Solution:

For an ideal gas obeying \( PV^{\frac{1}{2}} = \text{constant} \):

1. Use the ideal gas law: \( PV = nRT \), so \( P = \frac{nRT}{V} \).
2. Substitute \( P = \frac{nRT}{V} \) in \( PV^{\frac{1}{2}} = \text{constant} \):

\[
\frac{nRT}{V} \cdot V^{\frac{1}{2}} = \text{constant} \Rightarrow nRT \cdot V^{-\frac{1}{2}} = \text{constant}
\]

3. At initial state, \( T = T \) and \( V = V \):

\[
T V^{-\frac{1}{2}} = \text{constant}
\]

4. When \( V \) changes to \( 4V \):

\[
T' (4V)^{-\frac{1}{2}} = T V^{-\frac{1}{2}}
\]

5. Simplify:

\[
T' \cdot \frac{1}{2} = T \Rightarrow T' = 2T
\]

So, the final temperature \( T' = 2T \).