Ideal gas: Heat capacity: Difference between revisions
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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]] | ||
Revision as of 14:41, 4 December 2008
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_V = \left. \frac{\partial U}{\partial T} \right\vert_V }
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 U} is the internal energy. Given that an ideal gas has no interatomic potential energy, the only term that is important is the kinetic energy of an ideal gas, which is equal to 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 3/2 RT} . Thus
- 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 = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R }
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 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) }
for an ideal gas this becomes:
- 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.C_p -C_V \right.=R}
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 R} is the molar gas constant.
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