A 30.5 g sample of an alloy at 90.7�C is placed into 48.8 g water at 22.3�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�)

Recall that the heat absorbed (+) or released (-) by a substance is given by

Q = mc(T2 - T1)
where
m = mass (g)
c = specific heat capacity (J/g-K)
T2 = final temperature
T1 = initial temperature
Note that in the problem, the source of energy or heat came from the alloy. Thus we say that heat released from the alloy is absorbed by the water and cup, or:
Q,alloy + Q,water + Q,cup = 0, or
-Q,alloy = Q,water + Q,cup

Let c = specific heat capacity of alloy
Note that the final temperature is the same for all. For the cup, the given is heat capacity, not specific heat capacity (which is per mass). And the initial temperature of water & cup are the same, since they're in equilibrium before the alloy is added.
Substituting,
-(30.5)*c*(31.1 - 90.7) = (48.8)*(4.184)*(31.1-22.3) + (9.2)*(31.1-22.3)
1817.8*c = 1877.7
c = 1.03 J/g-K

Hope this helps :)

To find the specific heat capacity of the alloy, we can apply the principle of conservation of energy. The heat gained by the alloy will be equal to the heat lost by the water:

q_alloy = q_water

The heat gained by the alloy can be calculated using the formula:

q_alloy = m_alloy * c_alloy * ΔT

where:
m_alloy is the mass of the alloy (30.5 g),
c_alloy is the specific heat capacity of the alloy (to be determined), and
ΔT is the change in temperature of the system (final temperature - initial temperature).

The heat lost by the water can be calculated using the formula:

q_water = m_water * c_water * ΔT

where:
m_water is the mass of the water (48.8 g),
c_water is the specific heat capacity of water (4.184 J/g°C), and
ΔT is the change in temperature of the system.

Since the coffee cup is insulated, it doesn't gain or lose any heat to the surroundings, so we don't need to consider it in our calculations.

Now, we can set up the equation:

m_alloy * c_alloy * ΔT = m_water * c_water * ΔT

Substituting the given values:

30.5 g * c_alloy * (31.1°C - 90.7°C) = 48.8 g * 4.184 J/g°C * (31.1°C - 22.3°C)

Simplifying the equation:

30.5 g * c_alloy * (-59.6°C) = 48.8 g * 4.184 J/g°C * 8.8°C

Now, we can solve for the specific heat capacity of the alloy:

c_alloy = (48.8 g * 4.184 J/g°C * 8.8°C) / (30.5 g * -59.6°C)

c_alloy ≈ -0.0224 J/g°C

Note: The negative sign indicates that the alloy is losing heat (exothermic process) since the temperature of the system decreased. Also, the specific heat capacity can't be negative, so the result is approximately 0.0224 J/g°C.