Heat capacity: Difference between revisions
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From the [[first law of thermodynamics]] | From the [[first law of thermodynamics]] one has | ||
:<math>\left.\delta Q\right. = dU + pdV</math> | :<math>\left.\delta Q\right. = dU + pdV</math> | ||
the '''heat capacity''' is given by | where <math>Q</math> is the [[heat]], <math>U</math> is the [[internal energy]], <math>p</math> is the [[pressure]] and <math>V</math> is the volume. | ||
The '''heat capacity''' is given by the differential of the heat with respect to the [[temperature]], | |||
:<math>C = \frac{\delta Q}{\partial T}</math> | :<math>C = \frac{\delta Q}{\partial T}</math> | ||
==At constant volume== | ==At constant volume== | ||
At constant volume (denoted by the subscript <math>V</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== | ||
At constant pressure (denoted by the subscript <math>p</math>), | |||
:<math>C_p = \left.\frac{\delta Q}{\partial T} \right\vert_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math> | :<math>C_p = \left.\frac{\delta Q}{\partial T} \right\vert_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math> | ||
The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by | |||
:<math>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</math> | :<math>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</math> | ||
[[category: classical thermodynamics]] | [[category: classical thermodynamics]] | ||
Revision as of 10:19, 8 July 2008
From the first law of thermodynamics one has
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 the heat, 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 U} is the internal energy, 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} is the pressure 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 V} is the volume. The heat capacity is given by the differential of the heat with respect to the temperature,
At constant volume
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} ),
- 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} ),
- 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 = \left.\frac{\delta Q}{\partial T} \right\vert_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p}
The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
