# Molar heat capacity

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The molar heat capacity is the amount of energy required to change the temperature of an amount of substance by a certain amount per moles of the substance and change in temperature.

Adding heat to a substance changes its temperature in accordance to $$\Delta Q=nc_M\Delta T,$$

where $\Delta Q$ is the change in heat, $n$ is the number of moles of substance, $c_M$ is the molar heat capacity, and $\Delta T$ is the change in temperature.

At constant volume, $c_M=c_V$. At constant pressure, $c_M=c_P$.

For an ideal gas, $c_P=c_V+R$ where $R=$ the ideal gas constant. For an incompressible substance, $c_P=c_V$.

In adiabatic compression ($\Delta Q=0$) of an ideal gas, $PV^\gamma$ stays constant, where $\gamma=\frac{c_V+R}{c_V}$.