What is the specific heat of the unknown metal sample?
An irregular lump of an unknown metal has a measured density of 2.97 g/mL. The metal is heated to a temperature of 173 °C and placed in a graduated cylinder filled with 25.0 mL of water at 25.0 °C. After the system has reached thermal equilibrium, the volume in the cylinder is read at 34.0 mL, and the temperature is recorded as 40.6 °C. What is the specific heat of the unknown metal sample? Assume no heat is lost to the surroundings.
An irregular lump of an unknown metal has a measured density of 2.97 g/mL. The metal is heated to a temperature of 173 °C and placed in a graduated cylinder filled with 25.0 mL of water at 25.0 °C. After the system has reached thermal equilibrium, the volume in the cylinder is read at 34.0 mL, and the temperature is recorded as 40.6 °C. What is the specific heat of the unknown metal sample? Assume no heat is lost to the surroundings.
2 Answers
Explanation:
Write down given information for both substances (mass, specific heat, change in temperature)
For Water :
For Metal :
Use the formula
This (
Rearrange the formula
#C_m ~~ "0.457 J/g"^@ "C"#
The displacement of water gives the volume and thus mass of the metal:
#"34.0 - 25.0 mL" xx "2.97 g"/"mL" = "26.73 g"#
By conservation of energy,
#q_m + q_w = 0# where
#m# is metal and#w# is water.
Thus
#q_m = -q_w#
and
#m_mC_mDeltaT_m = -m_wC_wDeltaT_w# where
#C# is the constant-pressure specific heat capacity in#"J/g"^@ "C"# ,#m_i# the mass, and#DeltaT# the change in temperature in#""^@ "C"# .
They both reach the same thermal equilibrium temperature, so
#m_mC_m(T_f - T_m) = -m_wC_w(T_f - T_w)#
Therefore, assuming the specific heat capacity of water stays constant in the temperature range, and that its density is that at
#color(blue)(C_m) = -(m_wC_w(T_f - T_w))/(m_m(T_f - T_m))#
#= -("25.0 mL" xx "0.9919880 g"/"mL" cdot "4.184 J/g"^@ "C" cdot (40.6^@ "C" - 25.0^@ "C"))/("26.73 g" cdot (40.6^@ "C" - 173^@ "C"))#
#= color(blue)("0.457 J/g"^@ "C")#