In the lab, an experimenter mixes 9.0 g of water (initially at 23.4°C) with 80 g of a solid metal (initially at 350°C). At thermal equilibrium, he measures a final temp of 55.4°C. What must the specific heat of the metal be?

heat lost by metal + heat gained by water = 0

[mass metal x specific heat metal x (Tfinal-Tinitial)] + [mass H2O x specific heat H2O x (Tfinal-Tinitial)] = 0
Substitute and solve for the only unknown in the set up.

To find the specific heat of the metal, we can use the principle of conservation of energy. The heat gained by the water is equal to the heat lost by the metal.

The formula to calculate the heat gained or lost is:

Q = mcΔT

Where:
- Q is the heat gained or lost
- m is the mass
- c is the specific heat
- ΔT is the change in temperature

First, let's calculate the heat gained by the water.

Given:
- Mass of water (m) = 9.0 g
- Initial temperature of water (Ti) = 23.4°C
- Final temperature of water (Tf) = 55.4°C

ΔT_water = Tf - Ti = 55.4°C - 23.4°C = 32°C

Now, we can calculate the heat gained by the water using the specific heat capacity of water (c_water).

Specific heat capacity of water (c_water) = 4.18 J/g°C (approximately)

Q_water = m_water * c_water * ΔT_water
= 9.0 g * 4.18 J/g°C * 32°C
= 1200.96 J

Next, let's calculate the heat lost by the metal.

Given:
- Mass of the metal (m) = 80 g
- Initial temperature of the metal (Ti) = 350°C
- Final temperature of the system (Tf) = 55.4°C (at thermal equilibrium)

ΔT_metal = Tf - Ti = 55.4°C - 350°C = -294.6°C

Note that we use a negative sign for the change in temperature because the metal is losing heat.

Now, we can calculate the specific heat of the metal (c_metal) using the equation:

Q_water = -Q_metal

Q_metal = m_metal * c_metal * ΔT_metal

Since the metal loses heat, we can say:

Q_metal = -1200.96 J

Therefore, we can write the equation as:

-1200.96 J = 80 g * c_metal * -294.6°C

Simplifying, we find:

c_metal = -1200.96 J / (80 g * -294.6°C)

c_metal ≈ 0.165 J/g°C

Therefore, the specific heat of the metal is approximately 0.165 J/g°C.