A gas is suddenly heated from A to B from a pressure of 2.0...

A gas is suddenly heated from A to B from a pressure of 2.0 atm and temperature $300 \mathrm{K}$ to a pressure of 5.0 atm. The volume stays constant at $2.0 \mathrm{L}$. From $\mathrm{B}$ the gas expands isothermally to a volume of $5.0 \mathrm{L}$ at $\mathrm{C}$. The gas then returns to its original state at A by an isobaric compression at a pressure of $2.0 \mathrm{atm}$ (a) Sketch these changes on a pressurevolume diagram. (b) Find the temperature of the gas at $B$. (c) Find the internal energy change and the thermal energy given to or taken from the gas from A to B. (d) For the leg from $C$ to A find (i) the work done, (ii) the change in internal energy and (iii) the thermal energy given to or taken from the gas.

A gas is suddenly heated from A to B from a pressure of 2.0 atm and temperature $300 \mathrm{K}$ to a pressure of 5.0 atm. The volume stays constant at $2.0 \mathrm{L}$. From $\mathrm{B}$ the gas expands isothermally to a volume of $5.0 \mathrm{L}$ at $\mathrm{C}$. The gas then returns to its original state at A by an isobaric compression at a pressure of
$2.0 \mathrm{atm}$
(a) Sketch these changes on a pressurevolume diagram.
(b) Find the temperature of the gas at $B$.
(c) Find the internal energy change and the thermal energy given to or taken from the gas from A to B.
(d) For the leg from $C$ to A find (i) the work done, (ii) the change in internal energy and (iii) the thermal energy given to or taken from the gas.

Key Concepts

Pressure-Volume Diagram

A pressure-volume diagram is a graphical representation of a gas's state changes, plotting pressure against volume. It provides a visual method to analyze and compare different thermodynamic processes, such as constant volume, isothermal, and isobaric changes, and it helps in understanding work done by or on the gas as the area under the curve in the diagram.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and amount of gas. This law is fundamental in thermodynamics for predicting how a gas will behave under various changes in conditions, and it is essential for calculating unknown variables when a gas undergoes processes like heating, expansion, or compression.

Isochoric Process

An isochoric process is one in which the volume remains constant. In this process, because there is no change in volume, no work is done by the system. Consequently, any change in the internal energy of the gas is solely due to the heat transferred, making it an important case when applying the first law of thermodynamics.

Isothermal Process

An isothermal process is characterized by a constant temperature throughout the change. For ideal gases, this implies that the internal energy remains constant, and the work performed by the gas during expansion or compression is exactly balanced by the heat transferred to or from the system. This process is key in understanding reversible transformations and energy exchange without internal energy changes.

Isobaric Process

An isobaric process occurs at constant pressure. In such a process, the work done by or on the gas can be directly computed as the product of the pressure and the change in volume. The change in internal energy of the gas is related to the heat transferred and the work performed, making this process a clear example of energy conservation principles within thermodynamics.

First Law of Thermodynamics

The first law of thermodynamics, stated as ?U = Q + W, is a principle of energy conservation that relates the change in internal energy (?U) of a system to the heat added (Q) and the work done (W) by or on the system. This law is crucial for analyzing energy exchanges during various thermodynamic processes, allowing for the calculation of unknown energy components when the others are known.

Transcript

00:01 Here is an example of a device with an ideal gas undergoing a cycle.

00:08 Looking at the cycle, this could be a heat engine where work is being done during the entire cycle.

00:18 So we're going to analyze some things about certain legs.

00:22 There are three of them.

00:24 And we will start with writing down the important items.

00:29 One is the ideal gas law, which we all know and love, is pressure times volume, equals number of moles, times the gas constant, times the absolute temperature.

00:46 But we will usually, you need the first law of thermodynamics as well, which is simply energy conservation.

00:57 The first law says that the change in internal energy of the gas is equal to the work done by the gas.

01:10 Let's see.

01:11 No, let's get the signs right.

01:14 The internal energy of the gas change is equal to the heat absorbed or exhausted by the gas minus the work done by the gas.

01:26 So the convention is we are calculating the work done by the gas, which can be seen on the chart by whether the gas is expanding or contracting.

01:41 So we will look at some of the quantities during each of the legs in the process -the -engine cycle.

01:52 The first leg is a -2 -b, and we can see ostensibly that the work done is zero.

02:01 Since there is no change to the volume.

02:11 And what we are after is the internal energy change, which is equal to the amount of heat absorbed or expelled.

02:24 Now, another important thing that goes with the ideal gas law is that the change in internal energy is solely dependent upon the number of moles and the change in temperature.

02:39 It is usually considered that the number of moles does not change, but the temperature changes.

02:45 So what we'll need to do is determine the temperature at point b.

02:51 We are told that the temperature at point a starts at 300 kelvin.

02:57 So we need to find the temperature at point b.

03:00 And to do that, we can certainly use the ideal gas law at the pressure at point b.

03:08 Ratio with pressure at point a is the ratio of the temperatures.

03:21 So a simple thing to find the temperature at point b.

03:26 Now, typically, it's a good idea to put all your quantities in si units.

03:33 Here, because we have a ratio, i am ignoring for now the conversion of the pressures into pascal's.

03:41 But the pressure, the temperature at point b turns out to be 750 kelvin.

04:00 This is typical when the pressure goes up like that, the temperature had to have gone up.

04:08 And now we can calculate the change in eternal energy.

04:13 Now, they should have told us what type of gas we had, whether it was monatomic or diatomic or something else.

04:21 But since they didn't tell us, we are going to assume the specific heat capacity for a monotomic gas, which is three halves are, and we now have the temperature change, but we don't know the number of moles.

04:43 So we have one more thing we're going to have to determine from the ideal gas law, and there it probably pays to use point a.

05:03 And it also helps to convert to pascal's, which is roughly almost just 10 to the fifth times higher than the pressure in atmospheres.

05:34 A little bit different than that.

05:36 So let's go ahead and put into the ideal gas law to solve for the number of moles.

06:02 And we want our volume in cubic meters...

Please give Ace some feedback