Question

A metal brick found in an excavation was sent to a testing lab for nondestructive identification. The lab weighed the brick and found its mass to be \(3.00 \mathrm{~kg}\). The brick was heated to a temperature of \(300 .{ }^{\circ} \mathrm{C}\) and dropped into an insulated copper calorimeter of mass \(1.50 \mathrm{~kg}\) containing \(2.00 \mathrm{~kg}\) of water at \(20.0^{\circ} \mathrm{C}\). The final temperature at equilibrium was \(31.7^{\circ} \mathrm{C}\). By calculating the specific heat from this data, can you identify the metal of which the brick is made?

Short answer

Answer: The metal brick found in the excavation is made of Bismuth.

Step-by-step solution

Write down the formula for heat transfer

We will begin by using the formula for heat transfer, which states that the heat gained by one object must be equal to the heat lost by another object when they are in thermal equilibrium. In this case, we have: \(Q_{gain} = Q_{lost}\) Specific heat is defined as the amount of heat required to change the temperature of 1 kg of material by 1 degree Celsius. The formula for heat transfer in terms of specific heat, mass, and temperature change is: \(Q = mcΔT\) where \(Q\) is the heat transfer, \(m\) is the mass, \(c\) is the specific heat, and \(ΔT\) is the change in temperature.

Set up the equations for heat gained and lost

According to the heat transfer principle, the heat gained by water and the calorimeter is equal to the heat lost by the metal brick: \(Q_{metal} = Q_{water} + Q_{calorimeter}\) We need to substitute the equations for heat transfer using specific heat, mass, and temperature change for water, calorimeter, and metal: \(m_{metal}c_{metal}(T_f - T_{metal}) = m_{water}c_{water}(T_f - T_{water}) + m_{calorimeter}c_{calorimeter}(T_f - T_{calorimeter})\)

Plug in the known values

We will now substitute the values given in the exercise and note that since the calorimeter is made of copper, we can use the specific heat of copper, which is \({c_{calorimeter}} = 387\,J(kg\,K)^{-1}\) \(3.00c_{metal}(31.7 - 300) = 2.00(4186)(31.7 - 20.0) + 1.50(387)(31.7 - 20.0)\)

Solve for the specific heat of the metal brick

Now, we will solve the equation in step 3 for \(c_{metal}\): \(-3.00c_{metal}(268.3) = 2.00(4186)(11.7) + 1.50(387)(11.7)\) Calculate the right side of the equation: \(2.00(4186)(11.7)+1.50(387)(11.7)= 97906.2+6832.15=104738.35\) Now, divide both sides by -804.9 to solve for \(c_{metal}\): \(c_{metal}=\frac{104738.35}{-804.9}=-130\,\mathrm{J\,(kg\,K)^{-1}}\)

Identify the metal

Based on the calculated specific heat of the metal brick, which is approximately \(130\,\mathrm{J\,(kg\,K)^{-1}}\), we can conclude that the metal brick is made of Bismuth. The specific heat capacity of Bismuth is close to the calculated value, which is around \(125\,\mathrm{J\,(kg\,K)^{-1}}\). In conclusion, the metal brick found in the excavation is made of Bismuth, based on its specific heat capacity.