Question

a 50.00 g sample of an unknown metal is heated to 45.00°c. it is then placed in a coffee - cup calorimeter filled with water. the calorimeter and the water have a combined mass of 250.0 g and an overall specific heat of 1.035 cal/g·°c. the initial temperature of the calorimeter is 10.00°c. the system reaches a final temperature of 11.08°c when the metal is added. which of the following equations is the correct one to use to calculate the specific heat of the metal? ( c_{p,metal}=\frac{m_{metal}delta t_{metal}}{m_{cal}c_{p,cal}delta t_{cal}} ) ( c_{p,metal}=\frac{-m_{cal}c_{p,cal}delta t_{cal}}{m_{metal}delta t_{metal}} )

Explanation:

Step1: Apply the principle of heat transfer

According to the principle of heat transfer \(q_{lost}=q_{gained}\). The heat lost by the metal \(q_{metal}=m_{metal}C_{p,metal}\Delta T_{metal}\), and the heat gained by the calorimeter - water system \(q_{cal}=m_{cal}C_{p,cal}\Delta T_{cal}\). So, \(m_{metal}C_{p,metal}\Delta T_{metal}=-m_{cal}C_{p,cal}\Delta T_{cal}\) (the negative sign indicates heat loss).

Step2: Solve for \(C_{p,metal}\)

Rearrange the equation \(m_{metal}C_{p,metal}\Delta T_{metal}=-m_{cal}C_{p,cal}\Delta T_{cal}\) for \(C_{p,metal}\). We get \(C_{p,metal}=\frac{-m_{cal}C_{p,cal}\Delta T_{cal}}{m_{metal}\Delta T_{metal}}\)

Answer:

\(C_{p,metal}=\frac{-m_{cal}C_{p,cal}\Delta T_{cal}}{m_{metal}\Delta T_{metal}}\) (the second option in the given choices)