Energy Conservation

In chapter 1, the First Law of thermodynamics for a cyclic system shows that the cyclic integral of heat interactions is equal to the cyclic integral of all work processes. We now extend the application of this principle to non-cyclic processes. Thus, when a closed system undergoes a process which changes its state from 1 to 2, it is usually the case that % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq % aH0oazcaWGrbGaeyOeI0Yaa8qmaeaacqaH0oazcaWGxbGaeyiyIKRa % aGimaaWcbaGaaGymaaqaaiaaikdaa0Gaey4kIipaaSqaaiaaigdaae % aacaaIYaaaniabgUIiYdaaaa!459C!$$\int_1^2 {\delta Q - \int_1^2 {\delta W \ne 0} } ]$$ or % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuaiabgk % HiTiaadEfacqGHGjsUcaaIWaaaaa!3B12!$$Q - W \ne 0]$$ However, the value of the quantity (Q − W) is the same for all paths between states 1 and 2. The proof of this is as follows.