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.
To find the specific heat capacity of the unknown metal, we can use the principle of heat transfer and apply the equation:
q = m * c * ΔT
where q is the heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
In our scenario, the heat lost by the water is transferred to the unknown metal until thermal equilibrium is achieved. Let's break down the steps:
1. Calculate the heat lost by the water:
q_water = m_water * c_water * ΔT_water
Given:
m_water = 50.0 g (mass of water)
c_water = 4.18 J/g/°C (specific heat capacity of water)
ΔT_water = 88.6°C - 87.1°C (change in temperature of water)
Plug in the given values to calculate q_water.
2. Calculate the heat gained by the metal:
q_metal = m_metal * c_metal * ΔT_metal
Given:
m_metal = 12.9 g (mass of metal)
c_metal = unknown (specific heat capacity of metal)
ΔT_metal = 87.1°C - 26.5°C (change in temperature of metal)
Let's assume that the heat gained by the metal equals the heat lost by the water after accounting for the 10 percent loss through the cup. Therefore, we can write the equation:
q_water * 0.9 = q_metal
Substitute the expressions for q_water and q_metal using the calculated values.
3. Rearrange the equation to solve for c_metal:
We know that q_water = q_metal / 0.9, so we can substitute this value into the equation above.
Then solve for c_metal:
c_metal = (q_metal / 0.9) / (m_metal * ΔT_metal)
Simplify the equation using the given values and the calculated q_metal.
This will give you the specific heat capacity of the unknown metal.