Answer:
[tex]T_f=42.3^{\circ}C[/tex]
Explanation:
The heat released by the iron is absorbed by the gold. This quantity is calculated with the formula [tex]Q=mc\Delta T[/tex], which tells us the amount of heat (energy) Q needed to increase the temperature of a material of mass m, specific heat c by a temperature difference[tex]\Delta T[/tex].
Since the heat released (which we will take as negative Q) by the iron (Fe) is absorbed (positive Q then) by the gold (Au) we write [tex]Q_{Au}=-Q_{Fe}[/tex], which means [tex]m_{Au}c_{Au}\Delta T_{Au}=-m_{Fe}c_{Fe}\Delta T_{Fe}[/tex], which is to say [tex]m_{Au}c_{Au}(T_{fAu}-T_{0Au})=-m_{Fe}c_{Fe}(T_{fFe}-T_{0Fe})[/tex].
We know that thermal equilibrium has been reached, which means that at the end everything has the same temperature, so we have [tex]T_{fAu}=T_{fFe}=T_f[/tex], and we can substitute [tex]m_{Au}c_{Au}(T_f-T_{0Au})=-m_{Fe}c_{Fe}(T_f-T_{0Fe})[/tex] and do [tex]m_{Au}c_{Au}T_f-m_{Au}c_{Au}T_{0Au}=-m_{Fe}c_{Fe}T_f+m_{Fe}c_{Fe}T_{0Fe}[/tex], to get finally [tex]m_{Fe}c_{Fe}T_f+m_{Au}c_{Au}T_f=m_{Fe}c_{Fe}T_{0Fe}+m_{Au}c_{Au}T_{0Au}[/tex], which means [tex](m_{Fe}c_{Fe}+m_{Au}c_{Au})T_f=m_{Fe}c_{Fe}T_{0Fe}+m_{Au}c_{Au}T_{0Au}[/tex], giving us the final formula [tex]T_f=\frac{m_{Fe}c_{Fe}T_{0Fe}+m_{Au}c_{Au}T_{0Au}}{m_{Fe}c_{Fe}+m_{Au}c_{Au}}[/tex].
We can get from tables that [tex]c_{Fe}=0.450J/g^{\circ}C[/tex] and [tex]c_{Au}=0.129J/g^{\circ}C[/tex], so we put everything in a calculator and we get [tex]T_f=\frac{(24.5g)(0.450J/g^{\circ}C)(46.4^{\circ}C)+(13.6g)(0.129J/g^{\circ}C)(16.7^{\circ}C)}{(24.5g)(0.450J/g^{\circ}C)+(13.6g)(0.129J/g^{\circ}C)}=42.3^{\circ}C[/tex]
