(II) The P V diagram in Fig. 31 shows two possible states of...

(II) The $P V$ diagram in Fig. 31 shows two possible states of a system containing 1.55 moles of a monatomic ideal gas. $\left(P_{1}=P_{2}=455 \mathrm{N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{m}^{3}, \quad V_{2}=8.00 \mathrm{m}^{3} .\right)$ (a) Draw the process which depicts an isobaric expansion from state 1 to state $2,$ and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume $V_{2}$ followed by an isoyolumetric increase in temperature to state $2,$ and label this process $\mathrm{B}$ . $(d)$ Find the change in internal energy of the gas for the two-step process B.

(II) The $P V$ diagram in Fig. 31 shows two possible states of
a system containing 1.55 moles of a monatomic ideal
gas. $\left(P_{1}=P_{2}=455 \mathrm{N} / \mathrm{m}^{2}, \quad V_{1}=2.00 \mathrm{m}^{3}, \quad V_{2}=8.00 \mathrm{m}^{3} .\right)$
(a) Draw the process which depicts an isobaric expansion
from state 1 to state $2,$ and label this process A. (b) Find the
work done by the gas and the change in internal energy of the
gas in process A. (c) Draw the two-step process which depicts
an isothermal expansion from state 1 to the volume $V_{2}$
followed by an isoyolumetric increase in temperature to
state $2,$ and label
this process $\mathrm{B}$ . $(d)$
Find the change
in internal energy
of the gas for the
two-step process B.

Key Concepts

Pressure-Volume (PV) Diagram

A PV diagram is a graphical representation of the state of a thermodynamic system, where pressure is plotted against volume. It helps visualize the path taken during various processes, with the area under a process curve representing the work done by or on the system. This tool is fundamental in understanding and analyzing how different processes change the state of a system.

Isobaric Process

An isobaric process is one in which the pressure of the system remains constant while the volume changes. In these processes, the work done by the system is calculated simply as the constant pressure multiplied by the change in volume. This concept is key when evaluating heat transfer, work, and the corresponding change in internal energy under constant pressure conditions.

Isothermal Process

An isothermal process occurs at a constant temperature for the entire process. For ideal gases, this means that the internal energy does not change, and the work done is exactly balanced by the heat exchange with the surroundings. This process is critical in understanding how energy is transferred while maintaining thermal equilibrium, and it is typically analyzed using the logarithmic relation between initial and final volumes.

Isochoric Process

An isochoric process is defined by a constant volume, meaning that no work is performed as the system's volume does not change. In these processes, any heat exchanged results in a change solely in the internal energy of the system, making them important for understanding the direct relationship between heat and internal energy without the complication of work.

Work Calculation in Thermodynamic Processes

Calculating work in thermodynamic processes involves understanding the relation between pressure and volume change. The work done by a gas is typically represented as the area under the process curve on a PV diagram. Different processes have different work expressions; for instance, an isobaric process uses W = P?V, while an isothermal process uses a logarithmic relation derived from the ideal gas law.

Internal Energy of an Ideal Gas

For an ideal gas, the internal energy is a function solely of its temperature and is given by a specific expression that depends on the degrees of freedom (for a monatomic gas, U = (3/2)nRT). This concept is central to thermodynamics, as it connects how changes in temperature affect the energy stored in the gas, independent of the volume or pressure changes when specific thermodynamic processes are considered.

First Law of Thermodynamics

The first law of thermodynamics, a statement of energy conservation, asserts that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system. This law provides the fundamental framework for analyzing energy transfers in any thermodynamic process and is vital for understanding the interplay between heat, work, and changes in internal energy.

Transcript

00:01 So we are given a mono -atomic gas of 1 .55 moles.

00:08 And we are also given two states, a and b, corresponding to pressure 455 newton per meter square and volume 2 meter cube and 8 meter cubed.

00:27 So in the graph, in the graph, these points would correspond to 2 comma 455, which would be here, and 8 comma 455, it should be here.

00:41 So let's use a different color.

00:43 This is v1 and this is v2 blue.

00:49 Now that we have this, we want to find out for a, the isobaric expansion from point state 1 to state 2.

01:00 So isobaric means constant pressure, which means the y component must be constant, which simply means that the process will go like this.

01:10 Because why is constant, our process must be parallel to our x -axis.

01:16 This is our process.

01:19 For part b, we want to find out the amount of work done.

01:22 We know that work is simply integral pdv.

01:26 And because the pressure is constant here, this is 202.

01:30 The pressure is 455 newton per meter square times.

01:37 The volume change is 8 minus 2 meter cubed.

01:42 Final eight minus initial two.

01:45 If you notice, the units become newton meter and newton meter is a jewel.

01:50 And this number turns out to be 2 ,730 joules.

01:54 This is the amount of work done.

01:59 Now for part c, we want to think of a process where the system goes from 1 to the volume 8 state 1 to volume 8 meter cubed through an isothermal expansion.

02:15 An isothermal expansion usually has pb equal to constant, which means it's of the form x times y is equal to constant, that is y is equal to k by x.

02:26 So the process must look something like this.

02:30 This is the curve.

02:32 This would be the curve for y, x, y, is equal to constant.

02:36 So our process must correspond to, it must start from here, and it must go like this all the way until it hits 8 meter cubed.

02:47 Now at this point, let's use a different color.

02:53 At this point, we want an iso volumetric expansion.

02:58 That is, we want the volume to be constant, and we want our process to go back to state two, which means we simply have this process.

03:08 Because the volume is constant, it must be parallel to y -axis.

03:12 So instead of having this process, this one process, let's call it a...