Answer
We can rank the cylinders in order of temperature, from highest to lowest:
$b = d \gt a = c = e = f$
Work Step by Step
Let $P_0 = 50~kPa$
Let $V_0 = 2~L$
Let $N_0 = 3\times 10^{23}$
We can find a general expression for the temperature:
$P_0~V_0 = N_0~k~T$
$T = \frac{P_0~V_0}{N_0~k}$
We can find an expression for the temperature in each case.
(a) $T = \frac{(2P_0)~(2V_0)}{(2N_0)~k} = 2\times \frac{P_0~V_0}{N_0~k}$
(b) $T = \frac{(4P_0)~(2V_0)}{(2N_0)~k} = 4\times \frac{P_0~V_0}{N_0~k}$
(c) $T = \frac{(P_0)~(4V_0)}{(2N_0)~k} = 2\times \frac{P_0~V_0}{N_0~k}$
(d) $T = \frac{(2P_0)~(2V_0)}{(N_0)~k} = 4\times \frac{P_0~V_0}{N_0~k}$
(e) $T = \frac{(2P_0)~(V_0)}{(N_0)~k} = 2\times \frac{P_0~V_0}{N_0~k}$
(f) $T = \frac{(P_0)~(2V_0)}{(N_0)~k} = 2\times \frac{P_0~V_0}{N_0~k}$
We can rank the cylinders in order of temperature, from highest to lowest:
$b = d \gt a = c = e = f$
