Deriving the Temperature-Volume Relation for $PV^2 = \text{constant}$

We are given a process for an ideal gas where the relation between pressure ($P$) and volume ($V$) is $PV^2 = K$, where $K$ is a constant.

From the Ideal Gas Law, we know that for a fixed amount of gas ($n$), $PV = nRT$, where $R$ is the ideal gas constant and $T$ is the absolute temperature.

We can express pressure ($P$) from the Ideal Gas Law as: $P = \frac{nRT}{V}$

Now, substitute this expression for $P$ into the given process equation ($PV^2 = K$): $ \left(\frac{nRT}{V}\right) V^2 = K $ $ nRT V = K $

Since $n$ and $R$ are constants, we can rewrite the equation as: $ TV = \frac{K}{nR} $ Let $\frac{K}{nR} = \text{constant}'$. Therefore, the process follows the relation: $ TV = \text{constant}' $

Applying the Constant Relation to Initial and Final States

This means that the product of temperature and volume remains constant throughout the process. Applying this to the initial state (1) and the final state (2): $ T_1 V_1 = \text{constant}' $ $ T_2 V_2 = \text{constant}' $

Equating these two expressions: $ T_1 V_1 = T_2 V_2 $

To find the relationship between the temperature ratio and the volume ratio, rearrange the equation: $ \frac{T_1}{T_2} = \frac{V_2}{V_1} $

This result matches option C.