Ideal gas: Heat capacity - Revision history

Carl McBride: Removed resultbox as it seems to break the MathJax

2012-05-11T14:34:17Z

Removed resultbox as it seems to break the MathJax

← Older revision		Revision as of 16:34, 11 May 2012
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	where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy \| kinetic energy of an ideal gas]], which is equal to <math>(3/2)RT</math>. Thus		where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy \| kinetic energy of an ideal gas]], which is equal to <math>(3/2)RT</math>. Thus

	:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>.		:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>

	At constant [[pressure]] one has		At constant [[pressure]] one has
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	thus		thus

	~~{{resultbox\|~~<math>C_p = C_v + R = \frac{5}{2} R</math>}}		:<math>C_p = C_v + R = \frac{5}{2} R</math>

	where <math>R</math> is the [[molar gas constant]].		where <math>R</math> is the [[molar gas constant]].

Carl McBride: Test of resultbox template.

2010-04-20T15:08:01Z

Test of resultbox template.

← Older revision		Revision as of 17:08, 20 April 2010
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	thus		thus

	:<math>C_p = C_v + R = \frac{5}{2} R</math>		{{resultbox\|<math>C_p = C_v + R = \frac{5}{2} R</math>}}

	where <math>R</math> is the [[molar gas constant]].		where <math>R</math> is the [[molar gas constant]].

Carl McBride: Minor touches.

2008-12-04T15:46:17Z

Minor touches.

← Older revision		Revision as of 17:46, 4 December 2008
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	:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>		:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>

	where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy \| kinetic energy of an ideal gas]], which is equal to <math>3/2 RT</math>. Thus		where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy \| kinetic energy of an ideal gas]], which is equal to <math>(3/2)RT</math>. Thus

	:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>		:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>.

	At constant [[pressure]] one has		At constant [[pressure]] one has
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	thus		thus

	:<math>C_p = C_v + R </math>		:<math>C_p = C_v + R = \frac{5}{2} R</math>

	where <math>R</math> is the [[molar gas constant]].		where <math>R</math> is the [[molar gas constant]].

Carl McBride: Further re-write.

2008-12-04T15:43:49Z

Further re-write.

← Older revision		Revision as of 17:43, 4 December 2008
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	:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>		:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>

			At constant [[pressure]] one has

	~~One has~~		:<math>C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math>

	~~:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>~~		we can see that, just as before, one has

	~~for an~~ [[ideal gas]] ~~this becomes~~:		:<math>\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R </math>

			and from the [[Equation of State: Ideal Gas \| equation of state of an ideal gas]]

			:<math>p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R</math>

			thus

			:<math>C_p = C_v + R </math>

	~~:<math>\left.C_p -C_V \right.=R</math>~~
	where <math>R</math> is the [[molar gas constant]].		where <math>R</math> is the [[molar gas constant]].
	==References==		==References==

Carl McBride: Slight re-write.

2008-12-04T13:41:53Z

Slight re-write.

← Older revision		Revision as of 15:41, 4 December 2008
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			The [[heat capacity]] at constant volume is given by

			:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>

			where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy \| kinetic energy of an ideal gas]], which is equal to <math>3/2 RT</math>. Thus

			:<math>C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>


			One has

	:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>		:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>

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	:<math>\left.C_p -C_V \right.=R</math>		:<math>\left.C_p -C_V \right.=R</math>
			where <math>R</math> is the [[molar gas constant]].
	==References==		==References==
	#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1		#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
	#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11		#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11
	[[Category: Ideal gas]]		[[Category: Ideal gas]]

Carl McBride: Ideal gas: Specific heats moved to Ideal gas: Heat capacity

2007-06-21T11:29:04Z

Ideal gas: Specific heats moved to Ideal gas: Heat capacity

← Older revision	Revision as of 13:29, 21 June 2007
(No difference)

Carl McBride at 11:28, 21 June 2007

2007-06-21T11:28:45Z

← Older revision		Revision as of 13:28, 21 June 2007
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	:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial E}{\partial V}\right\vert_T \right) </math>		:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) </math>

			for an [[ideal gas]] this becomes:

	:<math>\left.C_p -C_V \right.=R</math>		:<math>\left.C_p -C_V \right.=R</math>
	==References==		==References==

Carl McBride at 12:57, 28 May 2007

2007-05-28T12:57:28Z

← Older revision		Revision as of 14:57, 28 May 2007
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	:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial E}{\partial V}\right\vert_T \right) </math>		:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial E}{\partial V}\right\vert_T \right) </math>
	:<math>\left.C_p -C_V \right.=1</math>		:<math>\left.C_p -C_V \right.=R</math>
	==References==		==References==
	#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1		#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
	#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11		#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11
	[[Category: Ideal gas]]		[[Category: Ideal gas]]

Carl McBride at 12:31, 24 May 2007

2007-05-24T12:31:48Z

← Older revision		Revision as of 14:31, 24 May 2007
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	:<math>\left.~~c_p~~ -~~c_v~~ \right.=1</math>		:<math>C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial E}{\partial V}\right\vert_T \right) </math>
			:<math>\left.C_p -C_V \right.=1</math>
	==References==		==References==
			#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
	#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11		#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11
	[[Category: Ideal gas]]		[[Category: Ideal gas]]

Carl McBride at 12:00, 24 May 2007

2007-05-24T12:00:00Z

← Older revision		Revision as of 14:00, 24 May 2007
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	:<math>\left.c_p -c_v \right.=1</math>		:<math>\left.c_p -c_v \right.=1</math>
			==References==
			#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11
	[[Category: Ideal gas]]		[[Category: Ideal gas]]