Assumptions:
we are doing a little kiddy version and thus using specific heat capacities defined as 278.15 K, as constants.
The system is isobaric.
Geometry is assumed yo be such that everything just nicely behaves (LMAO)
Iron has a specific heat capacity of 0.45 J*(kg^-1)*(K^-1) , molar weight of 0.055845 (Kg)*(mol^-1)
Tinitial=473.15 K
M=0.055845 kg
Copper has a specific heat Capacity of 0.39 J*(kg^-1)*(K^-1), molar weight of 0.063546 (krxg)*(mol^-1)
Tinitial=473.1
m=0.063546 kg
Water has a specific heat capacity of 4.1379 J*(kg^-1)*(K^-1), molar weight of 0.01801528 (Kg)*(mol^-1)
Tinitial=293.15 K
m=1.801528 kg
Thus we get
(0.45*473.15*0.055845)+(0.39*473.15*0.063546)+(4.1379*293.15*1.801528) = 2,208.9 J fot the total existential heat of the system
Solve the following for 'x' as you assume all substances reach the same temperature.
(0.45*x*0.055845)+(0.39*x*0.063546)+(4.1379*x*1.801528)=2,208.9
7.5044*x=2,208.9
x=294.34 K
So, like 21 degrees Centigrade at equilibrium for part one. For part two, you need to take into account that heat capacity also has a time dimension which wasn't relevant to the above calculation I might check back in to maybe do part two but I just finished taking my morning shit and have Church to get ready for.
I just solved the temperature and got the same answer, so this checks out.
The second part of the question is not possible to solve without more information. There's two problems with it:
1. The rate of heat transfer between the substances depends on the surface area, and therefore the shape, of the metal. A sphere will lose heat at a lower rate than a cube, which will lose heat at a lower rate than a radiator-shaped piece of metal.
2. The temperature distribution in the substance will not be uniform. The outside of the metal will be at a lower temperature than the center. Temperature could refer to the lowest temperature in the body, or the highest, or the average, or the temperature at some arbitrary point.
Even knowing these things, actually solving the problem is significantly harder than the first. Solving the equilibrium temperature is a high school chemistry problem. Solving the temperature at the second part analytically would stump anyone who hasn't studied graduate level differential equations.
The problem with solving at a certain time is twofold: The first is that the temperature in any given body has multiple methods of heat transfer through the system: Conduction through the body towards its surface, conduction from the body to the water, and convection from the body to the water. Take note that the water is a body just like the iron and copper is, just one that transfers heat through convection as well as conduction. The temperature changing at any one of these points will affect the rate of heat transfer in the other parts of the system. I don't have to attempt to solve this to tell you that this is modelled by a differential equation. Just knowing how these generally go I don't think it's even possible to solve this analytically. You would have to approximate it using numerical methods because the resulting differential equation is not going to be solvable.
The second problem is that there are multiple of these bodies in question. In this case we have three: the iron, the copper, and the water. When the temperature of one changes, this affects the rates of heat transfer in the other two. This is a case of a three body problem (though not one involving gravity, as it usually goes), and again it yields an unsolvable differential equation.
So you have a set of unsolvable differential equations modelling heat transfer nested within an unsolvable differential equation modelling a three body problem. I'm not saying it's impossible to procure an answer from this, but that it will take some truly arcane techniques to do so.
