Find \( V^{\prime} \), when \( V=12, P=2, T=600 \) and \( T^{\prime}=400, P=8 \) \( \frac{P V}{T}=\frac{P^{\prime} V^{\prime}}{T^{\prime}} \)

To solve for \( V^{\prime} \), we can use the ideal gas law relationship given in your equation: \[ \frac{P V}{T} = \frac{P^{\prime} V^{\prime}}{T^{\prime}} \] Plugging in the values you provided: \[ \frac{2 \times 12}{600} = \frac{8 \times V^{\prime}}{400} \] Calculating the left side: \[ \frac{24}{600} = \frac{1}{25} \] Now, the equation becomes: \[ \frac{1}{25} = \frac{8 \times V^{\prime}}{400} \] Now, simplify the right side: \[ \frac{8 \times V^{\prime}}{400} = \frac{8}{400} \times V^{\prime} = \frac{1}{50} \times V^{\prime} \] Setting both sides equal to each other gives: \[ \frac{1}{25} = \frac{1}{50} \times V^{\prime} \] To isolate \( V^{\prime} \), multiply both sides by 50: \[ 2 = V^{\prime} \] Thus, \( V^{\prime} = 2 \).