A piece of metal of mass 18 g at 107◦C is placed in a calorimeter containing 56.5 g of water at 25◦C. The final temperature of the mixture is 56.3◦C. What is the specific heat capacity of the metal? Assume that there is no energy lost to the surroundings.
I assume the heat capacity of the calorimeter is zero.
heat lost by metal + heat gained by waer = 0
[mass metal x specific heat metal x (Tfinal-Tinitial)] + [mass H2O x specific heat H2O x (Tfinal-Tinitial)] = 0.
Substitute and solve for specific heat metal.
To find the specific heat capacity of the metal, we can use the equation:
Q = mcΔT
Where:
Q = heat gained or lost by the substance (in this case, the metal)
m = mass of the substance (in this case, the metal)
c = specific heat capacity of the substance
ΔT = change in temperature
First, let's find the heat gained by the water using the equation:
Q_water = mcΔT
Where:
Q_water = heat gained by the water
m = mass of the water
c = specific heat capacity of water (4.186 J/g°C)
ΔT = change in temperature
Given:
mass of water (m_w) = 56.5 g
initial temperature of water (T1_w) = 25°C
final temperature of mixture (T2) = 56.3°C
Now, let's calculate the heat gained by the water:
Q_water = (m_w)(c_water)(ΔT_water)
= (56.5 g)(4.186 J/g°C)(56.3 - 25)°C
Next, let's calculate the heat lost by the metal using the equation:
Q_metal = mcΔT
Where:
Q_metal = heat lost by the metal
m = mass of the metal
c = specific heat capacity of the metal (what we're trying to find)
ΔT = change in temperature
Given:
mass of metal (m_m) = 18 g
initial temperature of metal (T1_m) = 107°C
final temperature of mixture (T2) = 56.3°C
Let's insert the values and calculate the heat lost by the metal:
Q_metal = (m_m)(c_metal)(ΔT_metal)
= (18 g)(c_metal)(56.3 - 107)°C
Since there is no energy lost to the surroundings, the heat gained by the water is equal to the heat lost by the metal. Therefore, we have:
Q_water = Q_metal
Substituting the formula for the heat gained by the water and the heat lost by the metal:
(m_w)(c_water)(ΔT_water) = (m_m)(c_metal)(ΔT_metal)
Now we can solve for the specific heat capacity of the metal (c_metal).
c_metal = [(m_w)(c_water)(ΔT_water)] / [(m_m)(ΔT_metal)]
Substituting the given values:
c_metal = [(56.5 g)(4.186 J/g°C)(56.3 - 25)°C] / [(18 g)(56.3 - 107)°C]
Now, you can calculate the specific heat capacity of the metal using this equation.
To determine the specific heat capacity of the metal, we can use the principle of energy conservation.
The heat gained by the metal is equal to the heat lost by the water. The formula for heat transfer is given by:
Q = mcΔT
Where:
Q is the heat transfer (in joules),
m is the mass (in grams),
c is the specific heat capacity (in J/g°C),
ΔT is the change in temperature (in °C).
First, let's calculate the heat gained by the metal using the given information.
m1 = mass of the metal = 18 g
ΔT1 = temperature change of the metal = final temperature - initial temperature = 56.3°C - 107°C = -50.7°C
Next, let's calculate the heat lost by the water using the given information.
m2 = mass of the water = 56.5 g
ΔT2 = temperature change of the water = final temperature - initial temperature = 56.3°C - 25°C = 31.3°C
Since the heat gained by the metal is equal to the heat lost by the water, we can set up the following equation:
m1c1ΔT1 = m2c2ΔT2
Substituting the known values, we have:
18 g * c1 * (-50.7°C) = 56.5 g * c2 * 31.3°C
Simplifying the equation:
-913.8 c1 = 1768.45 c2
Now solve for c1 (specific heat capacity of the metal):
c1 = (1768.45 * c2) / -913.8
To get the specific heat capacity of the metal, we need to find the specific heat capacity of water (c2). The specific heat capacity of water is approximately 4.18 J/g°C.
Substitute c2 = 4.18 J/g°C:
c1 = (1768.45 * 4.18) / -913.8
c1 ≈ -8.097 J/g°C
Since specific heat capacity is always positive, we can take the absolute value:
|c1| ≈ 8.097 J/g°C
Therefore, the specific heat capacity of the metal is approximately 8.097 J/g°C.