A piece of copper metal is initially at 100 C. It is dropped into a coffee cup calorimeter containing 50.0 g of water at a temperature of 20 C. After stirring, the final temperature of both copper and water is 25 C Assuming no heat losses, and that the specific heat (capacity) of water is 4.18 J (gK), what is the heat capacity of the copper in J/K?

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To calculate the heat capacity of the copper in joules per Kelvin (J/K), we need to use the equation:

Q = m * c * ∆T

Where:
Q = heat absorbed or released (in joules, J)
m = mass of the substance (in grams, g)
c = specific heat capacity of the substance (in J/(g*K))
∆T = change in temperature (in Kelvin, K)

Given:
Initial temperature of copper (T1) = 100°C
Final temperature of copper and water (T2) = 25°C
Mass of water (m_water) = 50.0 g
Specific heat capacity of water (c_water) = 4.18 J/(g*K)

First, we calculate the heat absorbed by the water:
Q_water = m_water * c_water * ∆T

∆T = T2 - T1
∆T = 25°C - 20°C = 5°C

Q_water = 50.0 g * 4.18 J/(g*K) * 5 K
Q_water = 1045 J

Since heat is conserved in this system, the heat released by the copper is equal to the heat absorbed by the water:

Q_copper = -Q_water

Thus, the heat capacity of the copper is:
C_copper = -Q_copper / ∆T

C_copper = -1045 J / 5 K
C_copper = -209 J/K

Therefore, the heat capacity of the copper is -209 J/K.

To find the heat capacity of the copper in joules per kelvin (J/K), we can use the principle of conservation of energy. In this case, we will assume that the only heat transfer occurs between the copper and the water, and there are no heat losses to the surroundings.

The heat transferred between the copper and the water can be calculated using the formula:

Q = m * c * ΔT

Where:
Q is the heat transferred (in joules)
m is the mass of the substance (in grams)
c is the specific heat capacity of the substance (in J/(g·K))
ΔT is the change in temperature (in kelvin)

First, let's calculate the heat transferred between the water and the copper. The mass of the water (m) is given as 50.0 g, the specific heat capacity of water (c) is 4.18 J/(g·K), and the change in temperature (ΔT) is 25°C - 20°C = 5K.

Q_water-to-copper = m_water * c_water * ΔT
Q_water-to-copper = 50.0 g * 4.18 J/(g·K) * 5 K
Q_water-to-copper = 1045 J

Since the heat transferred from the copper to the water is equal in magnitude but opposite in sign to the heat transferred from the water to the copper, we can say:

Q_copper-to-water = -Q_water-to-copper
Q_copper-to-water = -1045 J

The heat capacity (C) is defined as the amount of heat needed to raise the temperature of an object by 1 kelvin:

C = Q / ΔT

For the copper:

C_copper = Q_copper-to-water / ΔT
C_copper = -1045 J / 5 K
C_copper = -209 J/K

Therefore, the heat capacity of the copper is -209 J/K.