calculate the minimum number of grams of propane, C3H8(g), that must be combusted to provide the energy necessary to convert 6.05 kg of ice at -24.0 ∘C to liquid water at 71.5 ∘C.
To calculate the minimum number of grams of propane needed to provide the energy necessary to convert ice to liquid water, we'll follow these steps:
Step 1: Calculate the energy required to heat the ice from -24.0 °C to 0 °C.
Step 2: Calculate the energy required to melt the ice at 0 °C.
Step 3: Calculate the energy required to heat the liquid water from 0 °C to 71.5 °C.
Step 4: Calculate the total energy required.
Step 5: Use the standard enthalpy of combustion of propane to determine the mass of propane needed.
Let's begin step by step:
Step 1: Calculate the energy required to heat the ice from -24.0 °C to 0 °C.
The specific heat capacity of ice (c) is 2.09 J/g⋅°C.
First, calculate the temperature change:
ΔT = 0 °C - (-24.0 °C) = 24.0 °C
Next, calculate the energy using the formula:
q = m * c * ΔT
Assuming the mass of ice (m) is 6.05 kg, convert it to grams:
m = 6.05 kg * 1000 g/kg = 6050 g
q = 6050 g * 2.09 J/g⋅°C * 24.0 °C
q = 303,240 J
Step 2: Calculate the energy required to melt the ice at 0 °C.
The heat of fusion (ΔHfus) for water is 334 J/g.
q = m * ΔHfus
q = 6050 g * 334 J/g
q = 2,019,700 J
Step 3: Calculate the energy required to heat the liquid water from 0 °C to 71.5 °C.
The specific heat capacity of liquid water (c) is 4.18 J/g⋅°C.
ΔT = 71.5 °C - 0 °C = 71.5 °C
q = m * c * ΔT
q = 6050 g * 4.18 J/g⋅°C * 71.5 °C
q = 1,781,897 J
Step 4: Calculate the total energy required.
Total energy = q1 + q2 + q3
Total energy = 303,240 J + 2,019,700 J + 1,781,897 J
Total energy = 4,104,837 J
Step 5: Use the standard enthalpy of combustion of propane to determine the mass of propane needed.
The standard enthalpy of combustion of propane (ΔHc) is 2220 kJ/mol.
Convert the total energy required to kilojoules:
Total energy = 4,104,837 J * 1 kJ/1000 J = 4104.837 kJ
Convert ΔHc to joules:
ΔHc = 2220 kJ/mol * 1000 J/1 kJ = 2,220,000 J/mol
Calculate the number of moles of propane needed:
moles of propane = Total energy / ΔHc
moles of propane = 4104.837 kJ / 2,220,000 J/mol
moles of propane = 1.85 mol
Finally, calculate the mass of propane needed:
mass of propane = moles of propane * molar mass of propane
The molar mass of propane (C3H8) is:
C: 3 * 12.01 g/mol = 36.03 g/mol
H: 8 * 1.008 g/mol = 8.064 g/mol
molar mass of propane = 36.03 g/mol + 8.064 g/mol
molar mass of propane = 44.094 g/mol
mass of propane = 1.85 mol * 44.094 g/mol
mass of propane = 81.47 g
Therefore, the minimum number of grams of propane (C3H8(g)) needed is approximately 81.47 grams.
To calculate the minimum number of grams of propane required for the given process, we need to determine the amount of heat energy needed to raise the temperature of the ice to its melting point and then heat it further to reach the final temperature.
Let's break down the calculation into two steps:
Step 1: Heating the ice to its melting point
First, we need to calculate the heat energy required to raise the temperature of the ice (solid) from -24.0 °C to 0 °C using the specific heat capacity of ice.
The specific heat capacity of ice is approximately 2.09 J/g°C.
The temperature change is ΔT = 0 °C - (-24.0 °C) = 24.0 °C.
The heat energy required to raise the temperature of ice can be calculated using the formula:
q1 = m * c * ΔT
Where:
q1 is the heat energy in Joules,
m is the mass of ice in grams,
c is the specific heat capacity of ice in J/g°C, and
ΔT is the temperature change in °C.
Step 2: Melting the ice to water and heating to the final temperature
Next, we need to calculate the heat energy required to convert the ice at 0 °C to liquid water at 0 °C and then raise the temperature of the water to 71.5 °C.
The enthalpy of fusion of water is approximately 333.5 J/g.
The specific heat capacity of water is approximately 4.18 J/g°C.
The temperature change from 0 °C to 71.5 °C is ΔT = 71.5 °C - 0 °C = 71.5 °C.
The heat energy required to convert ice to water and heat it can be calculated as follows:
q2 = (m * Hf) + (m * c * ΔT)
Where:
q2 is the heat energy in Joules,
m is the mass of ice in grams,
Hf is the enthalpy of fusion in J/g,
c is the specific heat capacity of water in J/g°C, and
ΔT is the temperature change in °C.
Finally, to determine the minimum amount of propane required, we need to convert the calculated heat energy to grams of propane using the molar mass and the molar heat of combustion of propane.
1. Calculate the heat energy required:
q_total = q1 + q2
2. Convert the heat energy in Joules to grams of propane:
q_total (J) * (1 mol C3H8 / molar heat of combustion C3H8, in J) * (molar mass C3H8, in g / 1 mol C3H8) = mass of propane in grams
To obtain the molar heat of combustion of propane and the molar mass of propane, you can refer to reliable sources or use the following approximate values:
Molar heat of combustion of propane ≈ 2220 kJ/mol
Molar mass of propane (C3H8) ≈ 44.1 g/mol
Plug in the values and calculate the minimum number of grams of propane required.
Look up the heat of combustion of propane (j/g). I think it is 50kj/gram
mass(50kj/g)=6.05kg(cice)(24)+6.05(Hfice)+6.05(cwater)(71.5)
solve for mass.