Isothermal - Revision history

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energy>Jmdonev at 18:24, 22 October 2021

2021-10-22T18:24:36Z

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	[[Category:Done ~~2018~~-06-15]]		[[Category:Done 2021-10-29]]
	[[File:isothermal PV cycle .jpg\|thumb\|400px\|right\| The [[Pressure volume diagram]] of an isothermal process.]]		[[File:isothermal PV cycle .jpg\|thumb\|400px\|right\| The [[Pressure volume diagram]] of an isothermal process.]]
	<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes—whether it be the [[pressure]], [[volume]] and/or contents—without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> ~~This then looks like~~:		<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes—whether it be the [[pressure]], [[volume]] and/or contents—without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> The total energy of the system (and the enthalpy) often changes in isothermal processes though. The first law of thermodynamics can be expressed mathematically as:

	<center><math>\Delta U = Q + W = 0</math></center>		<center><math>\Delta U = Q + W = 0</math></center>

	and ~~consequently,~~		Which can be simplified to show that the amount of heat and work is exactly equal when there's no change in temperature:

	<center><math>Q = -W </math></center>		<center><math>Q = -W </math></center>
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	*<math>\Delta U</math> is the change in internal energy		*<math>\Delta U</math> is the change in internal energy
	*<math>Q</math> is [[heat]]		*<math>Q</math> is [[heat]] entering the system
	*<math>W</math> is [[work]]		*<math>W</math> is [[work]] done on the system

	~~What these equations mean is~~ that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <math>W</math> value), then heat must be removed from the system in accordance (negative <math>Q</math> value). In contrast, if a container is allowed to expand (negative <math>W</math>), then heat must be added to the system in order to keep the temperature constant. To calculate work, an integration must be done to the formula <math>W = -∫pdV</math>. This can also be thought of as calculating the area under the curve. However, due to the shape of the curve, it's not as simple of a calculation—in comparison to an [[isobar\|isobaric]] process, for example. The formula below is the integrated equation, and will calculate the work done for any isothermal process:		This expression means that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <math>W</math> value), then heat must be removed from the system in accordance (negative <math>Q</math> value). In contrast, if a container is allowed to expand (negative <math>W</math>), then heat must be added to the system in order to keep the temperature constant. To calculate work, an integration must be done to the formula <math>W = -∫pdV</math>. This can also be thought of as calculating the area under the curve. However, due to the shape of the curve, it's not as simple of a calculation—in comparison to an [[isobar\|isobaric]] process, for example. The formula below is the integrated equation, and will calculate the work done for any isothermal process:

	<center><math>W = -nRTln\frac{V_{f}}{V_{i}} = -p_{i}V_{i}ln\frac{V_{f}}{V_{i}} = -p_{f}V_{f}ln\frac{V_{f}}{V_{i}}</math> <ref>R. Knight, Physics for scientists and engineers. San Fransisco: Pearson Addison Wesley, 2008, p. 512.</ref></center>		<center><math>W = -nRTln\frac{V_{f}}{V_{i}} = -p_{i}V_{i}ln\frac{V_{f}}{V_{i}} = -p_{f}V_{f}ln\frac{V_{f}}{V_{i}}</math> <ref>R. Knight, Physics for scientists and engineers. San Fransisco: Pearson Addison Wesley, 2008, p. 512.</ref></center>
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	*<math>V_{f}</math> is the final volume		*<math>V_{f}</math> is the final volume

			The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes since the temperature remains constant until the phase change is complete.
	The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.

	The following video from UC Berkley's chemistry department explains the idea of an isothermal process with visuals.		The following video from UC Berkley's chemistry department explains the idea of an isothermal process with visuals.
	<html>		<html>
	<iframe width="820" height="480" src="https://www.youtube.com/embed/8y5KX4kzt0A" frameborder="0" allowfullscreen></iframe></html>		<iframe width="820" height="480" src="https://www.youtube.com/embed/8y5KX4kzt0A" frameborder="0" allowfullscreen></iframe></html>


			===Full Integration of the Work Equation===
			<center><math>W = -∫pdV = -∫\frac{nRT}{V}dV = -nRT∫\frac{dV}{V} = -nRT({lnV_f}-{lnV_i}) = nRT({lnV_i}-{lnV_f}) </math>
			<math>W = nRTln(\frac{V_i}{V_f}) = -nRTln(\frac{V_f}{V_i})</math>

	==References==		==References==
	{{reflist}}[[Category:Uploaded]]		{{reflist}}[[Category:Uploaded]]

Jmdonev: 1 revision imported

2018-06-25T14:30:30Z

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Jmdonev at 01:19, 9 June 2018

2018-06-09T01:19:22Z

← Older revision		Revision as of 01:19, 9 June 2018
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	[[Category:Done 2018-05-18]]		[[Category:Done 2018-06-15]]
	[[File:isothermal PV cycle .jpg\|thumb\|400px\|right\| The [[PV diagram]] of an isothermal process.]]		[[File:isothermal PV cycle .jpg\|thumb\|400px\|right\| The [[Pressure volume diagram]] of an isothermal process.]]

