A 30.5 g sample of an alloy at 94.8°C is placed into 52.1 g water at 22.8°C in an insulated coffee cup. The heat capacity of the coffee cup (without the water) is 9.2 J/K. If the final temperature of the system is 31.1°C, what is the specific heat capacity of the alloy? (c of water is 4.184 J/g×K)
To find the specific heat capacity of the alloy, we need to use the equation:
q = mcΔT
Where:
q is the heat transferred (in joules)
m is the mass of the system (in grams)
c is the specific heat capacity (in J/g×K)
ΔT is the change in temperature (in °C or K)
In this case, we have the following information:
Mass of alloy (m1) = 30.5 g
Initial temperature of alloy (T1) = 94.8 °C
Mass of water (m2) = 52.1 g
Initial temperature of water (T2) = 22.8 °C
Final temperature of system (Tf) = 31.1 °C
Specific heat capacity of coffee cup (c_cup) = 9.2 J/K
Specific heat capacity of water (c_water) = 4.184 J/g×K
First, let's find the heat transferred from the alloy to the water:
q1 = mcΔT
q1 = (30.5 g)(c)(31.1 °C - 94.8 °C)
Next, let's find the heat transferred from the water to the alloy cup:
q2 = mcΔT
q2 = (52.1 g)(4.184 J/g×K)(31.1 °C - 22.8 °C)
Since the system is insulated, the total heat lost by the alloy is equal to the total heat gained by the water and the cup:
q1 + q2 = 0
Let's rearrange the equation to solve for the specific heat capacity of the alloy (c):
(30.5 g)(c)(31.1 °C - 94.8 °C) + (52.1 g)(4.184 J/g×K)(31.1 °C - 22.8 °C) + (52.1 g)(9.2 J/K) = 0
Now, let's solve for c by rearranging the equation:
c = -[ (52.1 g)(4.184 J/g×K)(31.1 °C - 22.8 °C) + (52.1 g)(9.2 J/K) ] / (30.5 g)(31.1 °C - 94.8 °C)
Calculating this expression will give us the specific heat capacity of the alloy.