A 12.9 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.
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To determine the specific heat capacity of the unknown metal, we can use the principle of conservation of energy.
The heat gained by the metal equals the heat lost by the water:
Heat gained by the metal = Heat lost by the water
We can calculate the heat lost by the water using the equation:
Q = mcΔT
Where:
Q is the heat lost or gained (in joules)
m is the mass of the substance (in grams)
c is the specific heat capacity of the substance (in J/g/°C)
ΔT is the change in temperature (in °C)
Let's calculate the heat lost by the water:
The initial temperature of the water is 88.6°C, and it cools down to the final temperature of 87.1°C.
Water lost: ΔT = 87.1°C - 88.6°C = -1.5°C
The mass of the water is given as 50.0 grams.
Q_water = m_water * c_water * ΔT
Q_water = (50.0 g) * (4.18 J/g/°C) * (-1.5°C)
Now, we need to determine the heat gained by the metal:
The initial temperature of the metal is 26.5°C, and it warms up to the final temperature of 87.1°C.
Metal gained: ΔT = 87.1°C - 26.5°C = 60.6°C
The mass of the metal is given as 12.9 grams.
Q_metal = m_metal * c_metal * ΔT
Now, we can use the fact that 10 percent of the heat lost by the water is transferred through the cup:
Heat transferred through the cup = 0.1 * Q_water
Since the heat gained by the metal is equal to the heat lost by the water, we can set up the following equation:
Q_metal = (1 - 0.1) * Q_water = 0.9 * Q_water
Now, we can substitute the expressions for Q_water and Q_metal into the equation:
(m_metal * c_metal * ΔT) = 0.9 * ((m_water * c_water * ΔT))
Simplifying the equation:
m_metal * c_metal = (0.9 * m_water * c_water)
Plugging in the values:
m_metal * c_metal = (0.9 * (50.0 g) * (4.18 J/g/°C))
m_metal * c_metal = 188.1 J/°C
Finally, we can solve for the specific heat capacity of the unknown metal by dividing both sides of the equation by the mass of the metal:
c_metal = (188.1 J/°C) / (12.9 g)
c_metal ≈ 14.6 J/g/°C
Therefore, the specific heat capacity of the unknown metal is approximately 14.6 J/g/°C.
To solve this problem, 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.
Step 1: Calculate the heat lost by the water.
The formula to calculate the heat lost by the water is:
Qw = mass × specific heat capacity × change in temperature
Given:
mass of water (mw) = 50.0 g
specific heat capacity of water (cwater) = 4.18 J/g/°C
initial temperature of water (Tw1) = 88.6°C
final temperature of water (Tw2) = 87.1°C
Using the formula, we can calculate the heat lost by the water:
Qw = mw × cwater × (Tw1 - Tw2)
Qw = 50.0 g × 4.18 J/g/°C × (88.6°C - 87.1°C)
Step 2: Calculate the heat gained by the metal.
The formula to calculate the heat gained by the metal is:
Qm = mass × specific heat capacity × change in temperature
Given:
mass of the metal (mm) = 12.9 g
specific heat capacity of the metal (cmetal) = ?
To find the specific heat capacity of the metal, we need to rearrange the formula and solve for cmetal:
cmetal = Qm / (mm × change in temperature)
Since the heat gained by the metal is equal to the heat lost by the water, we have:
Qm = Qw
Therefore,
cmetal = Qw / (mm × change in temperature)
Once we calculate Qw, we can substitute the values and find the specific heat capacity of the metal.