	<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes—whether it be the [[pressure]], [[volume]] and/or contents—without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> This then looks like:		<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes—whether it be the [[pressure]], [[volume]] and/or contents—without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> This then looks like:

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	*<math>W</math> is [[work]]		*<math>W</math> is [[work]]

	What these equations mean is that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <math>W</math> value), then heat must be removed from the system in accordance (negative <math>Q</math> value). In contrast, if a container is allowed to expand (negative <math>W</math>), then heat must be added to the system in order to keep the temperature constant.		What these equations mean is that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <math>W</math> value), then heat must be removed from the system in accordance (negative <math>Q</math> value). In contrast, if a container is allowed to expand (negative <math>W</math>), then heat must be added to the system in order to keep the temperature constant. To calculate work, an integration must be done to the formula <math>W = -∫pdV</math>. This can also be thought of as calculating the area under the curve. However, due to the shape of the curve, it's not as simple of a calculation—in comparison to an [[isobar\|isobaric]] process, for example. The formula below is the integrated equation, and will calculate the work done for any isothermal process:

			<center><math>W = -nRTln\frac{V_{f}}{V_{i}} = -p_{i}V_{i}ln\frac{V_{f}}{V_{i}} = -p_{f}V_{f}ln\frac{V_{f}}{V_{i}}</math> <ref>R. Knight, Physics for scientists and engineers. San Fransisco: Pearson Addison Wesley, 2008, p. 512.</ref></center>

			where:

			* <math>n</math> is the number of moles
			*<math>R</math> is the [[ideal gas constant]]
			*<math>p_{i}</math> is the initial pressure
			*<math>V_{f}</math> is the final volume


	The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.		The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.

Jmdonev: 1 revision imported

2018-05-18T22:53:15Z

1 revision imported

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Jmdonev at 03:11, 18 May 2018

2018-05-18T03:11:43Z

← Older revision		Revision as of 03:11, 18 May 2018
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	[[Category:Done ~~2015~~-09-06]]		[[Category:Done 2018-05-18]]
	~~<onlyinclude>'''Isothermal''' refers to a process in which a~~ [[~~system]] changes - be it the [[pressure]], [[volume]] or contents - without the [[temperature]] changing~~.~~</onlyinclude> From the point of view of the~~ [[~~first law of thermodynamics]], this means that the [[internal energy~~]] of ~~the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule~~]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> This then looks like:		[[File:isothermal PV cycle .jpg\|thumb\|400px\|right\| The [[PV diagram]] of an isothermal process.]]

	<center><m>\Delta U = Q + W = 0</m></center>		<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes—whether it be the [[pressure]], [[volume]] and/or contents—without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> This then looks like:

			<center><math>\Delta U = Q + W = 0</math></center>

	and consequently,		and consequently,

	<center><m>Q = -W </m></center>		<center><math>Q = -W </math></center>

	where:		where:

	*<m>\Delta U</m> is the change in internal energy		*<math>\Delta U</math> is the change in internal energy
	*<m>Q</m> is [[heat]]		*<math>Q</math> is [[heat]]
	*<m>W</m> is [[work]]		*<math>W</math> is [[work]]

	What these equations mean is that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <m>W</m> value), then heat must be removed from the system in accordance (negative <m>Q</m> value). In contrast, if a container is allowed to expand (negative <m>W</m>), then heat must be ~~input~~ in order to keep the temperature constant.		What these equations mean is that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <math>W</math> value), then heat must be removed from the system in accordance (negative <math>Q</math> value). In contrast, if a container is allowed to expand (negative <math>W</math>), then heat must be added to the system in order to keep the temperature constant.

	The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.		The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.
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	==References==		==References==
	{{reflist}}		{{reflist}}[[Category:Uploaded]]

J.williams: 1 revision imported

2015-09-18T16:52:30Z

1 revision imported

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Jmdonev at 18:30, 17 September 2015

2015-09-17T18:30:51Z

New page

[[Category:Done 2015-09-06]]
<onlyinclude>'''Isothermal''' refers to a process in which a [[system]] changes - be it the [[pressure]], [[volume]] or contents - without the [[temperature]] changing.</onlyinclude> From the point of view of the [[first law of thermodynamics]], this means that the [[internal energy]] of the system is unchanged, since temperature is a measure of the average [[kinetic energy]] of [[molecule]]s within the system.<ref name=gould>H. Gould and J. Tobochnik, "Temperature," in ''Statistical and Thermal Physics'', 1st ed., Princeton, NJ: Princeton University Press, 2010, ch.2, sec.4, pp. 35-38</ref> This then looks like:

<center><m>\Delta U = Q + W = 0</m></center>

and consequently,

<center><m>Q = -W </m></center>

where:

*<m>\Delta U</m> is the change in internal energy
*<m>Q</m> is [[heat]]
*<m>W</m> is [[work]]

What these equations mean is that the work input to a system must be ''exactly'' balanced by a heat output, and vice versa. If an insulated container containing [[air]] is compressed (decreasing its volume, positive <m>W</m> value), then heat must be removed from the system in accordance (negative <m>Q</m> value). In contrast, if a container is allowed to expand (negative <m>W</m>), then heat must be input in order to keep the temperature constant.

The [[Carnot efficiency]] explaining the maximum [[thermal efficiency]] of a [[heat engine]] is derived by using isothermal processes, in which a thermodynamic cycle is completed with the use of 2 isothermal and 2 [[adiabatic]] processes.<ref name=Knight>R. D. Knight, "The Limits of Efficiency" in ''Physics for Scientists and Engineers: A Strategic Approach,'' 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.5, pp. 540-542</ref> [[Phase change]]s are an example of isothermal processes, since the temperature remains constant until the phase change is complete.

The following video from UC Berkley's chemistry department explains the idea of an isothermal process with visuals.
<html>
<iframe width="820" height="480" src="https://www.youtube.com/embed/8y5KX4kzt0A" frameborder="0" allowfullscreen></iframe></html>

==References==
{{reflist}}