State the meaning of two objects being in thermal equilibrium.
A mass $X$ of ice at $0\,^{\circ}\text{C}$ is put into a beaker containing a mass $M$ of water at Celsius temperature $t$. The beaker is completely insulated and has negligible heat capacity. After a while, the ice that was added reaches thermal equilibrium with the water already in the beaker. The specific latent heat of fusion of water is $L$. The specific heat capacity of water is $c$. The final Celsius temperature of the system is $\theta$. Give expressions, in terms of some or all of $X$, $M$, $t$, $\theta$, $L$ and $c$, for the thermal energy:
$E_1$, gained by the ice as it changes into water at $0\,^{\circ}\text{C}$.
$E_2$, lost as the water’s Celsius temperature falls from $t$ to $\theta$.
$E_3$, gained as the melted ice’s Celsius temperature rises from $0\,^{\circ}\text{C}$ to $\theta$.
Use your answers in (b) to deduce that the system’s final Celsius temperature $\theta$ is given by $\theta = \frac{Mct - XL}{c(M + X)}$.