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# The First Law of Thermodynamics. Heat Capacity

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• The first law of thermodynamics: $$Q = \Delta U + A,$$ where $\Delta U$ is the increment of the internal energy of the system.
• Work performed by gas: $$A = \int pdV.$$
• Internal energy of an ideal gas: $$U = \frac{m}{M}C_V T = \frac{m}{M} \frac{RT}{\gamma - 1} = \frac{pV}{\gamma - 1}.$$
• Molar heat capacity in a polytropic process ($!pV^n = const$): $$C = \frac{R}{\gamma -1} - \frac{R}{n-1} = \frac{(n-\gamma)R}{(n-1)(\gamma-1)}.$$
• Internal energy of one mole of a Van der Waals gas: $$U = C_V T - \frac{a}{V_M}.$$

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