure 15.12(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure 15.12(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a diagram, as Figure 15.12(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for

PDF page 690: the change in volume, or the area under the curve over that interval. Thus the total area under the curve equals the total work done. (b) Work must be done on the system to follow the reverse path. This is interpreted as a negative area under the curve. We can see where this leads by considering Figure 15.11(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in Figure 15.11(b), then work is done on the system. The area under the curve in that case is negative, because is negative. diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in Figure 15.12(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure 15.12(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a diagram, as Figure 15.12(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for

PDF page 692: diagram. For comparison, both are shown starting from the same point A. The upper curve ending at point B is an isothermal process—that is, one in which temperature is kept constant. If the gas behaves like an ideal gas, as is often the case, and if no phase change occurs, then . Since is constant, is a constant for an isothermal process. We ordinarily expect the temperature of a gas to decrease as it expands, and so we correctly suspect that heat transfer must occur from the surroundings to the gas to keep the temperature constant during an isothermal expansion. To show this more rigorously for the special case of a monatomic ideal gas, we note that the average kinetic energy of an atom in such a gas is given by

PDF page 693: where is the number of atoms in the gas. This relationship means that the internal energy of an ideal monatomic gas is constant during an isothermal process—that is, . If the internal energy does not change, then the net heat transfer into the gas must equal the net work done by the gas. That is, because here, . We must have just enough heat transfer to replace the work done. An isothermal process is inherently slow, because heat transfer occurs continuously to keep the gas temperature constant at all times and must be allowed to spread through the gas so that there are no hot or cold regions. Note that diatomic gases, such as those in air, have different formulas for average kinetic energy of an atom and total internal energy. Also shown in Figure 15.13(a) is a curve AC for an adiabatic process, defined to be one in which there is no heat transfer—that is, . Processes that are nearly adiabatic can be achieved either by using very effective insulation or by performing the process so fast that there is little time for heat transfer. Temperature must decrease during an adiabatic expansion process, since work is done at the expense of internal energy:

PDF page 690: the change in volume, or the area under the curve over that interval. Thus the total area under the curve equals the total work done. (b) Work must be done on the system to follow the reverse path. This is interpreted as a negative area under the curve. We can see where this leads by considering Figure 15.11(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in Figure 15.11(b), then work is done on the system. The area under the curve in that case is negative, because is negative. diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in Figure 15.12(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure 15.12(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a diagram, as Figure 15.12(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for

PDF page 690: the change in volume, or the area under the curve over that interval. Thus the total area under the curve equals the total work done. (b) Work must be done on the system to follow the reverse path. This is interpreted as a negative area under the curve. We can see where this leads by considering Figure 15.11(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in Figure 15.11(b), then work is done on the system. The area under the curve in that case is negative, because is negative. diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in Figure 15.12(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure 15.12(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a diagram, as Figure 15.12(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for

PDF page 690: the change in volume, or the area under the curve over that interval. Thus the total area under the curve equals the total work done. (b) Work must be done on the system to follow the reverse path. This is interpreted as a negative area under the curve. We can see where this leads by considering Figure 15.11(a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in Figure 15.11(b), then work is done on the system. The area under the curve in that case is negative, because is negative. diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in Figure 15.12(a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in Figure 15.12(b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a diagram, as Figure 15.12(c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for

PDF page 692: diagram. For comparison, both are shown starting from the same point A. The upper curve ending at point B is an isothermal process—that is, one in which temperature is kept constant. If the gas behaves like an ideal gas, as is often the case, and if no phase change occurs, then . Since is constant, is a constant for an isothermal process. We ordinarily expect the temperature of a gas to decrease as it expands, and so we correctly suspect that heat transfer must occur from the surroundings to the gas to keep the temperature constant during an isothermal expansion. To show this more rigorously for the special case of a monatomic ideal gas, we note that the average kinetic energy of an atom in such a gas is given by

PDF page 693: where is the number of atoms in the gas. This relationship means that the internal energy of an ideal monatomic gas is constant during an isothermal process—that is, . If the internal energy does not change, then the net heat transfer into the gas must equal the net work done by the gas. That is, because here, . We must have just enough heat transfer to replace the work done. An isothermal process is inherently slow, because heat transfer occurs continuously to keep the gas temperature constant at all times and must be allowed to spread through the gas so that there are no hot or cold regions. Note that diatomic gases, such as those in air, have different formulas for average kinetic energy of an atom and total internal energy. Also shown in Figure 15.13(a) is a curve AC for an adiabatic process, defined to be one in which there is no heat transfer—that is, . Processes that are nearly adiabatic can be achieved either by using very effective insulation or by performing the process so fast that there is little time for heat transfer. Temperature must decrease during an adiabatic expansion process, since work is done at the expense of internal energy:

PDF page 693: where is the number of atoms in the gas. This relationship means that the internal energy of an ideal monatomic gas is constant during an isothermal process—that is, . If the internal energy does not change, then the net heat transfer into the gas must equal the net work done by the gas. That is, because here, . We must have just enough heat transfer to replace the work done. An isothermal process is inherently slow, because heat transfer occurs continuously to keep the gas temperature constant at all times and must be allowed to spread through the gas so that there are no hot or cold regions. Note that diatomic gases, such as those in air, have different formulas for average kinetic energy of an atom and total internal energy. Also shown in Figure 15.13(a) is a curve AC for an adiabatic process, defined to be one in which there is no heat transfer—that is, . Processes that are nearly adiabatic can be achieved either by using very effective insulation or by performing the process so fast that there is little time for heat transfer. Temperature must decrease during an adiabatic expansion process, since work is done at the expense of internal energy: