Correct Answer - Option 4 : 297 K

__Concept:__

According to the First Law of Thermodynamics:

**“Energy is neither created nor destroyed, but only gets transformed from one form to another”. **

Due to temperature difference, there will be heat transfer from a hot body to a cold body until an equilibrium temperature is reached.

Since the container is insulated, no heat escapes to the surrounding.

Heat lost by the hot body **m**_{1}c_{1}(T_{1} - T_{f}) will be absorbed by the cold body **m**_{2}c_{2}(T_{f} – T_{2}).

where m = mass of the body, c = specific heat of the body, T_{1} = Initial temperature of the hot body, T_{2} = Initial temperature of the cold body,

and T_{f }= Final temperature of the mixture.

__Calculation:__

__Given:__

m_{1} = 10 kg, c_{1} = 0.8 kJ/kg-K ⇒ 800 J/kg-K, T_{1} = 373 K

m_{2} = 40 kg, c_{2} = 4 kJ/kg-K ⇒ 4000 J/kg-K, T_{2} = 293 K

∵ Heat lost by hot body = Heat gained by cold body

⇒ m_{1}c_{1}(T_{1} - T_{f}) = m_{2}c_{2}(T_{f} – T_{2})

\( \Rightarrow {T_f} = \frac{{{{\rm{m}}_1}{c_1}{T_1} + {m_2}{c_2}{T_2}}}{{{m_1}{c_1} + {m_2}{c_2}}}\)

\( \Rightarrow {T_f} = \frac{{\left( {10 \times 800 \times 373} \right) + \left( {40 \times 4000 \times 293} \right)}}{{\left( {10 \times 800} \right) + \left( {40 \times 4000} \right)}}\)

∴ **T**_{f} = 296.80 ≈ 297 K