Beginning with a pressure of 2.20 ×10^5 Pa and a volume of 6.34 ×10^-3 m^3, an ideal monatomic g

Beginning with a pressure of 2.20 ×10^5 Pa and a volume of...

Key Concepts

Monatomic Gas Characteristics

A monatomic gas consists of single-atom molecules, which simplifies its thermodynamic behavior. For such gases, the degrees of freedom are limited, and specific heat capacities (C_v and C_P) are determined accordingly. The ratio of the specific heats (? = C_P/C_v) is a constant value, which is particularly useful in analyzing adiabatic processes involving ideal monatomic gases.

Ideal Gas Law

The ideal gas law (PV = nRT) provides a relationship among pressure, volume, temperature, and the amount of gas. It is a crucial equation in thermodynamics that allows one to interrelate these properties in many processes, including adiabatic, isochoric, and isobaric changes, by providing a basis for calculating changes in state variables.

Work in Thermodynamic Processes

Work in a thermodynamic process is the energy transferred when a force moves the boundary of a system. It depends on the path taken by the process; for example, in an adiabatic process, work is directly related to the change in internal energy, whereas in an isobaric process it is computed as the product of the constant pressure and the change in volume. Understanding these differences is essential for comparing the energies involved in different thermodynamic paths.

Isochoric Process

An isochoric process is one in which the volume of the gas remains constant throughout the process. Since work is defined as the product of pressure and change in volume, no work is performed during an isochoric process. Any heat added or removed from the system solely changes its internal energy and temperature.

Adiabatic Process

An adiabatic process is one in which there is no heat exchange between the system and its surroundings. In this type of process for an ideal gas, any change in the internal energy of the gas is solely due to work done on or by the system. The relationship PV^? = constant, where ? is the heat capacity ratio, is a key characteristic of an adiabatic process and is used to relate pressure and volume changes during the process.

Isobaric Process

An isobaric process is characterized by a constant pressure during the process. In such processes, the work done by or on the system is given by the product of the constant pressure and the change in volume (W = P?V). This type of process is common in many practical scenarios where pressure remains fixed while the system undergoes volume change.

Transcript

00:01 Initial pressure of ideal gas is 2 .20 times 10 to the power 5 pascal so pi equal to 2 .20 times 10 to the power 5 pascal so final pressure is 8 .15 times 10 to the power 4 pascal initial volume of gases 6 .34 times 10 to the power minus 3 cubic meter and for ideal monotamic gas we have gamma equal to 5 upon now in the first process that is a diabetic process we can write final pressure times final volume to the power gamma is equal to initial pressure times initial volume to the power gamma so this gives us final volume to the power gamma is equal to initial pressure upon final pressure into vi to the power gamma now by taking power of one upon gamma on both side so this will give us final volume equal to initial pressure upon final pressure hold to the power one upon gamma into initial volume now by plugging different values in the relation final volume equal to here initial pressure is 2 .20 times 10 to power 5 pascal upon final pressure is 8 .15 times 10 to the power 4 pascal whole to the power 1 upon gamma here gamma is 5 upon 3 into into initial volume is 60 .3 4 4 times 10 to the power minus 3 cubic meter this gives us final volume equal to 11 .5 times 10 to the power minus 3 cubic meter the relation for work done in a debatic process is work done equal to initial volume into initial pressure minus final pressure into final volume upon 1 minus upon gamma minus 1 now by plugging different values here here initial pressure is 2 .20 times 10 to the power 5 pascal into initial volume is 60 .34 times 10 to the power minus 3 cubic meter minus initial pressure is 8 .15 times 10 to the power 4 pascal into initial into final volume is 11 .5 times 10 to the power minus 3 cubic meter upon gamma minus 1 here gamma is 5 upon 3 minus 1 so this gives us work done in a diabetic process equal to 6 and 6 at 6 0 .325 june now we want to calculate the work done in the alternative process in the alternative process the gas is cooled izoquorically first to the final pressure then it is azeuberically expanded to its final volume so again here in the isochoric cooling we have initial volume equal to final volume that is 60 .3 4 times 10 to the power minus 3 cubic meter so here initial pressure is 2 .20 times 10 to the power 5 pascal and after cooling to the final pressure we have final pressure equal to 8.

06:43 8 .15 times 10 to the power 4 pascal asd is no change in volume so delta v is equal to 0 this gives us work done as equal to 0 because of no change in volume now in the isabaric process isabic expanding we can calculate work done by the relation w equal to p into delta v here pressure is a constant at 8 .15 times 10 to the power 4 pascal into final volume is 11 .5 times 10 to the power minus 3 cubic meter minus initial volume is 6 .3 4 times 10 to the power minus 3 cubic meter this gives us work done equal to 420 .54 joules now we can calculate the total work in the alternative process so the total work in the alternative process equal to work done in the izochoric cooling that is zero plus work done in the isobaric expanding which is 420 .5 4 jule so this gives us total work equal to 420 .5 for jule now we want to calculate the difference in the work done when work is done on the guess idebatically and in the alternative process so the difference in work done equal to 686 .325 jule minus 420 .54 jule...

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