A sample of metal with a specific heat of .502 J/g°c is heated to 100.0°C and then placed a 50.0 g sample of water at 20.0°C. The final temperature of the system is 76.2°C. What is the mass of the metal used in process?

Question

Naomi Aronson · Accepted Answer

The mass of the metal is 459 g.

We must identify the heat transfers that are happening here. One is the heat transferred from the metal as it cools (#q_1#). The second is the heat transferred to the water as it warms (#q_2#). Per the Law of Conservation of Energy, the sum of the two heat transfers must be zero. #q_1 + q_2 = 0# The formula for the heat absorbed by or released from a substance is #color(blue)(bar(ul(|color(white)(a/a)q = mcΔTcolor(white)(a/a)|)))" "# where #q# is the quantity of heat #m# is the mass of the substance #c# is the specific heat capacity of the material #ΔT# is the temperature change This gives us #color(white)(ml)q_1 color(white)(mm)+color(white)(mll) q_2 color(white)(mm)= 0# #m_1c_1ΔT_1 + m_2c_2ΔT_2 = 0# In this problem, we have #m_1color(white)(l) = ?# #c_1color(white)(m) = "0.502 J·°C"^"-1""g"^"-1"# #ΔT_1 = T_"f" - T_"i" = "(76.2 - 100.0) °C" = "-23.8 °C"# #m_2color(white)(l) = "50.0 g"# #c_2color(white)(m)= "4.184 J·°C"^"-1""g"^"-1"# #ΔT_2 = T_"f" - T_"i" = "(76.2 - 20.0) °C" = "26.2 °C"# #q_1 = m_1 × "0.502 J"·color(red)(cancel(color(black)("°C"^"-1")))"g"^"-1" × ("-23.8" color(red)(cancel(color(black)("°C")))) = "-11.95"color(white)(l)m_1 color(white)(l)"J·g"^"-1"# #q_2 = "50.0 g" × "4.184 J"·color(red)(cancel(color(black)("°C"^"-1")))"g"^"-1" × 26.2 color(red)(cancel(color(black)("°C"))) = "5481 J"# #q_1 + q_2 = "-11.95"m_1 color(red)(cancel(color(black)("J")))·"g"^"-1" + 5481 color(red)(cancel(color(black)("J"))) = 0# #m_1 = 5481/("11.95 g"^"-1") = "459 g"#

Naomi Aronson · Answer

undefined Use the formula $ q_{\text{metal}} = -q_{\text{water}} $ and solve for the mass of the metal. $ q_{\text{metal}} = m_{\text{metal}} \times C_{\text{metal}} \times \Delta T_{\text{metal}} $, $ q_{\text{water}} = m_{\text{water}} \times C_{\text{water}} \times \Delta T_{\text{water}} $. Substituting the given values, you can find the mass of the metal.