Heat capacity: Difference between revisions
Carl McBride (talk | contribs) m (→At constant volume: Added internal link) |
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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
- 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 \left.\delta Q\right. = dU + pdV}
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,
- 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}}
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
- 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}