A

Step 1: Given the law for the gas, we have VP^2 = k, where k is a constant.
Step 2: Initially, at volume V, let the pressure be P and the temperature be T. Thus, we can write the relationship as:
\( V P^2 = k \)
Step 3: When the gas expands to a volume 2V, let the new pressure be P'. According to the law, we have:
\( 2V (P')^2 = k \)
Step 4: Since both equal k, we can set the equations equal to each other:
\( V P^2 = 2V (P')^2 \)
Step 5: Dividing both sides by V (assuming V \neq 0):
\( P^2 = 2 (P')^2 \)
Step 6: Taking the square root:
\( P = \sqrt{2} P' \)
Step 7: Using the Ideal Gas Law, which states \( PV = nRT \):
Initially: \( P V = n R T \)
After expansion: \( P' (2V) = n R T' \)
Step 8: Therefore, \( P' = \frac{nRT'}{2V} \).
Step 9: Substitute P' into the expression for P found earlier, we have:
\( P = \sqrt{2} \left(\frac{nR T'}{2V}\right) \)
Step 10: Setting the two expressions for P equal:
\( \sqrt{2} \left(\frac{nR T'}{2V}\right) = \frac{nR T}{V} \)
Step 11: Canceling out nR/V across gives:
\( \sqrt{2} \frac{T'}{2} = T \)
Step 12: Rearranging for T':
\( T' = \frac{2T}{\sqrt{2}} \)
Step 13: Simplifying yields:
\( T' = \sqrt{2} T \)
Therefore, when the gas expands to the volume 2V, the temperature becomes \( T' = \sqrt{2} T \).