Heat capacity: Difference between revisions
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The '''heat capacity''' is defined as the differential of [[heat]] with respect to the [[temperature]] <math>T</math>, | |||
The '''heat capacity''' is | |||
:<math>C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}</math> | :<math>C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}</math> | ||
where <math>S</math> is the [[entropy]]. | where <math>Q</math> is [[heat]] and <math>S</math> is the [[entropy]]. | ||
==At constant volume== | ==At constant volume== | ||
From the [[first law of thermodynamics]] one has | |||
:<math>\left.\delta Q\right. = dU + pdV</math> | |||
thus at constant volume, denoted by the subscript <math>V</math>, then <math>dV=0</math>, | |||
:<math>C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math> | :<math>C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math> | ||
==At constant pressure== | ==At constant pressure== | ||
Revision as of 12:56, 2 December 2008
The heat capacity is defined as the differential of heat with respect to the temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is heat and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the entropy.
At constant volume
From the first law of thermodynamics one has
thus at constant volume, denoted by the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV=0} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V }
At constant pressure
At constant pressure (denoted by the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ),
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p}