Specific heat - Revision history

Nian at 13:29, 30 March 2026

2026-03-30T13:29:55Z

← Older revision		Revision as of 13:29, 30 March 2026
Line 9:		Line 9:
	--></nomobile><!--		--></nomobile><!--

			--><noinclude><!--
	-->{{#vardefine:secno\|1}}<!--		-->{{#vardefine:secno\|1}}<!--
	~~--><noinclude><!--~~
	-->{{#vardefine:eqno\|0}}<!--		-->{{#vardefine:eqno\|0}}<!--
	--></noinclude><!--		--></noinclude><!--

Nian at 09:55, 30 March 2026

2026-03-30T09:55:01Z

← Older revision		Revision as of 09:55, 30 March 2026
Line 1:		Line 1:
	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]
	<noinclude>		<noinclude><!--
	[[Category:Compressible flow:Topic]]		-->[[Category:Compressible flow:Topic]]<!--
	</noinclude>		--></noinclude><!--

	<nomobile>		--><nomobile><!--
	__TOC__		-->__TOC__<!--
	</nomobile>		--></nomobile><!--

	~~{{#vardefine:secno\|1}}~~
	~~{{#vardefine:eqno\|0}}~~

			-->{{#vardefine:secno\|1}}<!--
			--><noinclude><!--
			-->{{#vardefine:eqno\|0}}<!--
			--></noinclude><!--
			-->
	For thermally perfect and calorically perfect gases		For thermally perfect and calorically perfect gases

Nian at 09:47, 30 March 2026

2026-03-30T09:47:18Z

← Older revision		Revision as of 09:47, 30 March 2026
Line 10:		Line 10:

	{{#vardefine:secno\|1}}		{{#vardefine:secno\|1}}
	{{#vardefine:eqno\|7}}		{{#vardefine:eqno\|0}}

	For thermally perfect and calorically perfect gases		For thermally perfect and calorically perfect gases

Nian at 09:45, 30 March 2026

2026-03-30T09:45:19Z

← Older revision		Revision as of 09:45, 30 March 2026
Line 5:		Line 5:
	</noinclude>		</noinclude>

			<nomobile>
	__TOC__		__TOC__
			</nomobile>

	{{#vardefine:secno\|1}}		{{#vardefine:secno\|1}}

Nian at 07:00, 30 March 2026

2026-03-30T07:00:24Z

← Older revision		Revision as of 07:00, 30 March 2026
Line 35:		Line 35:
	{{NumEqn\|<math>		{{NumEqn\|<math>
	C_p=C_v+R		C_p=C_v+R
	</math>}}		</math>\|label=eq-specific-heat-b}}

	Dividing ~~Eqn. \ref~~{eq~~:specificheat:~~b} by <math>C_v</math> gives		Dividing {{EquationNote\|label=eq-specific-heat-b}} by <math>C_v</math> gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{C_p}{C_v}=1+\frac{R}{C_v}		\frac{C_p}{C_v}=1+\frac{R}{C_v}
	</math>}}		</math>\|label=eq-specific-heat-c}}

	Introducing the ratio of specific heats defined as		Introducing the ratio of specific heats defined as
Line 55:		Line 55:
	</math>}}		</math>}}

	In the same way, dividing ~~Eqn. \ref~~{eq~~:specificheat:~~b} with <math>C_p</math> gives		In the same way, dividing {{EquationNote\|label=eq-specific-heat-b}} with <math>C_p</math> gives

	{{NumEqn\|<math>		{{NumEqn\|<math>

Nian at 06:56, 30 March 2026

2026-03-30T06:56:56Z

← Older revision		Revision as of 06:56, 30 March 2026
Line 6:		Line 6:

	__TOC__		__TOC__

			{{#vardefine:secno\|1}}
			{{#vardefine:eqno\|7}}

	For thermally perfect and calorically perfect gases		For thermally perfect and calorically perfect gases

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\begin{aligned}		\begin{aligned}
	&C_p=\frac{dh}{dT}\\		&C_p=\frac{dh}{dT}\\
	&C_v=\frac{de}{dT}		&C_v=\frac{de}{dT}
	\end{aligned}		\end{aligned}
	</math>		</math>\|label=eq-specific-heat}}

	From the definition of enthalpy and the equation of state <math>p=\rho RT</math>		From the definition of enthalpy and the equation of state <math>p=\rho RT</math>

	<math ~~display="block"~~>		{{NumEqn\|<math>
	h=e+\frac{p}{\rho}=e+RT		h=e+\frac{p}{\rho}=e+RT
	</math>		</math>\|label=eq-enthalpy}}

	Differentiate ~~Eqn. \ref~~{eq:enthalpy} with respect to temperature gives		Differentiate {{EquationNote\|label=eq-enthalpy}} with respect to temperature gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT}		\frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT}
	</math>		</math>}}

	Inserting the specific heats gives		Inserting the specific heats gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	C_p=C_v+R		C_p=C_v+R
	</math>		</math>}}

	Dividing Eqn. \ref{eq:specificheat:b} by <math>C_v</math> gives		Dividing Eqn. \ref{eq:specificheat:b} by <math>C_v</math> gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{C_p}{C_v}=1+\frac{R}{C_v}		\frac{C_p}{C_v}=1+\frac{R}{C_v}
	</math>		</math>}}

	Introducing the ratio of specific heats defined as		Introducing the ratio of specific heats defined as

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\gamma=\frac{C_p}{C_v}		\gamma=\frac{C_p}{C_v}
	</math>		</math>\|label=eq-gamma}}

	Now, inserting ~~Eqn. \ref~~{eq:gamma} in Eqn. \ref{eq:specificheat:c} gives		Now, inserting {{EquationNote\|label=eq-gamma}} in Eqn. \ref{eq:specificheat:c} gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	C_v=\frac{R}{\gamma-1}		C_v=\frac{R}{\gamma-1}
	</math>		</math>}}

	In the same way, dividing Eqn. \ref{eq:specificheat:b} with <math>C_p</math> gives		In the same way, dividing Eqn. \ref{eq:specificheat:b} with <math>C_p</math> gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p}		1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p}
	</math>		</math>}}

	and thus		and thus

	<math ~~display="block"~~>		{{NumEqn\|<math>
	C_p=\frac{\gamma R}{\gamma-1}		C_p=\frac{\gamma R}{\gamma-1}
	</math>		</math>}}

Nian at 19:16, 20 March 2026

2026-03-20T19:16:36Z

← Older revision		Revision as of 19:16, 20 March 2026
Line 1:		Line 1:
	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Thermodynamics]]
			<noinclude>
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
	~~[[Category:Thermodynamics]]~~		</noinclude>

	__TOC__		__TOC__

Nian at 19:03, 20 March 2026

2026-03-20T19:03:18Z

← Older revision		Revision as of 19:03, 20 March 2026
Line 1:		Line 1:
			[[Category:Compressible flow]]
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]

Nian at 18:44, 20 March 2026

2026-03-20T18:44:10Z

← Older revision		Revision as of 18:44, 20 March 2026
Line 1:		Line 1:
	[[Category:Compressible flow]]		[[Category:Compressible flow:Topic]]
	~~[[Category~~:Topic]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]

Nian at 15:44, 20 March 2026

2026-03-20T15:44:56Z

← Older revision		Revision as of 15:44, 20 March 2026
Line 1:		Line 1:
	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]