열역학

$\Delta Q=\Delta U+W\tag{1}$ $dU = nC_{v}dT\tag{2}$

등압 과정

등적 과정( $Δ V = 0 ⟹ W = 0$ ) $Δ Q = Δ U + W = n C_{v} d T$ 등온 과정 $Δ Q = Δ U + W$ $d U = n C_{v} d T = 0 ⟹ Δ U = 0$

W = \int d W = \int P d V d V = \int \frac{n RT}{V} d V = n RT ln \frac{V _{f}}{V _{i}}

$Δ Q = W = n RT ln \frac{V _{f}}{V _{i}}$

단열 과정 ( $d Q = 0$ )

d U = - P d V n C_{v} d T = - \frac{n RT}{V} d V - \int \frac{1}{T} d T = \frac{R}{C _{v}} \int \frac{1}{V} d V - ln (\frac{T _{f}}{T _{i}}) = (γ - 1) ln (\frac{V _{f}}{V _{i}}) ∴ T_{i} V_{i}^{γ - 1} = T_{f} V_{f}^{γ - 1} (5)

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