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The v and p just indicate that it is specific heat at a constant volume or constant pressure. I can't think of a situation where you would need to convert between the two or be extra wary which one the table is giving you. Usually, if you need specific heat, just refer to what the table gives you. You use C=(5/2)R and C=(3/2)R for ideal gases; (5/2)R is for the specific heat of gases at a constant pressure and (3/2)R for constant volume.
You would used Cv,m and Cp,m if you want to find Cv or Cp in calculating q (using q=CT). The values using R to calculate C are ways of estimating the heat capacities of certain molecules if you don't know it, but I don't really think he went over that.
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