What is the relationship between temperature and volume of gas in this process ?

Question

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Given Condition: The problem states that heat transferred equals the decrease in internal energy ( $d Q = - d U$ ).

First Law Equation: Substituting this into the first law ( $d Q = d U + d W$ ) gives $- d U = d U + P d V$ , which simplifies to $2 C_{v} d T + P d V = 0$ .

Substitution: Using the ideal gas equation $P = \frac{RT}{V}$ and Mayer's relation $R = (γ - 1) C_{v}$ , the equation becomes $2 C_{v} d T + (γ - 1) C_{v} T \frac{d V}{V} = 0$ .

Integration: Dividing by $C_{v} T$ and integrating gives $2 ln (T) + (γ - 1) ln (V) = co n s t an t$ , which simplifies to $ln (T^{2} V^{γ - 1}) = co n s t an t$ .

Final Result: Removing the logarithm gives $T^{2} V^{γ - 1} = co n s t an t$ , which simplifies to $T \cdot V^{(γ - 1) /2} = co n s t an t$ (Option C).

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Options:

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