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Is molar heat capacity and heat capacity

  1. at constant volume both represented by $C_v$?
  2. at constant pressure both represented by $C_p$?
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    $\begingroup$ Yes. Unfortunately, we use usually use the same symbol for both. The context of the problem and solution determines which one is being used. $\endgroup$ – Chet Miller Feb 13 '16 at 17:02
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The symbols, which are standardized in ISO/IEC 80000 Quantities and units (14 parts) and recommended by IUPAC in Quantities, Units and Symbols in Physical Chemistry (Green Book), distinguish the various related quantities:

$$\small \begin{array}{llll} \hline \text{Quantity} & \text{Quantity} & \text{Unit} & \text{Unit} \\ \text{name} & \text{symbol} & \text{name} & \text{symbol} \\ \hline \text{heat capacity} & C & \text{joule per kelvin} & \mathrm{J/K} \\ \text{heat capacity at constant pressure} & C_p & \text{joule per kelvin} & \mathrm{J/K} \\ \text{heat capacity at constant volume} & C_V & \text{joule per kelvin} & \mathrm{J/K} \\ \text{specific heat capacity} & c & \text{joule per kilogram kelvin} & \mathrm{J/( kg \cdot K )} \\ \text{specific heat capacity at constant pressure} & c_p & \text{joule per kilogram kelvin} & \mathrm{J/( kg \cdot K )} \\ \text{specific heat capacity at constant volume} & c_V & \text{joule per kilogram kelvin} & \mathrm{J/( kg \cdot K )} \\ \text{molar heat capacity} & C_\mathrm m & \text{joule per mole kelvin} & \mathrm{J/( mol \cdot K )} \\ \text{molar heat capacity at constant pressure} & C_{\mathrm m, p} & \text{joule per mole kelvin} & \mathrm{J/( mol \cdot K )} \\ \text{molar heat capacity at constant volume} & C_{\mathrm m, V} & \text{joule per mole kelvin} & \mathrm{J/( mol \cdot K )} \\ \hline \end{array}$$

However, in contrast to symbols for units, symbols for quantities are only recommendations. Unfortunately, quantity symbols are used inconsistently in the literature, sometimes in a sloppy way. In principle, a given symbol can indicate different quantities.

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