a metal sample weighing 45.0g and at a temperature of 99.1C was place in 38.6g of water in a calorimeter at 25.0C. the calorimeter reached a maximum temperature of 32.8C.
1. calculate the specific heat of the metal.
2. what is the approximate molar mass of the metal?
1) [mass metal x specific heat metal x Tfinal-Tinitial)] + [mass H2O x speicif heat water x (Tfinal-Tinitial) = 0
Only one unknown, specific heat metal. Solve for that.
2)Use the Law of Dulong and Petit
-23041 degrees celcius
Step 1: Calculate the heat gained by the water
To calculate the heat gained by the water, we can use the equation:
q = m * c * ΔT
Where:
q = heat gained/lost (in Joules)
m = mass of the substance (in grams)
c = specific heat capacity (in J/g°C)
ΔT = change in temperature (final temperature - initial temperature)
Given:
m_water = 38.6 g
c_water = 4.18 J/g°C (specific heat capacity of water)
ΔT_water = 32.8°C - 25.0°C = 7.8°C
Using the equation, we have:
q_water = 38.6 g * 4.18 J/g°C * 7.8°C
q_water ≈ 1211.39 J
Step 2: Calculate the heat lost by the metal
To calculate the heat lost by the metal, we can use the equation:
q = m * c * ΔT
Given:
m_metal = 45.0 g (mass of the metal)
c_metal = ? (specific heat capacity of the metal)
ΔT_metal = 99.1°C - 32.8°C = 66.3°C
Using the equation, we have:
q_metal = 45.0 g * c_metal * 66.3°C
Step 3: Equate the heat gained and the heat lost
Since the heat gained by the water is equal to the heat lost by the metal (assuming no heat is lost to the surroundings), we can set up the equation:
q_water = q_metal
Using the values we calculated in steps 1 and 2:
1211.39 J = 45.0 g * c_metal * 66.3°C
Simplifying the equation:
c_metal = 1211.39 J / (45.0 g * 66.3°C)
Step 4: Calculate the specific heat of the metal
Calculating the specific heat of the metal:
c_metal ≈ 0.512 J/g°C
Therefore, the specific heat of the metal is approximately 0.512 J/g°C.
To calculate the approximate molar mass of the metal, we need to know its specific heat capacity on a molar basis. Given that the molar mass of the metal is denoted as M, we can express the specific heat capacity of the metal on a molar basis as:
C_metal = c_metal / M
Since we know the specific heat capacity of the metal (0.512 J/g°C) calculated in step 4, we can rearrange the equation to solve for the molar mass:
M = c_metal / C_metal
Step 5: Calculate the approximate molar mass of the metal
Given:
C_metal = 24.3 J/mol°C (specific heat capacity on a molar basis for most metals)
Using the equation:
M = c_metal / C_metal
M ≈ 0.512 J/g°C / 24.3 J/mol°C
M ≈ 0.0210 mol/g
To convert molar mass from mol/g to g/mol, we take the reciprocal:
M ≈ 1 / 0.0210 g/mol
M ≈ 47.6 g/mol
Therefore, the approximate molar mass of the metal is approximately 47.6 g/mol.
To calculate the specific heat of the metal and the approximate molar mass of the metal, we can use the principle of energy conservation and the formula for heat transfer.
1. Calculate the specific heat of the metal:
The specific heat (c) is the amount of heat energy required to raise the temperature of a given mass of a substance by a certain amount. In this case, the metal absorbs heat from the water and the calorimeter. The equation for heat transfer is:
q = mcΔT
Where:
q is the heat absorbed or transferred (in Joules),
m is the mass of the substance (in grams),
c is the specific heat of the substance (in J/g°C),
ΔT is the change in temperature (in °C).
We can solve for the specific heat of the metal (c) using this equation and the given values:
qmetal = qwater + qcalorimeter
The heat transferred to the water is given by:
qwater = mwater * cwater * ΔTwater
Substituting the values:
mwater = 38.6g (mass of water)
cwater = 4.18 J/g°C (specific heat of water)
ΔTwater = Tf - Ti = 32.8°C - 25.0°C = 7.8°C (final temperature - initial temperature)
qwater = 38.6g * 4.18 J/g°C * 7.8°C = 1185.24 J
The heat transferred to the calorimeter can be calculated similarly:
qcalorimeter = mcalorimeter * ccalorimeter * ΔTcalorimeter
Substituting the values:
mcalorimeter = unknown
ccalorimeter = unknown
ΔTcalorimeter = Tf - Ti = 32.8°C - 25.0°C = 7.8°C (final temperature - initial temperature)
We'll need to know the specific heat and mass of the calorimeter to calculate qcalorimeter. The specific heat of the calorimeter can vary depending on the material it is made of. If that information is provided, we can proceed to calculate qcalorimeter. Otherwise, we can assume a reasonable value for the specific heat (e.g., 0.9 J/g°C for a metallic calorimeter) and continue with the calculations.
Once we have qwater and qcalorimeter, we can calculate the heat absorbed by the metal:
qmetal = qwater + qcalorimeter
Now we can solve for the specific heat of the metal (cmetal) using the given mass (mmetal) and the change in temperature (ΔTmetal), which is the final temperature minus the initial temperature:
cmetal = qmetal / (mmetal * ΔTmetal)
Substituting the values:
mmetal = 45.0g (mass of metal)
ΔTmetal = Tf - Ti = 32.8°C - 99.1°C = -66.3°C (final temperature - initial temperature)
Calculate the specific heat of the metal (cmetal) using the known values and the heat absorbed by the metal (qmetal).
2. Calculate the approximate molar mass of the metal:
The molar mass of a substance is the mass of one mole of that substance. We can use the formula:
Molar mass = (specific heat of the metal * mass of the metal) / (moles of the metal)
Rearrange the equation to solve for moles of the metal:
Moles of the metal = (specific heat of the metal * mass of the metal) / molar mass
We can substitute the known values into the equation to calculate the approximate molar mass of the metal.