A 13 gram sample of an unknown metal at 26.5°C is placed in a Styrofoam cup containing 50.0 grams of water at 88.6°C. The water cools down and the metal warms up until thermal equilibrium is achieved at 87.1°C. A 10 percent of heat lost by water is transferred through the cup, determine the specific heat capacity of the unknown metal. The specific heat capacity of water is 4.18 J/g/°C.

The sums of heats gained in the system is zero.

heatgained by water*.9+heatgainedmetal=0
notice the .9 to inidcated the 90 percent efficiency.

50*cw*(87.1-88.6).9+13Cm*(87.1-26.5)=0

solve for Cm

To determine the specific heat capacity of the unknown metal, we can use the principle of conservation of energy. The heat lost by the water will be equal to the heat gained by the metal.

First, let's calculate the heat lost by the water using the formula Q = m * c * ΔT, where:
- Q is the heat lost by the water (in joules)
- m is the mass of the water (in grams)
- c is the specific heat capacity of water (4.18 J/g/°C)
- ΔT is the change in temperature of the water (final temperature - initial temperature)

Initial temperature of water = 88.6°C
Final temperature of water = 87.1°C
Mass of water = 50.0 grams

Using these values, we can calculate the heat lost by water:
Q_water = 50.0 g * 4.18 J/g/°C * (87.1°C - 88.6°C)

Next, we need to account for the fact that only 10 percent of the heat lost by the water is transferred through the cup. To find the actual heat lost by the water, we need to divide Q_water by 0.10.
Actual Q_water = Q_water / 0.10

Now, we can find the heat gained by the metal using the same formula Q = m * c * ΔT, but this time for the metal. However, we need to solve for the specific heat capacity of the unknown metal (c_metal).

The mass of the metal is given as 13 grams.
The initial temperature of the metal is 26.5°C.
The final temperature of the metal is 87.1°C.

Using these values, we can solve for the heat gained by the metal:
Q_metal = 13 g * c_metal * (87.1°C - 26.5°C)

Now, since the heat lost by the water is equal to the heat gained by the metal, we can set the two equations equal to each other and solve for c_metal:

Actual Q_water = Q_metal

Substituting the formulas we derived earlier:
Q_water / 0.10 = 13 g * c_metal * (87.1°C - 26.5°C)

Now we can solve for c_metal:
c_metal = (Q_water / 0.10) / (13 g * (87.1°C - 26.5°C))

Using the values you provided and following these calculations, you should be able to find the specific heat capacity of the unknown metal.