Solution

Assuming we're working with ideal gases
We need the total work done by this mol of gas
We have two isochoric processes, 1-2 and 3-4 ($V_1=V_2$ and $V_3=V_4$). And two isobaric processes, 2-3 and 4-1 ($P_2=P_3$ and $P_1=P_4$), and we know that $T_2=T_4$
$W_{12}=0$, $W_{34}=0$ (isochoric processes)
so, the total work will be
$W_T=W_{23} +W_{41}$
$W_T=P_2(V_3 -V_2)+P_1(V_1-V_4)$
$W_T=P_2V_3 -P_2V_2 +P_1V_1 -P_1V_4$
but $P_2V_3=RT_3$, $P_1V_1=RT_1$, and $P_2V_2=P_4V_4=RT_2=RT_4$ (because $T_2=T_4$)
and from this relations, we get in the work
$W_T=RT_3 +RT_1 -2RT_2$
From Gay-Lussac's law
$\frac{P_1}{T_1}=\frac{P_2}{T_2}$, and $\frac{P_3}{T_3}=\frac{P_4}{T_4}$
Multiplying term by term
$\frac{P_1P_3}{T_1T_3}=\frac{P_2P_4}{T_2T_4}$
we get
$\frac{1}{T_1T_3}=\frac{1}{(T_2)^2}$
so $T_2=(T_1T_3)^\frac{1}{2}$
and susbtituting this into the total work
$W_T=RT_3 +RT_1 -2R(T_1T_3)^\frac{1}{2}$
$W_T=R[T_3 + T_1 -2(T_1T_3)^\frac{1}{2}]$
so
$W_T=R[(T_3)^\frac{1}{2} -(T_1)^\frac{1}{2}]^2$