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The concept of work is clear from mechanics, where $dW=F\cdot dx$, leading to $dW=PdV$. Also different forms of energy are defined in mechanics, e.g., $E=\frac{1}{2}m v^2$, etc. How are the energy and heat defined and measured in thermodynamics?

Because once we define them independently, we can state the first law of thermodynamics, $d E=\delta Q+\delta W$.

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Related: physics.stackexchange.com/q/8522/2451 and links therein. –  Qmechanic Nov 21 '13 at 9:13

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