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![**Deriving the Thermodynamic Definition for Entropy**
One way to "derive" the thermodynamic definition for entropy is simply to recognize that its thermodynamic definition must be a state function, and all thermodynamic state functions are worthy of giving a special name and carry special meaning.
**a) Starting with the First Law of Thermodynamics (expressed either of 2 ways):**
\[ \Delta U = q + w \]
\[ dU = dq + dw \]
**Show all the steps and assumptions/conditions required to arrive at the new expression below, which includes the definition of entropy:**
\[ \Delta S = \frac{q_{rev}}{T} = \frac{\Delta U}{T} - nR \ln \left(\frac{V_f}{V_i}\right) \]
**b) Using your result from part (a), explain why \(\frac{q_{rev}}{T}\) must be a state function.**
To be explicit, explain why entropy must be defined by (P, V, T) alone, and any change between the same two states, \((P_1, V_1, T_1)\) and \((P_2, V_2, T_2)\), regardless of path, will give the same change in entropy.](https://content.bartleby.com/qna-images/question/aba5746d-e0e8-46ca-a0bb-ec14000055d1/c7853821-0dec-4b4b-a7d2-d8387c2702b0/2zyqfb_processed.png)
Transcribed Image Text:**Deriving the Thermodynamic Definition for Entropy**
One way to "derive" the thermodynamic definition for entropy is simply to recognize that its thermodynamic definition must be a state function, and all thermodynamic state functions are worthy of giving a special name and carry special meaning.
**a) Starting with the First Law of Thermodynamics (expressed either of 2 ways):**
\[ \Delta U = q + w \]
\[ dU = dq + dw \]
**Show all the steps and assumptions/conditions required to arrive at the new expression below, which includes the definition of entropy:**
\[ \Delta S = \frac{q_{rev}}{T} = \frac{\Delta U}{T} - nR \ln \left(\frac{V_f}{V_i}\right) \]
**b) Using your result from part (a), explain why \(\frac{q_{rev}}{T}\) must be a state function.**
To be explicit, explain why entropy must be defined by (P, V, T) alone, and any change between the same two states, \((P_1, V_1, T_1)\) and \((P_2, V_2, T_2)\), regardless of path, will give the same change in entropy.
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