Repeat Prob. 8-106 by assuming the piston is made of 5 kg of copper initially at the average te

Repeat Prob. 8-106 by assuming the piston is made of 5 kg...

Key Concepts

Thermal Equilibrium

Thermal equilibrium is the state in which all parts of a system have reached the same temperature, resulting in no net heat flow between them. In processes involving partitions such as pistons, establishing thermal equilibrium ensures that energy exchanges between the components, like the gases on either side and the piston material, culminate in a uniform temperature distribution.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a given amount. For a material like copper, which is used in the piston, the finite heat capacity indicates that it can absorb or release heat during the process, affecting the rate and extent to which the system reaches thermal equilibrium.

Conduction in Metals

Conduction in metals refers to the efficient transfer of thermal energy through a material due to its free electrons. Copper, known for its high thermal conductivity, facilitates rapid heat transfer within the piston and between the piston and the adjacent gases, influencing the temperature gradients and the dynamics of achieving equilibrium.

Energy Conservation in Thermodynamic Systems

Energy conservation in thermodynamics is the principle that energy cannot be created or destroyed; it can only change forms. In a system consisting of gases and a copper piston, all energy exchanges in the form of work and heat transfer must account for the conservation of the total energy, allowing for the calculation of final equilibrium states when all components interact.

Piston Dynamics in Gas Systems

Piston dynamics in gas systems encompass the study of how a movable barrier mediates the exchange of energy and work between different gas compartments. The material properties of the piston, including its mass and heat capacity, determine how it influences and responds to the thermal and mechanical interactions between the gases, affecting the overall behavior and final equilibrium of the system.

Transcript

00:01 So in this problem, we ask to repeat the previous problem, assuming that this piston here is now made out of copper.

00:10 So it can store some thermal energy.

00:16 We assume that it doesn't deform so that it doesn't store any elastic potential energy.

00:23 So we proceed pretty much exactly like we did in the previous problem.

00:28 Now we have to worry about the heat capacity of the heat capacities.

00:37 So the temperature of the copper was initially the average temperature.

00:43 We're going to assume.

00:44 Again, there would be a temperature gradient, obviously, but we'll just assume that it's an average temperature.

00:50 Now, everything else is the same.

00:54 The heat capacities and the ideal gas constants for the gases are the same.

00:59 The temperatures of the gases and pressures of the gases are the same initially.

01:04 The universal gas constant is obviously the same.

01:07 But now we need to consider the heat capacity of this copper here.

01:11 And also the mass of the copper is 5 kilograms.

01:15 So the mass of the two gases is the same as in the previous problem.

01:20 But we also know that the change in internal energy in the tank as a whole has to be zero because there's no heat transfer and there's no work.

01:30 So, and it's the same as before with the two gases, but now we have to add in the internal energy change in the copper.

01:38 And again, all of these t2s are going to be the same, because in the end, this is all going to be in thermal equilibrium.

01:45 And so this equation gives us a, this equation here gives us a formula for solving for t2.

01:53 And that winds up being 56 .0 degrees c.

01:58 And in the previous problem, it was 57 .2 degrees.

02:01 So it's a little colder because the copper has absorbed some energy because again, its temperature rose.

02:12 Now, the entropy generated is the same as with the gases, but now we have to consider the entropy generated in the copper.

02:20 Now, the number of total moles of gas is not changed because the copper is obviously not gas.

02:28 So the pressure after it's equilibrated doesn't change either.

02:36 So that's 509 .4 kilopascals again.

02:42 The formulas for the entropy for the gases are the same as before.

02:46 But now we have the entropy for the copper.

02:49 And again, we have the mass of the copper, the heat capacity, the copper, and the temperature ratio.

02:55 And obviously the pressure in the copper is the same.

02:59 So that winds up being a value of 0 .021 kilowatres per kilogram kelvin...

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