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For an ideal gas, internal energy ($u$) is a function of temperature only. Using the ideal gas law ($PV = RT$), the work done during expansion at constant pressure is $PdV = RdT$. Here’s a social media post tailored for ,

Therefore, the extra energy required to do this work compared to the constant volume scenario is exactly the gas constant $R$. This leads to Mayer's conclusion: $$c_p = c_v + R$$ For an ideal gas, internal energy ($u$) is

Without Mayer's relation linking $c_p$ and $c_v$, we could not close the system of equations needed to solve for pressure and density changes in high-speed aerodynamic flows.