Question

A 10 g sheet of gold with a temperature of 18 ֯C is laid flat on a sheet of iron that weighs 20 g and has a temperature of 55.6 ֯C. Given that the specific heat capacities of Au and Fe are 0.129 J g -1 ֯C -1 and 0.444 J g-1 ֯C -1 , respectively, what is the final temperature of the combine metals? Assume that no heat is lost to the surrounding, and the heat gained by the gold is equal to the heat lost by iron. (Hint: Similar to examples treated in the videos, it is not necessary to convert the temperature into the unit of Kelvin since the unit of the specific heat capacity is in ֯C).

Answer 1

The final temperature of the combined gold and iron sheets at thermal equilibrium is approximately 50.75°C. This was determined by setting the heat gained by gold equal to the heat lost by iron and solving for the final temperature.

Step-by-step explanation:

To solve for the final temperature of the combined metals, we apply the principle of conservation of energy, which states that heat lost by one body must equal the heat gained by another when no heat is lost to the surroundings. The formula to use is q = mc\u0394T, where q is the heat transferred, m is the mass, c is the specific heat capacity, and \u0394T is the change in temperature. The heat gained by the gold (Au) should equal the heat lost by the iron (Fe), assuming no heat is lost to the surrounding environment. We set up the equation:

Heat gained by Au = Heat lost by Fe

m_{Au}\u00d7c_{Au}\u00d7(T_f - T_{i,Au}) = m_{Fe}\u00d7c_{Fe}\u00d7(T_{i,Fe} - T_f)

(10g\u00d70.129 J/g\u00b0C\u00d7(T_f - 18\u00b0C) = (20g\u00d70.444 J/g\u00b0C\u00d7(55.6\u00b0C - T_f)

We solve the equation for T_f (final temperature) to find the temperature at which both metals achieve thermal equilibrium.

1.29T_f - 23.22 = 8.88(55.6) - 8.88T_f

10.17T_f = 492.768 + 23.22

T_f = (492.768 + 23.22) / 10.17

T_f \u2248 50.75\u00b0C