Chemistry

The first law connects heat, work, and internal energy—and the key insight is that $c_p$ and $c_v$ differ only in the work term. When you heat a gas at constant pressure, it expands and does work on its surroundings; at constant volume, no expansion work occurs. By carefully applying $\Delta U = q - w$ to both scenarios and using the ideal gas law, you'll see that the extra heat needed at constant pressure is exactly the work $P\Delta V = R\Delta T$ per mole—revealing why $c_p$ exceeds $c_v$ by precisely $R$.

The first law connects heat, work, and internal energy—and the key insight is that $c_p$ and $c_v$ differ only in the work term. When you heat a gas at constant pressure, it expands and does work on its surroundings; at constant volume, no expansion work occurs. By carefully applying $\Delta U = q - w$ to both scenarios and using the ideal gas law, you'll see that the extra heat needed at constant pressure is exactly the work $P\Delta V = R\Delta T$ per mole—revealing why $c_p$ exceeds $c_v$ by precisely $R$.